Turn angle facts, perimeters and worded problems into an equation, solve it, then answer the question that was asked. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Here the algebra is not given — you write it. Perimeter, ages, money, angles and 'I think of a number' all become an equation with a clear letter. Define the letter in a sentence: 'let x be the smaller number'. Then every other amount is an expression in x, the story gives an equation, and you solve it.
Check in the original story, not just in the algebra. If x is a length, a negative answer is a red flag. A two-part question that says 'hence' wants your x substituted into an expression, not a fresh start.
Translate the sentence into algebra, then solve. Perimeter, angles on a line, and 'I think of a number' are the same two-step: form, then undo. Write the equation before you start shuffling numbers.
Objectives
Let n be the number you are asked to find.
“Double a number, then add 5, gives 17” is 2n + 5 = 17.
Undo the operations in reverse order.
Key points
Choose a letter for the unknown and write every other quantity in terms of it.
Use a maths fact (angles in a triangle, perimeter, 'the total is…') to build an equation.
Solve equations with brackets and with letters on both sides.
Substitute back to find the real quantity the question asked for, not just x.
Worked examples
Example 1
A number is doubled and then 3 is added. The result is 7. What is the number?
2 marks
Let the number be n. Then 2n + 3 = 7.
2n = 7 − 3 = 4.
n = 4 ÷ 2 = 2.
Answer: 2
Common slips
Slip: Let the number be n. Then 4n + 3 = 7. That changes a number from the question, so it answers a different question.
Slip: 5n = 7 − 3 = 4. That changes a number in the working, so the step no longer follows from the one before.
Slip: n = 8 ÷ 2 = 2. That does not match the working above it.
Example 2
Sam and Tia have 20 sweets altogether. Tia has 3 times as many sweets as Sam. How many sweets does Sam have?
2 marks
Let Sam have n sweets. Then Tia has 3n.
n + 3n = 4n = 20.
n = 20 ÷ 4 = 5.
Answer: 5
Common slips
Slip: Let Sam have n sweets. Then Tia has 5n. That changes a number from the question, so it answers a different question.
Slip: n + 6n = 4n = 20. That changes a number in the working, so the step no longer follows from the one before.
Slip: n = 24 ÷ 4 = 5. That does not match the working above it.
Example 3
4 times a number, plus 4, is equal to the number plus 22. What is the number?
3 marks
Let the number be n. Then 4n + 4 = n + 22.
Subtract n and 4 from both sides: 3n = 18.
n = 18 ÷ 3 = 6.
Answer: 6
Common slips
Slip: Let the number be n. Then 6n + 4 = n + 22. That changes a number from the question, so it answers a different question.
Slip: Subtract n and 7 from both sides: 3n = 18. That changes a number in the working, so the step no longer follows from the one before.
Slip: n = 22 ÷ 3 = 6. That does not match the working above it.
Questions and answers
GCSE Maths forming & solving equations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
2 times a number, plus 5, is equal to the number plus 13. What is the number?
Show answer
Answer: 8
Let the number be n. Then 2n + 5 = n + 13.
Subtract n and 5 from both sides: 1n = 8.
M1 — Let the number be n. Then 2n + 5 = n + 13.
A1 — Answer 8
Question 2 Easy2 marks
A number is trebled and then 7 is subtracted. The result is 8. What is the number?
Show answer
Answer: 5
Let the number be n. Then 3n − 7 = 8.
3n = 8 + 7 = 15.
M1 — Let the number be n. Then 3n − 7 = 8.
A1 — Answer 5
Question 3 Medium2 marks
3 times a number, plus 4, is equal to the number plus 8. What is the number?
Show answer
Answer: 2
Let the number be n. Then 3n + 4 = n + 8.
Subtract n and 4 from both sides: 2n = 4.
M1 — Let the number be n. Then 3n + 4 = n + 8.
A1 — Answer 2
Question 4 Medium2 marks
A number is multiplied by 4 and then 6 is added. The result is 38. What is the number?
Show answer
Answer: 8
Let the number be n. Then 4n + 6 = 38.
4n = 38 − 6 = 32.
M1 — Let the number be n. Then 4n + 6 = 38.
A1 — Answer 8
Question 5 Hard2 marks
4 times a number, plus 3, is equal to the number plus 12. What is the number?
Show answer
Answer: 3
Let the number be n. Then 4n + 3 = n + 12.
Subtract n and 3 from both sides: 3n = 9.
M1 — Let the number be n. Then 4n + 3 = n + 12.
A1 — Answer 3
Question 6 Hard2 marks
A number is multiplied by 5 and then 5 is subtracted. The result is 5. What is the number?
Show answer
Answer: 2
Let the number be n. Then 5n − 5 = 5.
5n = 5 + 5 = 10.
M1 — Let the number be n. Then 5n − 5 = 5.
A1 — Answer 2
Question 7 Exam-style3 marks
A rectangle has width n cm and length (n + 2) cm. Its perimeter is 20 cm. What is the length of the rectangle?