Year 6: Letters stand for numbers: simple formulae
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
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.