Year 6 Algebra: worked examples and questions with answers
By the end of this chapter you will be able to
- Substitute numbers into a simple formula, and write a formula in words and letters
- Write a missing number problem as an equation with a letter, and solve it by working backwards
- Describe the term-to-term rule of a linear sequence, and continue the sequence
- Write the nth term of a linear sequence, and use it to find a term or test whether a number is a term
- Find all the pairs of numbers that satisfy an equation with two unknowns, using a table
- List every combination of two items that makes a total, and count them
You should already know
- Order of operations, with brackets, multiplying and adding (Year 6, Chapter 2)
- Multiples, factors and place value (Year 6, Chapters 1 and 3)
- Counting on and back in steps, and finding the difference between two numbers (Years 3 to 5)
- Ratio and proportion, where amounts are linked by the same rule (Year 6, Chapter 7)
Where this leads
- Measures, area and volume, where formulae are used for shapes (Year 6, Chapter 9)
- Coordinates and graphs of linear rules, such as y = 2x + 1 (Year 6, Chapter 11)
- Algebra and equations at KS3 and GCSE
National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Use simple formulae, Generate and describe linear number sequences, Express missing number problems algebraically, Find pairs of numbers that satisfy an equation with two unknowns, Enumerate possibilities of combinations of two variables.
Exercise P Do you remember?
Non-calculatorMethod. Multiply first, then add.
- Multiply 6 by 7. Multiplication comes before addition.6 × 7 = 42
- Add 4. This is the last step.42 + 4 = 46
- Write the answer. 46
Check. 46 - 4 = 42, and 42 ÷ 6 = 7. ✓
Watch out. Do not add 4 before you multiply.
Method. Find one quarter first, then multiply by 3.
- Divide 20 by 4 to find one quarter. The bottom number tells us how many equal parts.20 ÷ 4 = 5
- Multiply the answer by 3. The top number tells us how many parts we want.5 × 3 = 15
- Write the answer. 15
Check. 15 ÷ 3 = 5, and 5 × 4 = 20. ✓
Watch out. Divide by the bottom number first, then multiply by the top number.
Method. Take the known number away from the total.
- Write the sum. The missing number is the one we add to 8.8 + ? = 15
- Take 8 from 15. Subtraction undoes addition.15 - 8 = 7
- Write the answer. 7
Check. 8 + 7 = 15. ✓
Watch out. Subtract the number you know from the total.
Method. Find the amount added each time, then add it again.
- Find the difference between two numbers next to each other. The sequence goes up by the same amount.4 - 2 = 2
- Add 2 to the last number. The next number is 8 + 2.8 + 2 = 10
- Add 2 again. The number after that is 10 + 2.10 + 2 = 12
- Write the answer. 10, 12
Check. Every number is 2 more than the one before: 2, 4, 6, 8, 10, 12. ✓
Watch out. Check the difference is the same for every pair.
Method. Multiply first, then subtract.
- Multiply 5 by 9. Multiplication comes before subtraction.5 × 9 = 45
- Take 12 away. This is the last step.45 - 12 = 33
- Write the answer. 33
Check. 33 + 12 = 45, and 45 ÷ 5 = 9. ✓
Watch out. Do the multiplication before the subtraction.
Letters stand for numbers: simple formulae
In algebra, a letter stands for a number. Letters let us write one rule that works for any numbers. A rule written with letters is called a formula.
For example, the perimeter of a rectangle is the total length of its four sides. If the rectangle has sides a and b, the formula is P = 2a + 2b. To use a formula, we substitute (put in) the numbers for the letters, then work out the answer.
Remember the order of operations. Work out brackets first, then multiply, then add or take away.
Key ideas
- A letter stands for a number. Write 4 × n as 4n.
- To substitute, put the number in place of the letter, then work out the answer.
- Multiply before you add or take away, unless there are brackets.
- A formula is a rule for any numbers. Check it with one number first.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.4n + 3
- Put 6 in place of n. Multiply first. Multiplication comes before addition and subtraction.4 × 6 = 24
- Add 3. This is the last step of the formula.24 + 3 = 27
- Write the answer. 27
Check. Work it out again: 4 × 6 = 24, then 24 + 3 = 27. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.P = 2a + 2b
- Put in the numbers: a = 9 and b = 4. Each letter is replaced by its number.P = 2 × 9 + 2 × 4
- Multiply first. Multiplication comes before addition.2 × 9 = 18, and 2 × 4 = 8
- Add the two results. This gives the perimeter.18 + 8 = 26
- Write the answer. 26
Check. The sides are 9, 4, 9 and 4 cm: 9 + 4 + 9 + 4 = 26. ✓
Watch out. Work out 2 × a and 2 × b separately, then add.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.9n + 2
- Put 5 in place of n. Multiply first. Multiplication comes before addition and subtraction.9 × 5 = 45
- Add 2. This is the last step of the formula.45 + 2 = 47
- Write the answer. 47
Check. Work it out again: 9 × 5 = 45, then 45 + 2 = 47. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Watch out
4n means 4 × n. When n = 6, 4n is 24, not 46. Many pupils add first and multiply second. Always multiply first.
Exam tip
Put brackets around a number that you substitute if it helps: 2(a + b) with a = 9 and b = 4 is 2 × (9 + 4) = 26.
Exercise 8A Substituting into formulae
Non-calculatorKey words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3a + 5
- Put 4 in place of a. Multiply first. Multiplication comes before addition and subtraction.3 × 4 = 12
- Add 5. This is the last step of the formula.12 + 5 = 17
- Write the answer. 17
Check. Work it out again: 3 × 4 = 12, then 12 + 5 = 17. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.7n - 9
- Put 6 in place of n. Multiply first. Multiplication comes before addition and subtraction.7 × 6 = 42
- Take away 9. This is the last step of the formula.42 - 9 = 33
- Write the answer. 33
Check. Work it out again: 7 × 6 = 42, then 42 - 9 = 33. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.2x + 8
- Put 5 in place of x. Multiply first. Multiplication comes before addition and subtraction.2 × 5 = 10
- Add 8. This is the last step of the formula.10 + 8 = 18
- Write the answer. 18
Check. Work it out again: 2 × 5 = 10, then 10 + 8 = 18. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.6y - 4
- Put 7 in place of y. Multiply first. Multiplication comes before addition and subtraction.6 × 7 = 42
- Take away 4. This is the last step of the formula.42 - 4 = 38
- Write the answer. 38
Check. Work it out again: 6 × 7 = 42, then 42 - 4 = 38. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3m + 10
- Put 9 in place of m. Multiply first. Multiplication comes before addition and subtraction.3 × 9 = 27
- Add 10. This is the last step of the formula.27 + 10 = 37
- Write the answer. 37
Check. Work it out again: 3 × 9 = 27, then 27 + 10 = 37. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.4p - 15
- Put 8 in place of p. Multiply first. Multiplication comes before addition and subtraction.4 × 8 = 32
- Take away 15. This is the last step of the formula.32 - 15 = 17
- Write the answer. 17
Check. Work it out again: 4 × 8 = 32, then 32 - 15 = 17. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.P = 2a + 2b
- Put in a = 12 and b = 5. Each letter is replaced by its number.P = 2 × 12 + 2 × 5
- Multiply first. Multiplication comes before addition.2 × 12 = 24, and 2 × 5 = 10
- Add the results. This gives P.24 + 10 = 34
- Write the answer. 34
Check. 34 is right, because 12 + 5 + 12 + 5 = 34. ✓
Watch out. Multiply each letter by 2 before you add.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.3(a + 4)
- Put in a = 6. Each letter is replaced by its number.3 × (6 + 4)
- Work out the brackets first. Brackets come before multiplying.6 + 4 = 10
- Multiply the result by 3. The 3 multiplies the whole bracket.3 × 10 = 30
- Write the answer. 30
Check. If you multiply first, 3 × 6 + 4 = 22, which is wrong, so the brackets matter. ✓
Watch out. The 3 multiplies the whole bracket, so work out the bracket first.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.A = l × w
- Put in l = 14 and w = 6. Each letter is replaced by its number.A = 14 × 6
- Multiply. This gives the area in square centimetres.14 × 6 = 84
- Write the answer. 84
Check. 84 ÷ 6 = 14, so the answer is right. ✓
Watch out. Area is length times width, so the unit is square centimetres.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.A = (b × h) ÷ 2
- Put in b = 9 and h = 8. Each letter is replaced by its number.A = (9 × 8) ÷ 2
- Work out the bracket first. The brackets come first.9 × 8 = 72
- Divide by 2. Half of the answer is the area of the triangle.72 ÷ 2 = 36
- Write the answer. 36
Check. 36 × 2 = 72 = 9 × 8. ✓
Watch out. Multiply the base and height first, then divide by 2.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3m + 2
- Put 6 in place of m. Multiply first. Multiplication comes before addition and subtraction.3 × 6 = 18
- Add 2. This is the last step of the formula.18 + 2 = 20
- Write the answer. 20
Check. Work it out again: 3 × 6 = 18, then 18 + 2 = 20. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Write the rule for each part, then add the bag charge to the notebook cost.
- Find the cost of the notebooks. Each notebook costs 35p, so n notebooks cost 35 pence for each notebook, added together.35 × n = 35n
- Add the bag charge. The 60p bag charge is paid once, so it is added on.35n + 60
- Use C for the total cost. The total cost is written as C = ...C = 35n + 60
Check. When n = 1, C = 35 + 60 = 95 pence, which is one notebook plus the bag. ✓
Watch out. The bag charge is added once, not once per notebook.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.35n + 60
- Put 4 in place of n. Multiply first. Multiplication comes before addition and subtraction.35 × 4 = 140
- Add 60. This is the last step of the formula.140 + 60 = 200
- Write the answer. 200
Check. Work it out again: 35 × 4 = 140, then 140 + 60 = 200. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put each number in and work out both sides. If the answers are different, the two are not the same.
- Put in n = 3 for the first expression. Each n is replaced by 3.2 × 3 + 5 = 6 + 5 = 11
- Put in n = 3 for the second expression. Use the same number for the second expression.2 + 5 × 3 = 2 + 15 = 17
- Compare the answers. Both answers must be equal for the expressions to be the same.11 is not the same as 17
- Write the answer. No
Check. 11 ≠ 17, so 2n + 5 and 2 + 5n are different expressions. ✓
Watch out. The order matters: 2n + 5 is not the same as 2 + 5n.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Find T when n = 5, then use the formula backwards to find n when T = 46.
- Work out n + 3 when n = 5. Put in n = 5 first.5 + 3 = 8
- Multiply by 4. The 4 multiplies the whole bracket.4 × 8 = 32
- Take 2 away to find T. This is the last step of the formula.32 - 2 = 30
Check. 30 is correct. ✓
Watch out. Work out the bracket first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Work backwards, undoing each step.
- Add 2 to both sides. This undoes the take away 2.46 + 2 = 48
- Divide both sides by 4. This undoes the multiply by 4.48 ÷ 4 = 12
- Take 3 from both sides. This undoes the add 3 inside the bracket.12 - 3 = 9
- Write the answer. 9
Check. 4 × (9 + 3) - 2 = 4 × 12 - 2 = 46. ✓
Watch out. Undo the last step first.
Missing numbers written with letters
Sometimes a number is missing from a problem. We can use a letter for the missing number, and write the words as an equation. An equation has an equals sign, and the two sides are equal.
Read the words slowly. Each part of the story becomes a part of the equation. Then solve the equation by working backwards. Undo the last step first, using the opposite operation: add undoes subtract, and multiply undoes divide.
The bar model shows three bags with the same number of sweets, b, and 4 loose sweets. Together they make 25 sweets.
Key ideas
- A letter can stand for a missing number.
- Write the words as an equation with an equals sign.
- Undo the steps in reverse order, using the opposite operation.
- Always check your answer by putting it back into the equation.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.n + 9 = 20
- Undo the add 9: take 9 from both sides. Taking 9 away from both sides keeps the equation true.n = 11
- The letter is on its own. There is no number in front to undo.n = 11
- Check: put the answer back into the equation. If both sides match, the answer is right.11 + 9 = 20 ✓
- Write the answer. n = 11
Check. 11 + 9 = 20. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Write the bar model as an equation, then undo the add and the multiply.
- Write the equation from the bar model. 3 bags of b, plus 4 more, make 25.3b + 4 = 25
- Take 4 from both sides. This leaves the bags on their own.3b = 21
- Divide both sides by 3. This finds one bag.b = 21 ÷ 3 = 7
- Check. Put b back in.3 × 7 + 4 = 25 ✓
- Write the answer. b = 7
Check. 3 × 7 + 4 = 25. ✓
Watch out. Each bag holds the same number. Do not divide the total by 4.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5p - 6 = 19
- Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.5p = 19 + 6 = 25
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.p = 25 ÷ 5 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 5 - 6 = 19 ✓
- Write the answer. p = 5
Check. 5 × 5 - 6 = 19. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Watch out
Do the opposite operation to both sides. In 2n + 5 = 13, take 5 from both sides first (to get 2n = 8), then divide by 2. Do not divide 5 by 2 first.
Exam tip
Estimate first. If the answer is bigger than the total, you have made a mistake. Then check: put the answer back in.
Exercise 8B Writing and solving equations
Non-calculatorKey words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.n + 8 = 17
- Undo the add 8: take 8 from both sides. Taking 8 away from both sides keeps the equation true.n = 9
- The letter is on its own. There is no number in front to undo.n = 9
- Check: put the answer back into the equation. If both sides match, the answer is right.9 + 8 = 17 ✓
- Write the answer. n = 9
Check. 9 + 8 = 17. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.k - 6 = 11
- Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.k = 11 + 6 = 17
- The letter is on its own. There is no number in front to undo.k = 17
- Check: put the answer back into the equation. If both sides match, the answer is right.17 - 6 = 11 ✓
- Write the answer. k = 17
Check. 17 - 6 = 11. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2x + 3 = 15
- Undo the add 3: take 3 from both sides. Taking 3 away from both sides keeps the equation true.2x = 12
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.x = 12 ÷ 2 = 6
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 6 + 3 = 15 ✓
- Write the answer. x = 6
Check. 2 × 6 + 3 = 15. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.4y + 1 = 29
- Undo the add 1: take 1 from both sides. Taking 1 away from both sides keeps the equation true.4y = 28
- Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.y = 28 ÷ 4 = 7
- Check: put the answer back into the equation. If both sides match, the answer is right.4 × 7 + 1 = 29 ✓
- Write the answer. y = 7
Check. 4 × 7 + 1 = 29. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.3z - 2 = 10
- Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.3z = 10 + 2 = 12
- Undo the × 3: divide both sides by 3. Dividing both sides by 3 leaves the letter on its own.z = 12 ÷ 3 = 4
- Check: put the answer back into the equation. If both sides match, the answer is right.3 × 4 - 2 = 10 ✓
- Write the answer. z = 4
Check. 3 × 4 - 2 = 10. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5w - 7 = 18
- Undo the take away 7: add 7 to both sides. Adding the same number to both sides keeps the equation true.5w = 18 + 7 = 25
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.w = 25 ÷ 5 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 5 - 7 = 18 ✓
- Write the answer. w = 5
Check. 5 × 5 - 7 = 18. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Write the bar model as an equation, then undo the add and the multiply.
- Write the equation from the bar model. 3 bags of b, plus 4 more, make 25.3b + 4 = 25
- Take 4 from both sides. This leaves the bags on their own.3b = 21
- Divide both sides by 3. This finds one bag.b = 21 ÷ 3 = 7
- Check. Put b back in.3 × 7 + 4 = 25 ✓
- Write the answer. b = 7
Check. 3 × 7 + 4 = 25. ✓
Watch out. Each bag holds the same number. Do not divide the total by 4.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.6n - 4 = 26
- Undo the take away 4: add 4 to both sides. Adding the same number to both sides keeps the equation true.6n = 26 + 4 = 30
- Undo the × 6: divide both sides by 6. Dividing both sides by 6 leaves the letter on its own.n = 30 ÷ 6 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.6 × 5 - 4 = 26 ✓
- Write the answer. n = 5
Check. 6 × 5 - 4 = 26. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2n + 9 = 31
- Undo the add 9: take 9 from both sides. Taking 9 away from both sides keeps the equation true.2n = 22
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.n = 22 ÷ 2 = 11
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 11 + 9 = 31 ✓
- Write the answer. n = 11
Check. 2 × 11 + 9 = 31. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Put the number in place of the letter and see whether both sides are the same.
- Write the equation. We will put the number in and see if both sides are equal.3x + 2 = 16
- Put x = 4 into the left side. Work out the left side only.3 × 4 + 2 = 14
- Compare with the right side. If the answer is the same as the right side, it is a solution.14 ≠ 16
- Write the answer. No
Check. Solve it to be sure: 3x + 2 = 16 gives x = 14/3, which is not 4. ✓
Watch out. Both sides must be equal. A number that looks close is not a solution.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Put the number in place of the letter and see whether both sides are the same.
- Write the equation. We will put the number in and see if both sides are equal.2n + 5 = 13
- Put n = 9 into the left side. Work out the left side only.2 × 9 + 5 = 23
- Compare with the right side. If the answer is the same as the right side, it is a solution.23 ≠ 13
- Write the answer. No
Check. Solve it to be sure: 2n + 5 = 13 gives n = 4. ✓
Watch out. Both sides must be equal. A number that looks close is not a solution.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2n + 8 = 26
- Undo the add 8: take 8 from both sides. Taking 8 away from both sides keeps the equation true.2n = 18
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.n = 18 ÷ 2 = 9
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 9 + 8 = 26 ✓
- Write the answer. n = 9
Check. 2 × 9 + 8 = 26. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5n = 60
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.n = 60 ÷ 5 = 12
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 12 = 60 ✓
- Write the answer. n = 12
Check. 5 × 12 = 60. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Multiply both sides by 4 to undo the divide.
- Write the equation. The letter is being divided by 4.n ÷ 4 = 7
- Multiply both sides by 4. This undoes the divide by 4.n = 7 × 4 = 28
- Write the answer. 28
Check. 28 ÷ 4 = 7. ✓
Watch out. Multiply, not divide, to undo a division.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Work backwards: undo the doubling, then the add 7, then the multiply by 3.
- Undo the doubling: divide 50 by 2. The last step was multiply by 2.50 ÷ 2 = 25
- Undo the add 7: take 7 from 25. The step before was add 7.25 - 7 = 18
- Undo the multiply by 3: divide 18 by 3. The first step was multiply by 3.18 ÷ 3 = 6
- Write the answer. 6
Check. 6 × 3 = 18, then 18 + 7 = 25, then 25 × 2 = 50. ✓
Watch out. Write the equation as 2(3n + 7) = 50 first. Then undo one step at a time.
Linear number sequences: the term-to-term rule
A sequence is a list of numbers that follow a rule. Each number in the list is a term.
In a linear sequence, the same amount is added each time, or the same amount is taken away each time. The rule that says how to get from one term to the next is the term-to-term rule. For example, 5, 9, 13, 17 goes up by 4 each time, so the rule is add 4.
Look at the dot patterns below. Each pattern has 2 more dots than the one before it, so the number of dots goes up in a linear sequence.
Key ideas
- Find the difference between two terms next to each other.
- If the difference is always the same, the sequence is linear, and the difference is the rule.
- To go forwards, add (or take away) the difference. To go backwards, do the opposite.
- A sequence that goes up by 4 and a sequence that goes down by 4 are both linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 4 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.9 - 5 = 4
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.9 - 5 = 4, 13 - 9 = 4, 17 - 13 = 4
- Apply the rule: add 4. This gives the next term.17 + 4 = 21
- Do the same again for the term after that. The same rule works every time.21 + 4 = 25
- Write the answer. 21, 25
Check. Start from the last term: 17 + 4 = 21. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.36 - 40 = -4
- Check that every pair has the same difference. Then the rule is the same all the way along.4, 4, 4 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.subtract 4
Check. Start at 40 and apply the rule: 40 - 4 = 36. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.
- Take away the first term from 38. We want to know how far the term is from the start.38 - 3 = 35
- Divide by the difference, 5. The sequence goes up by the same amount each time.35 ÷ 5 = 7
- It is a whole number that is 0 or more, so the number is in the sequence. The position is one more than the number of jumps.position = 7 + 1 = 8, so it is term number 8
- Write the answer. Yes
Check. Check: 3 + 7 × 5 = 38. ✓
Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.
Watch out
A linear sequence must have the same difference all the way along. For 1, 4, 9, 16 the differences are 3, 5, 7, so it is not linear.
Exam tip
Say the rule in words: “add 5” or “subtract 4”. Then check it with the first two terms.
Exercise 8C Terms, rules and patterns
Non-calculatorKey words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 5 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.11 - 6 = 5
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.11 - 6 = 5, 16 - 11 = 5, 21 - 16 = 5
- Apply the rule: add 5. This gives the next term.21 + 5 = 26
- Write the answer. 26
Check. Start from the last term: 21 + 5 = 26. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.7 - 2 = 5
- Check that every pair has the same difference. Then the rule is the same all the way along.5, 5, 5 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.add 5
Check. Start at 2 and apply the rule: 2 + 5 = 7. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The missing term is halfway between its two neighbours, so add them and halve.
- Look at the gap: it is one jump from each side. So the missing term is halfway between the two terms.30, __, 22
- Add the two terms. The two jumps together make the whole gap.30 + 22 = 52
- Halve the answer. This gives the middle term.52 ÷ 2 = 26
- Write the answer. 26
Check. 26 - 30 = -4 and 22 - 26 = -4: both jumps are the same. ✓
Watch out. Check that both jumps are the same size. If they are not, the sequence is not linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The missing term is halfway between its two neighbours, so add them and halve.
- Look at the gap: it is one jump from each side. So the missing term is halfway between the two terms.12, __, 20
- Add the two terms. The two jumps together make the whole gap.12 + 20 = 32
- Halve the answer. This gives the middle term.32 ÷ 2 = 16
- Write the answer. 16
Check. 16 - 12 = 4 and 20 - 16 = 4: both jumps are the same. ✓
Watch out. Check that both jumps are the same size. If they are not, the sequence is not linear.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.45 - 50 = -5
- Check that every pair has the same difference. Then the rule is the same all the way along.5, 5, 5 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.subtract 5
Check. Start at 50 and apply the rule: 50 - 5 = 45. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to subtract 8 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.92 - 100 = -8
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.92 - 100 = -8, 84 - 92 = -8
- Apply the rule: subtract 8. This gives the next term.84 - 8 = 76
- Write the answer. 76
Check. Start from the last term: 84 - 8 = 76. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 2 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.6 - 4 = 2
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.6 - 4 = 2, 8 - 6 = 2, 10 - 8 = 2
- Apply the rule: add 2. This gives the next term.10 + 2 = 12
- Write the answer. 12
Check. Start from the last term: 10 + 2 = 12. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference between each pair of neighbouring terms. If it changes, the sequence is not linear.
- Find the differences. Subtract each term from the next one.4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9
- Compare the differences. A linear sequence has the same difference every time.3, 5, 7, 9 are not all the same
- Write the answer. No
Check. Adding 3 gives 4, but adding 3 again gives 7, not 9. So the rule is not linear. ✓
Watch out. The squares 1, 4, 9, 16, 25 go up by more and more each time.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Check that every difference is the same.
- Find the differences. Each pair of neighbouring terms is compared.5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3
- Compare the differences. They are all 3, so the sequence is linear and the rule is add 3.the differences are all 3
- Write the answer. Yes
Check. 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11. ✓
Watch out. All the differences must be the same.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work backwards: take the difference away from the term you know.
- Find the difference. The sequence goes up by 5 each time.17 - 12 = 5
- Go back one term. Take the difference away from 12.12 - 5 = 7
- Write the answer. 7
Check. 7 + 5 = 12, and 12 + 5 = 17. ✓
Watch out. Going backwards means doing the opposite operation.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Start at 3 and add 6 each time.
- Start with the first term. The first term is given.3
- Add 6 to get the next term. Apply the rule once.3 + 6 = 9
- Add 6 again. Keep applying the same rule.9 + 6 = 15, then 15 + 6 = 21, then 21 + 6 = 27
- Write the answer. 3, 9, 15, 21, 27
Check. Each term is 6 more than the one before. ✓
Watch out. Start from the first term, not from zero.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.
- Take away the first term from 100. We want to know how far the term is from the start.100 - 4 = 96
- Divide by the difference, 4. The sequence goes up by the same amount each time.96 ÷ 4 = 24
- It is a whole number that is 0 or more, so the number is in the sequence. The position is one more than the number of jumps.position = 24 + 1 = 25, so it is term number 25
- Write the answer. Yes
Check. Check: 4 + 24 × 4 = 100. ✓
Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Compare the first terms and the rule. Both sequences go up by 7.
- Write out the first terms of each sequence. This shows the pattern clearly.A: 2, 9, 16, 23, ... B: 5, 12, 19, 26, ...
- Look at the remainders when each term is divided by 7. Every term of A leaves 2 when divided by 7. Every term of B leaves 5.2 and 5 are different, so the two lists can never match
- Write the answer. No
Check. Sequence A never shares a term with sequence B, because both go up by 7 but start at different remainders. ✓
Watch out. Two sequences with the same difference can still never meet if they start in different places.
The nth term of a linear sequence
The nth term is a rule that gives any term of a sequence. The letter n is the position of the term. The first term has n = 1, the second term has n = 2, and so on.
For a linear sequence, the number in front of n is the difference. For 5, 8, 11, 14 the difference is 3, so the rule starts with 3n. Then we find the number to add so that the first term is right: 3 × 1 = 3, but the first term is 5, so we add 2. The nth term is 3n + 2.
We can also use the nth term to test whether a number is in the sequence. Solve the equation, and check that n is a whole number.
| Position n | Term | Working |
|---|---|---|
| 1 | 5 | 3 × 1 + 2 = 5 |
| 2 | 8 | 3 × 2 + 2 = 8 |
| 3 | 11 | 3 × 3 + 2 = 11 |
| 4 | 14 | 3 × 4 + 2 = 14 |
Key ideas
- The number in front of n is the difference between terms.
- Put n = 1 into the rule to check that the first term is right.
- To find a term in position n, put that number in place of n.
- To test a number, solve the equation. n must be a whole number that is 1 or more.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.8 - 5 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 5, so the rule must give 5 when n = 1.3 × 1 + ? = 5, so ? = 5 - 3 = 2
- Check with the second term. Put n = 2 in the rule: the answer must be 8.3 × 2 + 2 = 8. ✓
- Write the nth term. 3n + 2
Check. Put n = 2 in 3n + 2: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3n + 2
- Put 30 in place of n. Multiply first. Multiplication comes before addition and subtraction.3 × 30 = 90
- Add 2. This is the last step of the formula.90 + 2 = 92
- Write the answer. 92
Check. Work it out again: 3 × 30 = 90, then 90 + 2 = 92. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 101?4n + 3 = 101
- Take 3 away from both sides. This leaves the part with n.4n = 98
- Divide both sides by 4. This gives n, the position of the term.n = 98 ÷ 4 = 49/2
- Decide. The position must be a whole number, 1 or more.n = 49/2, which is not a whole number, so 101 is not a term
- Write the answer. No
Check. Try the nearest whole numbers: 4 × 24 + 3 and 4 × 25 + 3 are not 101. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Watch out
In 3n + 2 the first term is not 3. Put n = 1 in: 3 × 1 + 2 = 5. Always check the rule with n = 1 before you use it.
Exam tip
A descending sequence can be written with a minus sign in front of n. For 35, 30, 25 the nth term is 40 - 5n, because the difference is 5.
Exercise 8D Writing and using the nth term
Non-calculatorKey words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.5 - 2 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 2, so the rule must give 2 when n = 1.3 × 1 + ? = 2, so ? = 2 - 3 = -1
- Check with the second term. Put n = 2 in the rule: the answer must be 5.3 × 2 - 1 = 5. ✓
- Write the nth term. 3n - 1
Check. Put n = 2 in 3n - 1: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.11 - 7 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 7, so the rule must give 7 when n = 1.4 × 1 + ? = 7, so ? = 7 - 4 = 3
- Check with the second term. Put n = 2 in the rule: the answer must be 11.4 × 2 + 3 = 11. ✓
- Write the nth term. 4n + 3
Check. Put n = 2 in 4n + 3: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.13 - 10 = 3
- Start the rule with 3n, because the difference is 3. The number in front of n is the difference.3n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 10, so the rule must give 10 when n = 1.3 × 1 + ? = 10, so ? = 10 - 3 = 7
- Check with the second term. Put n = 2 in the rule: the answer must be 13.3 × 2 + 7 = 13. ✓
- Write the nth term. 3n + 7
Check. Put n = 2 in 3n + 7: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.5 - 1 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 1, so the rule must give 1 when n = 1.4 × 1 + ? = 1, so ? = 1 - 4 = -3
- Check with the second term. Put n = 2 in the rule: the answer must be 5.4 × 2 - 3 = 5. ✓
- Write the nth term. 4n - 3
Check. Put n = 2 in 4n - 3: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.10 - 6 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 6, so the rule must give 6 when n = 1.4 × 1 + ? = 6, so ? = 6 - 4 = 2
- Check with the second term. Put n = 2 in the rule: the answer must be 10.4 × 2 + 2 = 10. ✓
- Write the nth term. 4n + 2
Check. Put n = 2 in 4n + 2: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time. Here it goes down.25 - 30 = -5
- Write the rule as 35 minus 5 times n. The number in front of n is 5, because the sequence goes down by 5.35 - 5n
- Check that n = 1 gives the first term. Put n = 1 in the rule: it must give 30.35 - 5 × 1 = 30. ✓
- Write the nth term. 35 - 5n
Check. Put n = 2 in 35 - 5n: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.3n + 4
- Put 10 in place of n. Multiply first. Multiplication comes before addition and subtraction.3 × 10 = 30
- Add 4. This is the last step of the formula.30 + 4 = 34
- Write the answer. 34
Check. Work it out again: 3 × 10 = 30, then 30 + 4 = 34. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Count the jumps from the first term, multiply by the difference, then add the first term.
- Find the difference. The sequence goes up or down by the same amount each time.difference = 3
- Count the jumps from the first term to term 50. There are 50 - 1 = 49 jumps.50 - 1 = 49 jumps
- Multiply the jumps by the difference. Each jump adds the difference.49 × 3 = 147
- Add the first term. This gives the term.6 + 147 = 153
- Write the answer. 153
Check. Jump back: 153 - 147 = 6, the first term. ✓
Watch out. There are one fewer jumps than the position. The 5th term has 4 jumps.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 100?7n + 2 = 100
- Take 2 away from both sides. This leaves the part with n.7n = 98
- Divide both sides by 7. This gives n, the position of the term.n = 98 ÷ 7 = 14
- Decide. The position must be a whole number, 1 or more.n = 14, a whole number, so 100 is term number 14
- Write the answer. Yes
Check. Put n = 14 in: 7 × 14 + 2 = 100. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 80?6n + 1 = 80
- Take 1 away from both sides. This leaves the part with n.6n = 79
- Divide both sides by 6. This gives n, the position of the term.n = 79 ÷ 6 = 79/6
- Decide. The position must be a whole number, 1 or more.n = 79/6, which is not a whole number, so 80 is not a term
- Write the answer. No
Check. Try the nearest whole numbers: 6 × 13 + 1 and 6 × 14 + 1 are not 80. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5n - 2 = 73
- Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.5n = 73 + 2 = 75
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.n = 75 ÷ 5 = 15
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 15 - 2 = 73 ✓
- Write the answer. n = 15
Check. 5 × 15 - 2 = 73. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.7 - 3 = 4
- Start the rule with 4n, because the difference is 4. The number in front of n is the difference.4n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 3, so the rule must give 3 when n = 1.4 × 1 + ? = 3, so ? = 3 - 4 = -1
- Check with the second term. Put n = 2 in the rule: the answer must be 7.4 × 2 - 1 = 7. ✓
- Write the nth term. 4n - 1
Check. Put n = 2 in 4n - 1: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Put n = 1 into Ali’s rule and compare with the first term.
- Write the first term of the sequence. The first term is 5.first term = 5
- Put n = 1 into 4n. This is the first term that Ali’s rule gives.4 × 1 = 4
- Compare the two numbers. They must be the same.4 is not 5
- Write the answer. No
Check. The difference is 4, but the first term is 5, so the rule needs + 1: 4n + 1. ✓
Watch out. Check the rule with n = 1 before you trust it.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. List the first few terms of each sequence, then find the first number that appears in both lists.
- Write out the first terms of sequence A, with the rule 3n + 5. Put n = 1, 2, 3 in turn.8, 11, 14, 17, 20, 23, ...
- Write out the first terms of sequence B, with the rule 5n + 1. Put n = 1, 2, 3 in turn.6, 11, 16, 21, 26, ...
- Find the first number that is in both lists. Compare the lists from the start.11 is in both (A when n = 2, B when n = 2)
- Write the answer. 11
Check. 3 × 2 + 5 = 11 and 5 × 2 + 1 = 11. ✓
Watch out. The first number in both lists is the smallest one. Check the next one, 26, does not come before it.
Pairs of numbers with two unknowns
An equation with two letters, such as a + b = 9, can have many answers. Each answer is a pair: a = 1 and b = 8 is one pair, and a = 5 and b = 4 is another.
To find all the pairs, choose a value for one letter, starting from 1. Work out the other letter. Then choose the next value, and keep going until the other letter would be less than 1. A table keeps the list in order, so that no pair is missed or counted twice.
If the question gives a second clue, such as a is bigger than b, use it to cross out pairs that do not fit.
Key ideas
- An equation with two unknowns has many pairs of answers.
- Make a table: choose one letter in order, and work out the other letter each time.
- Check each pair in the original equation.
- Count the rows of the table to find how many pairs there are.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 9
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 8 1 + 8 = 9 2 7 2 + 7 = 9 3 6 3 + 6 = 9 4 5 4 + 5 = 9 5 4 5 + 4 = 9 6 3 6 + 3 = 9 7 2 7 + 2 = 9 8 1 8 + 1 = 9 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 8 and 1
- Count the pairs. Each row of the table is one pair.8 pairs
- Write the answer. 8
Check. Every row was checked in the equation. The table has 8 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + 2y = 10
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 2 4 2 + 2 × 4 = 10 4 3 4 + 2 × 3 = 10 6 2 6 + 2 × 2 = 10 8 1 8 + 2 × 1 = 10 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 8 and 1
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a × b = 24
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 24 1 × 24 = 24 2 12 2 × 12 = 24 3 8 3 × 8 = 24 4 6 4 × 6 = 24 6 4 6 × 4 = 24 8 3 8 × 3 = 24 12 2 12 × 2 = 24 24 1 24 × 1 = 24 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 24 and 1
- Count the pairs. Each row of the table is one pair.8 pairs
- Write the answer. 8
Check. Every row was checked in the equation. The table has 8 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Watch out
Pairs are written in the order of the letters. (1, 8) and (8, 1) are two different pairs for a + b = 9, because a has a different value in each. Both are counted. Read the question to see if the letters have an order.
Exam tip
Use a table. Write the pairs in order, starting with the smallest value, so that you can see when the list is complete.
Exercise 8E Finding values and pairs
Non-calculatorKey words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 4 into the equation. Now there is only one letter left.a + b = 10
- Take 4 from 10. This is how much is left for the other part.10 - 4 = 6
- The number left is the value of the letter. This finds the other letter.b = 6
- Check the pair in the equation. Both sides must be equal.4 + 6 = 10 ✓
- Write the answer. b = 6
Check. 4 + 6 = 10. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put x = 7 into the equation. Now there is only one letter left.x + y = 15
- Take 7 from 15. This is how much is left for the other part.15 - 7 = 8
- The number left is the value of the letter. This finds the other letter.y = 8
- Check the pair in the equation. Both sides must be equal.7 + 8 = 15 ✓
- Write the answer. y = 8
Check. 7 + 8 = 15. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Add 9 to both numbers to find a. Then check the answer.
- Write the equation with b = 9. Put the known number in place of b.a - 9 = 3
- Add 9 to both sides. This undoes the take away 9.a = 3 + 9 = 12
- Write the answer. 12
Check. 12 - 9 = 3. ✓
Watch out. Add to find the bigger number when you subtract.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 1 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 1, then take it from 12. This is how much is left for the other part.12 - 2 = 10
- The number left is the value of the letter. This finds the other letter.b = 10
- Check the pair in the equation. Both sides must be equal.2 + 10 = 12 ✓
- Write the answer. b = 10
Check. 2 + 10 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 3 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 3, then take it from 12. This is how much is left for the other part.12 - 6 = 6
- The number left is the value of the letter. This finds the other letter.b = 6
- Check the pair in the equation. Both sides must be equal.6 + 6 = 12 ✓
- Write the answer. b = 6
Check. 6 + 6 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 5 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 5, then take it from 12. This is how much is left for the other part.12 - 10 = 2
- The number left is the value of the letter. This finds the other letter.b = 2
- Check the pair in the equation. Both sides must be equal.10 + 2 = 12 ✓
- Write the answer. b = 2
Check. 10 + 2 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 12
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 11 1 + 11 = 12 2 10 2 + 10 = 12 3 9 3 + 9 = 12 4 8 4 + 8 = 12 5 7 5 + 7 = 12 6 6 6 + 6 = 12 7 5 7 + 5 = 12 8 4 8 + 4 = 12 9 3 9 + 3 = 12 10 2 10 + 2 = 12 11 1 11 + 1 = 12 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 11 and 1
- Count the pairs. Each row of the table is one pair.11 pairs
- Write the answer. 11
Check. Every row was checked in the equation. The table has 11 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + y = 20, both odd
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 1 19 1 + 19 = 20, both odd 3 17 3 + 17 = 20, both odd 5 15 5 + 15 = 20, both odd 7 13 7 + 13 = 20, both odd 9 11 9 + 11 = 20, both odd 11 9 11 + 9 = 20, both odd 13 7 13 + 7 = 20, both odd 15 5 15 + 5 = 20, both odd 17 3 17 + 3 = 20, both odd 19 1 19 + 1 = 20, both odd - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 19 and 1
- Count the pairs. Each row of the table is one pair.10 pairs
- Write the answer. 10
Check. Every row was checked in the equation. The table has 10 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 14, a > b
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 8 6 8 + 6 = 14, and 8 is bigger 9 5 9 + 5 = 14, and 9 is bigger 10 4 10 + 4 = 14, and 10 is bigger 11 3 11 + 3 = 14, and 11 is bigger 12 2 12 + 2 = 14, and 12 is bigger 13 1 13 + 1 = 14, and 13 is bigger - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 13 and 1
- Count the pairs. Each row of the table is one pair.6 pairs
- Write the answer. 6
Check. Every row was checked in the equation. The table has 6 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + 3y = 15
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 3 4 3 + 3 × 4 = 15 6 3 6 + 3 × 3 = 15 9 2 9 + 3 × 2 = 15 12 1 12 + 3 × 1 = 15 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 12 and 1
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Add the two equations to find x. Then use either equation to find y.
- Write the two equations. Both are true for the same x and y.x + y = 9 and x - y = 3
- Add the two equations. The y terms cancel out: +y and -y.2x = 12
- Divide by 2. This gives x.x = 12 ÷ 2 = 6
- Find y. Put x = 6 into x + y = 9.6 + y = 9, so y = 3
- Write the answer. 6
Check. 6 + 3 = 9 and 6 - 3 = 3. ✓
Watch out. Check both equations with both numbers.
| x | y |
|---|---|
| 1 | 7 |
| 3 | 6 |
| 5 | 5 |
| 7 | 4 |
| 9 | ? |
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Use the equation x + 2y = 15 with x = 9.
- Write the equation. The table shows pairs for this equation.x + 2y = 15
- Put x = 9 into the equation. Now there is only y left.9 + 2y = 15
- Take 9 from 15. This leaves 2y.2y = 15 - 9 = 6
- Divide by 2. This gives y.y = 6 ÷ 2 = 3
- Write the answer. 3
Check. 9 + 2 × 3 = 15. ✓
Watch out. Check that the other rows of the table work the same way.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put the values in the equation and see if both sides are equal.
- Write the equation. We will test the pair.2a + b = 15
- Put in a = 5 and b = 6. Replace each letter by its number.2 × 5 + 6 = 10 + 6
- Compare the answer with 15. It must be exactly 15.10 + 6 = 16, and 16 is not 15
- Write the answer. No
Check. Since 16 ≠ 15, the pair is not a solution. ✓
Watch out. Check the whole equation, not one part of it.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a × b = 36, a < b
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 36 1 × 36 = 36, and 1 is smaller 2 18 2 × 18 = 36, and 2 is smaller 3 12 3 × 12 = 36, and 3 is smaller 4 9 4 × 9 = 36, and 4 is smaller - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 4 and 9
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Combinations of two variables
Sometimes two letters stand for amounts of two things, and the amounts must add up to a total. For example, if a is the number of 2p coins and b is the number of 5p coins, then the total value is 2a + 5b pence.
To find every combination, we list every possible number of one item, in order, and work out the number of the other item each time. Only the lists where the number of each item is a whole number that fits the total count as combinations.
Make a table, so that no combination is missed and none is counted twice.
Key ideas
- A combination uses a whole number of each item.
- Start with 0 (or 1, if each item must be used at least once) and count up in order.
- For each choice, work out how much is left, and check that it can be made with the other item.
- Count the rows of the table to find the number of combinations.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 5p coin is worth 5p.12p from 2p and 5p items
- Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 5p coins Check 1 2 1 × 2 + 2 × 5 = 12 6 0 6 × 2 + 0 × 5 = 12 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 12p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One pen is worth 3p. One pencil is worth 5p.26p from 3p and 5p items
- Choose how many pencils, starting at 1, and work out how much is left. The rest of the total must be made from pens.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of pens Number of pencils Check 2 4 2 × 3 + 4 × 5 = 26 7 1 7 × 3 + 1 × 5 = 26 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 26p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One red card is worth 9 points. One blue card is worth 4 points.49 points from 9 points and 4 points items
- Choose how many blue cards, starting at 1, and work out how much is left. The rest of the total must be made from red cards.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of red cards Number of blue cards Check 1 10 1 × 9 + 10 × 4 = 49 5 1 5 × 9 + 1 × 4 = 49 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 49 points, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Watch out
Check whether the question allows zero of one item. If it says “at least one of each”, start your list at 1, not 0.
Exam tip
Make a table with the two columns and a check column. Stop when the first item is too big to fit the total.
Exercise 8F Combinations and systems
Non-calculatorKey words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 5p coin is worth 5p. One 10p coin is worth 10p.25p from 5p and 10p items
- Choose how many 10p coins, starting at 0, and work out how much is left. The rest of the total must be made from 5p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 5p coins Number of 10p coins Check 1 2 1 × 5 + 2 × 10 = 25 3 1 3 × 5 + 1 × 10 = 25 5 0 5 × 5 + 0 × 10 = 25 - Count the ways. Each row is one way.3 ways
- Write the answer. 3
Check. Each row adds to 25p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 10p coin is worth 10p.30p from 2p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 10p coins Check 5 2 5 × 2 + 2 × 10 = 30 10 1 10 × 2 + 1 × 10 = 30 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 30p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One adult is worth 7 points. One child is worth 3 points.40 points from 7 points and 3 points items
- Choose how many children, starting at 1, and work out how much is left. The rest of the total must be made from adults.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of adults Number of children Check 1 11 1 × 7 + 11 × 3 = 40 4 4 4 × 7 + 4 × 3 = 40 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 40 points, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.3x + 2y = 12
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 2 3 3 × 2 + 2 × 3 = 12 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 2 and 3
- Count the pairs. Each row of the table is one pair.1 pairs
- Write the answer. 1
Check. Every row was checked in the equation. The table has 1 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One apple is worth 25p. One banana is worth 15p.150p from 25p and 15p items
- Choose how many bananas, starting at 1, and work out how much is left. The rest of the total must be made from apples.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of apples Number of bananas Check 3 5 3 × 25 + 5 × 15 = 150 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 150p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 10p coin is worth 10p.16p from 2p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 10p coins Check 3 1 3 × 2 + 1 × 10 = 16 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 16p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Start with all the counters as red, then look at what the blue ones add.
- Write the two clues. One clue is about the number of counters, the other about the points.r + b = 12 and 2r + 3b = 30
- If all 12 counters were red, the score would be 12 × 2 = 24. This is the starting score.24 points
- Each blue counter gives 1 more point than a red one. So the extra points come from the blue counters.3 - 2 = 1 extra point each
- Work out how many blue counters are needed. The score needs 6 more points.30 - 24 = 6, so b = 6 × 1 = 6
- Write the answer. 6
Check. 6 blue counters give 18 points and 6 red counters give 12 points, a total of 30. ✓
Watch out. Use the difference between the two counters, not the total of 12.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 5p coin is worth 5p. One 10p coin is worth 10p.40p from 5p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 5p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 5p coins Number of 10p coins Check 2 3 2 × 5 + 3 × 10 = 40 4 2 4 × 5 + 2 × 10 = 40 6 1 6 × 5 + 1 × 10 = 40 - Count the ways. Each row is one way.3 ways
- Write the answer. 3
Check. Each row adds to 40p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List the ways to make 12p with at least one of each coin, then count them.
- Try 1, 2 and 3 of the 3p coin, and see how much is left for the 2p coins. With at least one 2p coin, the 3p coins cannot use all the money.12 - 3 = 9, 12 - 6 = 6, 12 - 9 = 3
- Keep only the rows where the rest is a whole number of 2p coins. The rest must be even, so it can be split into 2p coins.9 is odd, 6 ÷ 2 = 3, and 3 is odd
- Count the ways. There is only one row that works.one way: 3 × 2p and 2 × 3p
- Write the answer. Sam is wrong, because there are not 4 ways.No
Check. 3 × 2 + 2 × 3 = 12, and no other row works. ✓
Watch out. Count only the rows that really work. Listing all the rows helps you check.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One pencil is worth 3p. One rubber is worth 7p.30p from 3p and 7p items
- Choose how many rubbers, starting at 1, and work out how much is left. The rest of the total must be made from pencils.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of pencils Number of rubbers Check 3 3 3 × 3 + 3 × 7 = 30 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 30p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Compare the number of 2p coins with the number of 5p coins.
- Write the two clues. One clue is the number of coins, the other is the value.a + b = 20 and 2a + 5b = 70
- If all 20 coins were 2p, the total would be 40p. This is the starting total.20 × 2 = 40
- Each 5p coin adds 3p more than a 2p coin. So the extra money comes from the 5p coins.5 - 2 = 3
- Find how many 5p coins give the extra 30p. The total needs 70p, so 70 - 40 = 30 more.30 ÷ 3 = 10
- Write the answer. 10
Check. 10 coins of 5p = 50p and 10 coins of 2p = 20p, a total of 70p, from 20 coins altogether. ✓
Watch out. Every 5p coin replaces a 2p coin, so it adds only 3p more.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Compare with 10 coins of 20p, then find how many 50p coins add the extra money.
- Write the two clues. One clue is the number of coins, the other is the value.a + b = 10 and 50a + 20b = 260
- If all 10 coins were 20p, the total would be 200p. This is the starting total.10 × 20 = 200
- Each 50p coin adds 30p more than a 20p coin. So the extra money comes from the 50p coins.50 - 20 = 30
- Find how many 50p coins give the extra 60p. The total needs 260p, so 260 - 200 = 60 more.60 ÷ 30 = 2
- Write the answer. 2
Check. 2 coins of 50p and 8 coins of 20p give 100p + 160p = 260p, from 10 coins. ✓
Watch out. Check that both clues are true: the number of coins and the total value.
Chapter summary
| Skill | Method |
|---|---|
| Letters and formulae | A letter stands for a number. To use a formula, substitute the numbers, then multiply before you add or take away. |
| Equations | Write the words as an equation. Undo each step in reverse order, using the opposite operation, then check. |
| Term-to-term rule | Find the difference between neighbouring terms. If it is always the same, the sequence is linear and the rule is add (or subtract) that amount. |
| nth term | Put the difference in front of n, then add or take away the number that makes the first term right: 3n + 2. |
| Testing a number | Solve a n + b = the number. If n is a whole number that is 1 or more, the number is a term. |
| Pairs with two unknowns | Choose values for one letter in order, work out the other, and make a table. Count the rows. |
| Combinations | List every whole-number choice of one item in order, work out the other item, and count the rows that work. |
Can you…?
- I can substitute numbers into a formula and work out the answer.
- I can write a missing-number problem as an equation and solve it by working backwards.
- I can describe the term-to-term rule of a linear sequence and continue it.
- I can write the nth term of a linear sequence and use it to find any term.
- I can test whether a number is a term of a linear sequence.
- I can find all the pairs of numbers that satisfy an equation with two unknowns.
- I can list and count the combinations of two items that make a total.
Key terms
- Letter
- A symbol, such as n, that stands for a number.
- Formula
- A rule that links letters, such as P = 2a + 2b.
- Substitute
- To put a number in place of a letter.
- Equation
- A statement with an equals sign, with equal amounts on each side, such as 3n + 2 = 14.
- Term
- One number in a sequence.
- Linear sequence
- A sequence in which the same amount is added (or taken away) each time.
- nth term
- A rule that gives any term of a sequence, using the position n.
- Pair
- Two numbers that together satisfy an equation.
- Combination
- One way of choosing an amount of each item to make a total.
Exercise 8Z Check-up (non-calculator)
Non-calculatorKey words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.5n + 3
- Put 8 in place of n. Multiply first. Multiplication comes before addition and subtraction.5 × 8 = 40
- Add 3. This is the last step of the formula.40 + 3 = 43
- Write the answer. 43
Check. Work it out again: 5 × 8 = 40, then 40 + 3 = 43. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the numbers in place of the letters, then work it out in the right order.
- Write the formula. It tells us which numbers to use.P = 2a + 2b
- Put in a = 11 and b = 3. Each letter is replaced by its number.P = 2 × 11 + 2 × 3
- Multiply first. Multiplication comes before addition.22 + 6
- Add. This gives P.22 + 6 = 28
- Write the answer. 28
Check. 11 + 3 + 11 + 3 = 28. ✓
Watch out. Multiply each letter by 2 before you add.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.4n + 5 = 33
- Undo the add 5: take 5 from both sides. Taking 5 away from both sides keeps the equation true.4n = 28
- Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.n = 28 ÷ 4 = 7
- Check: put the answer back into the equation. If both sides match, the answer is right.4 × 7 + 5 = 33 ✓
- Write the answer. n = 7
Check. 4 × 7 + 5 = 33. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.3x - 6 = 21
- Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.3x = 21 + 6 = 27
- Undo the × 3: divide both sides by 3. Dividing both sides by 3 leaves the letter on its own.x = 27 ÷ 3 = 9
- Check: put the answer back into the equation. If both sides match, the answer is right.3 × 9 - 6 = 21 ✓
- Write the answer. x = 9
Check. 3 × 9 - 6 = 21. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to add 5 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.12 - 7 = 5
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.12 - 7 = 5, 17 - 12 = 5
- Apply the rule: add 5. This gives the next term.17 + 5 = 22
- Do the same again for the term after that. The same rule works every time.22 + 5 = 27
- Write the answer. 22, 27
Check. Start from the last term: 17 + 5 = 22. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Find the difference and say whether the terms go up or down.
- Find the difference between two terms next to each other. This is the amount we add or take away.27 - 30 = -3
- Check that every pair has the same difference. Then the rule is the same all the way along.3, 3, 3 (sizes of the jumps)
- Write the rule. Say whether we add or subtract, and by how much.subtract 3
Check. Start at 30 and apply the rule: 30 - 3 = 27. ✓
Watch out. Say "add 4" or "subtract 4", not "plus 4" or "4 more" with no operation.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Find the difference: it is the number in front of n. Then find the number that makes the first term right.
- Find the difference between two terms next to each other. It is the same every time, because the sequence is linear.10 - 4 = 6
- Start the rule with 6n, because the difference is 6. The number in front of n is the difference.6n + ?
- Find the number that is added: put n = 1 in the rule. The first term is 4, so the rule must give 4 when n = 1.6 × 1 + ? = 4, so ? = 4 - 6 = -2
- Check with the second term. Put n = 2 in the rule: the answer must be 10.6 × 2 - 2 = 10. ✓
- Write the nth term. 6n - 2
Check. Put n = 2 in 6n - 2: this gives the second term of the sequence. ✓
Watch out. The number in front of n is the difference, but the rule also needs the number that is added (or taken away).
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. Work out how far the number is from the first term. If that is a whole number of jumps, the number is in the sequence.
- Take away the first term from 62. We want to know how far the term is from the start.62 - 5 = 57
- Divide by the difference, 4. The sequence goes up by the same amount each time.57 ÷ 4 = 57/4
- It is not a whole number (or it is below the first term), so the number is not in the sequence. Every term is the first term plus a whole number of jumps.57/4 is not a whole number
- Write the answer. No
Check. Terms are 5, 9, 13 ... and never reach 62. ✓
Watch out. The number must be exactly one of the terms. A number between two terms is not in the sequence.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 7
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 6 1 + 6 = 7 2 5 2 + 5 = 7 3 4 3 + 4 = 7 4 3 4 + 3 = 7 5 2 5 + 2 = 7 6 1 6 + 1 = 7 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 6 and 1
- Count the pairs. Each row of the table is one pair.6 pairs
- Write the answer. 6
Check. Every row was checked in the equation. The table has 6 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 5p coin is worth 5p.15p from 2p and 5p items
- Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 5p coins Check 0 3 0 × 2 + 3 × 5 = 15 5 1 5 × 2 + 1 × 5 = 15 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 15p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Total 22 marks. Allow about 22 minutes.
Exercise 8X SATs-style questions
Non-calculatorKey words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.5n - 3
- Put 7 in place of n. Multiply first. Multiplication comes before addition and subtraction.5 × 7 = 35
- Take away 3. This is the last step of the formula.35 - 3 = 32
- Write the answer. 32
Check. Work it out again: 5 × 7 = 35, then 35 - 3 = 32. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.4x + 6 = 30
- Undo the add 6: take 6 from both sides. Taking 6 away from both sides keeps the equation true.4x = 24
- Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.x = 24 ÷ 4 = 6
- Check: put the answer back into the equation. If both sides match, the answer is right.4 × 6 + 6 = 30 ✓
- Write the answer. x = 6
Check. 4 × 6 + 6 = 30. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. A term is one number in a sequence. The term-to-term rule says how to get from one term to the next. In a linear sequence the same amount is added (or taken away) each time.
Method. The rule is to subtract 4 each time. Use it to find the next terms.
- Find the difference between two terms next to each other. If the difference is always the same, the sequence is linear.9 - 13 = -4
- Check the other pairs of terms. They must have the same difference, or the rule is not linear.9 - 13 = -4, 5 - 9 = -4, 1 - 5 = -4
- Apply the rule: subtract 4. This gives the next term.1 - 4 = -3
- Write the answer. -3
Check. Start from the last term: 1 - 4 = -3. ✓
Watch out. Check that the difference is the same for every pair of neighbouring terms before you use it.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.4m + 10 = 42
- Undo the add 10: take 10 from both sides. Taking 10 away from both sides keeps the equation true.4m = 32
- Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.m = 32 ÷ 4 = 8
- Check: put the answer back into the equation. If both sides match, the answer is right.4 × 8 + 10 = 42 ✓
- Write the answer. m = 8
Check. 4 × 8 + 10 = 42. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. A formula is a rule written with letters, such as P = 2a + 2b. To substitute means to put a number in place of a letter.
Method. Put the number in place of the letter, multiply, then add or take away.
- Write the formula. The letter stands for a number we know, so we put it in.5n + 4
- Put 40 in place of n. Multiply first. Multiplication comes before addition and subtraction.5 × 40 = 200
- Add 4. This is the last step of the formula.200 + 4 = 204
- Write the answer. 204
Check. Work it out again: 5 × 40 = 200, then 200 + 4 = 204. ✓
Watch out. Multiply before you add or take away. Do not do the sum from left to right.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 16, both even
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 2 14 2 + 14 = 16, both even 4 12 4 + 12 = 16, both even 6 10 6 + 10 = 16, both even 8 8 8 + 8 = 16, both even 10 6 10 + 6 = 16, both even 12 4 12 + 4 = 16, both even 14 2 14 + 2 = 16, both even - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 14 and 2
- Count the pairs. Each row of the table is one pair.7 pairs
- Write the answer. 7
Check. Every row was checked in the equation. The table has 7 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. The nth term is a rule that gives any term. The letter n is the position of the term: the first term has n = 1, the second term has n = 2.
Method. Solve the equation for n. If n is a whole number that is 1 or more, the number is a term.
- Write the equation. We ask: for which n is the term equal to 150?7n + 1 = 150
- Take 1 away from both sides. This leaves the part with n.7n = 149
- Divide both sides by 7. This gives n, the position of the term.n = 149 ÷ 7 = 149/7
- Decide. The position must be a whole number, 1 or more.n = 149/7, which is not a whole number, so 150 is not a term
- Write the answer. No
Check. Try the nearest whole numbers: 7 × 21 + 1 and 7 × 22 + 1 are not 150. ✓
Watch out. The position n must be a whole number. A number between two terms is never a term.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 3p coin is worth 3p. One 5p coin is worth 5p.37p from 3p and 5p items
- Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 3p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 3p coins Number of 5p coins Check 4 5 4 × 3 + 5 × 5 = 37 9 2 9 × 3 + 2 × 5 = 37 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 37p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.6n - 2 = 118
- Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.6n = 118 + 2 = 120
- Undo the × 6: divide both sides by 6. Dividing both sides by 6 leaves the letter on its own.n = 120 ÷ 6 = 20
- Check: put the answer back into the equation. If both sides match, the answer is right.6 × 20 - 2 = 118 ✓
- Write the answer. n = 20
Check. 6 × 20 - 2 = 118. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Add the two equations to find a, then find b.
- Write the two equations. Both are true for the same a and b.a + b = 18 and a - b = 4
- Add the two equations. The b terms cancel: +b and -b.2a = 22
- Divide by 2. This gives a.a = 22 ÷ 2 = 11
- Find b. Put a = 11 into a + b = 18.11 + b = 18, so b = 7
- Check both equations. Both must be true.11 + 7 = 18 and 11 - 7 = 4
- Write the answer. The question asks for a.11
Check. 11 + 7 = 18 and 11 - 7 = 4. ✓
Watch out. Check both equations with both numbers.