GCSE Maths: Forming equations from words (Higher)
Part of the GCSE Maths linear equations chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Forming equations from words
Equations are most useful when a real problem is turned into one. A good method has five steps:
- Read the question and decide what you need to find.
- Choose a letter for the unknown and say what it stands for, with units. (“Let x be the cost of one ticket in pounds.”)
- Write an equation using the information. The two sides must have the same value.
- Solve the equation.
- Answer in words, with units, and check that your answer fits the story.
The hard step is turning words into algebra. Learn these phrases. Notice that the order matters for “less than” and “minus”.
| In words | In algebra |
|---|---|
| a number plus 5, 5 more than a number | x + 5 |
| 5 less than a number, a number minus 5 | x − 5 |
| 5 minus a number | 5 − x |
| 5 times a number, double a number (2 times) | 5x, 2x |
| a number divided by 4, a quarter of a number | x4 |
| 3 more than twice a number | 2x + 3 |
| twice the sum of a number and 3 | 2(x + 3) |
Ages. If Ali is x years old and his sister is 5 years older, she is x + 5. In 8 years’ time they will be x + 8 and x + 13. “Three times as old” means 3x.
Costs. If one ticket costs £x, then 4 tickets cost 4x pounds. A fixed charge such as a booking fee is added once, but a charge per item or per hour is multiplied by the number of items or hours.
Finally, check your answer in the words of the question, not just in your equation. If your equation was wrong, an answer that satisfies the equation can still be wrong for the story.
Key ideas
- Always begin with “Let x = …” and give the units.
- The two sides of the equation must be equal in value: use words like “is”, “gives”, “total” or “the same as” to find the = sign.
- “5 less than x” is x − 5 (not 5 − x).
- After solving, answer the question that was asked (with units) and check it against the story.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number I think of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.I multiply the number by 4 (4x) and subtract 7 (4x − 7). The result is 29.
Equation: 4x − 7 = 29 - Add 7 to both sides. This undoes the −7, because −7 + 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 7 + 7 = 29 + 7
4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 9.
Check. 4 × 9 = 36, and 36 − 7 = 29 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of hours the plumber works.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The call-out fee of £35 is charged once. The hourly charge is £25 for each hour, which is 25x. The bill is 35 + 25x = 110.
Equation: 25x + 35 = 110 - Subtract 35 from both sides. This undoes the +35, because +35 − 35 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.25x + 35 − 35 = 110 − 35
25x = 75 - Divide both sides by 25. 25x means 25 multiplied by x. The inverse of multiplying by 25 is dividing by 25, and we do it to both sides to keep the balance. This leaves x on its own.25x25 = 7525
x = 7525 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The plumber worked for 3 hours.
Check. 25 × 3 = 75, and 75 + 35 = 110 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Ali's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.His sister is 5 years older, so she is x + 5. Their ages add up to 31.
Equation: x + (x + 5) = 31 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
The equation is now x + x + 5 = 31 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x make 2x.
The equation is now 2x + 5 = 31 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 5 − 5 = 31 − 5
2x = 26 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 262
x = 262 = 13 - Write the solution. x is now on its own, so the equation tells us its value.x = 13
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Ali is 13 years old, and his sister is 18.
Check. 13 + 18 = 31, and 18 is 5 more than 13 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Jo thinks of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Jo adds 6 to her number: x + 6. She then doubles the whole result, so the bracket is multiplied by 2: 2(x + 6). The answer is 30.
Equation: 2(x + 6) = 30 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 6):
2 × x = 2x
2 × 6 = 12
The equation is now 2x + 12 = 30 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 12 − 12 = 30 − 12
2x = 18 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 182
x = 182 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Jo's number is 9.
Check. 9 + 6 = 15, and 2 × 15 = 30 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the son's age now, in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.His mother is three times as old: 3x. In 8 years the son will be x + 8 and his mother will be 3x + 8. At that time she will be twice as old as him: 3x + 8 = 2(x + 8).
Equation: 3x + 8 = 2(x + 8) - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 8):
2 × x = 2x
2 × 8 = 16
The equation is now 3x + 8 = 2x + 16 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 8 − 2x = 2x + 16 − 2x
x + 8 = 16 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 8 − 8 = 16 − 8
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The son is 8 years old and his mother is 24.
Check. In 8 years the son is 16 and his mother is 32. 32 = 2 × 16 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Watch out
“5 less than x” is x − 5. It is not 5 − x. For example, 5 less than 20 is 15 = 20 − 5. Also read what the question asks: after you find x, you may still need to use it (for example to find the sister’s age, not Ali’s).Exercise 9G Forming and solving equations from words
Non-calculatorMethod. Plus means add.
- Say it in symbols. “A number” is x, and “plus 5” means add 5.x + 5
- Check the order. For addition the order does not change the value.x + 5 = 5 + x
Check. If x = 10, the phrase gives 15 and x + 5 = 15. ✓
Method. “Less than” means take away from the number.
- Start with the number. We take 3 away from the number, not the other way round.x
- Subtract 3. 3 less than 10 is 7 = 10 − 3.x − 3
Check. If x = 10, 3 less than it is 7 and x − 3 = 7. ✓
Watch out. “3 less than x” is x − 3, not 3 − x.
Method. “Twice” means multiply by 2.
- Say it in symbols. Twice means two times, and we do not write the × sign in algebra.2 × x = 2x
- Check. If x = 10, twice the number is 20.2x = 20 ✓
Check. 2 × 10 = 20. ✓
Method. “Divided by” means a fraction, with the number on top.
- Write a division. The number is divided by 4.x ÷ 4
- Write it as a fraction. In algebra division is written as a fraction.x4
Check. If x = 20, 20 ÷ 4 = 5 and 204 = 5. ✓
Method. Do the doubling first, then add 3.
- Twice a number. 2x
- 3 more than that. Add 3 to the doubled number.2x + 3
Check. If x = 10: twice is 20, and 3 more is 23. 2 × 10 + 3 = 23. ✓
Method. The word “sum” means the adding happens first, so it goes in a bracket.
- The sum of a number and 3. Add first.x + 3
- Twice that. Multiply the whole sum by 2, so use a bracket.2(x + 3)
Check. If x = 10: the sum is 13, twice it is 26. 2(10 + 3) = 26. ✓
Watch out. Without the bracket, 2x + 3 would only double the x.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Max's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.He adds 7 to the number: x + 7. The result is 15.
Equation: x + 7 = 15 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 15 − 7
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Max's number is 8.
Check. 8 + 7 = 15 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Mia's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.She subtracts 6 from the number: x − 6. The result is 11.
Equation: x − 6 = 11 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 6 + 6 = 11 + 6
x = 17 - Write the solution. x is now on its own, so the equation tells us its value.x = 17
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Mia's number is 17.
Check. 17 − 6 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Half of the number is x2. Adding 3 gives x2 + 3 = 11.
Equation: x2 + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 + 3 − 3 = 11 − 3
x2 = 8 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 8
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 16.
Check. 16 ÷ 2 + 3 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 times the number is 3x. 5 less than that is 3x − 5. This equals 22.
Equation: 3x − 5 = 22 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 5 + 5 = 22 + 5
3x = 27 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 273
x = 273 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 9.
Check. 3 × 9 − 5 = 22 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of hours.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The fee is charged once (35). The hourly charge is 25x. The total is 135.
Equation: 25x + 35 = 135 - Subtract 35 from both sides. This undoes the +35, because +35 − 35 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.25x + 35 − 35 = 135 − 35
25x = 100 - Divide both sides by 25. 25x means 25 multiplied by x. The inverse of multiplying by 25 is dividing by 25, and we do it to both sides to keep the balance. This leaves x on its own.25x25 = 10025
x = 10025 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The plumber worked for 4 hours.
Check. 25 × 4 = 100 and 100 + 35 = 135 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Cara's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Ben is x + 4. The total of their ages is 30.
Equation: x + (x + 4) = 30 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 4) has nothing in front of it except +1, so the brackets can simply be removed: x + 4
The equation is now x + x + 4 = 30 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x make 2x.
The equation is now 2x + 4 = 30 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 4 − 4 = 30 − 4
2x = 26 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 262
x = 262 = 13 - Write the solution. x is now on its own, so the equation tells us its value.x = 13
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Cara is 13 years old.
Check. Cara is 13 and Ben is 17. 13 + 17 = 30 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one pen in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 pens cost 3x. The ruler costs 2. Altogether 11.
Equation: 3x + 2 = 11 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 2 − 2 = 11 − 2
3x = 9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 93
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One pen costs £3.
Check. 3 × £3 = £9 and £9 + £2 = £11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of minutes.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The monthly charge is 1200 pence. The calls cost 5x pence. The bill is 1500 pence.
Equation: 5x + 1200 = 1500 - Subtract 1200 from both sides. This undoes the +1200, because +1200 − 1200 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 1200 − 1200 = 1500 − 1200
5x = 300 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 3005
x = 3005 = 60 - Write the solution. x is now on its own, so the equation tells us its value.x = 60
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.There were 60 minutes of calls.
Check. 5 × 60 = 300p and 300 + 1200 = 1500p ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = her brother's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Zara is 3x years old. The sum of their ages is x + 3x = 36.
Equation: x + 3x = 36 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 3x make 4x.
The equation is now 4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Zara is 27 years old.
Check. Her brother is 9 and Zara is 27. 9 + 27 = 36 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one notebook in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Sam spends 20 − 5 = £15. The spending is 4 notebooks and a £3 pen: 4x + 3 = 15.
Equation: 4x + 3 = 15 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 15 − 3
4x = 12 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 124
x = 124 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One notebook costs £3.
Check. 4 × £3 = £12, plus £3 is £15, and £20 − £15 = £5 change ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the smallest number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Consecutive numbers go up by 1, so the numbers are x, x + 1 and x + 2. They add up to 51.
Equation: x + (x + 1) + (x + 2) = 51 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 1) has nothing in front of it except +1, so the brackets can simply be removed: x + 1
(x + 2) has nothing in front of it except +1, so the brackets can simply be removed: x + 2
The equation is now x + x + 1 + x + 2 = 51 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x + x make 3x and the numbers 1 + 2 make 3.
The equation is now 3x + 3 = 51 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 3 − 3 = 51 − 3
3x = 48 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 483
x = 483 = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The smallest number is 16.
Check. 16 + 17 + 18 = 51 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of months.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Gym A costs 20 + 10x pounds. Gym B costs 50 + 5x pounds. We want these to be equal.
Equation: 10x + 20 = 5x + 50 - Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.10x + 20 − 5x = 5x + 50 − 5x
5x + 20 = 50 - Subtract 20 from both sides. This undoes the +20, because +20 − 20 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 20 − 20 = 50 − 20
5x = 30 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 305
x = 305 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The total costs are the same after 6 months.
Check. Gym A: 20 + 10 × 6 = £80. Gym B: 50 + 5 × 6 = £80. The same ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Translate the words carefully, then solve.
- Check Hana's equation. “5 less than a number” means we take 5 away from the number, so the number comes first.5 less than 20 is 15 = 20 − 5, so “5 less than x” is x − 5.
- Write the correct equation. The result is 17.x − 5 = 17
- Solve it. Add 5 to both sides.x = 17 + 5 = 22
Check. 22 − 5 = 17. ✓ (With Hana's equation, 5 − x = 17 would give a negative number, x = −12, which does not match the story.)
Watch out. “Less than” reverses the order of the words.
Method. Write an equation for “the same answer” and see whether it can be true.
- Write Jay's and Ria's answers. Both start with the same number x.Jay: 2x + 3. Ria: 2x + 5.
- Set them equal. They get the same answer when the two expressions are equal.2x + 3 = 2x + 5
- Subtract 2x from both sides. The x terms cancel.3 = 5
- Conclude. 3 = 5 is false, whatever x is.There is no number that gives the same answer, so Jay and Ria can never get the same answer. (Ria's is always 2 more.)
Check. Try x = 4: Jay gets 11 and Ria gets 13. They differ by 2 whatever the number. ✓
Watch out. A false statement such as 3 = 5 means no solution, not x = 3 or x = 5.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the daughter's age now, in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The mother is 3x. In 8 years the daughter is x + 8 and the mother is 3x + 8, and 3x + 8 is twice x + 8.
Equation: 3x + 8 = 2(x + 8) - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 8):
2 × x = 2x
2 × 8 = 16
The equation is now 3x + 8 = 2x + 16 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 8 − 2x = 2x + 16 − 2x
x + 8 = 16 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 8 − 8 = 16 − 8
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The daughter is 8 years old now.
Check. Now: daughter 8, mother 24. In 8 years: 16 and 32, and 32 = 2 × 16 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Increased by 3 and multiplied by 5: 5(x + 3). Multiplied by 8 and then 3 subtracted: 8x − 3.
Equation: 5(x + 3) = 8x − 3 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x + 3):
5 × x = 5x
5 × 3 = 15
The equation is now 5x + 15 = 8x − 3 - Remove the x term from the left side: subtract 5x from both sides. We want all the x terms on one side. The left side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x + 15 − 5x = 8x − 3 − 5x
15 = 3x − 3 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.15 + 3 = 3x − 3 + 3
18 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.183 = 3x3
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 6.
Check. 5 × (6 + 3) = 45 and 8 × 6 − 3 = 45 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.