The sum of two numbers is 64. One of the numbers is 10 more than the other. What are the two numbers?

Step-by-step solution

Idea: Use one letter for the smaller number; then the larger one is that letter + 10. Equating the linear polynomial for the sum to 64 gives a linear equation.

1. Let the smaller number be x. Then the larger number is x + 10.½ mark
2. Their sum is 64: x + (x + 10) = 64, that is, 2x + 10 = 64. (2x + 10 is a linear polynomial; setting it equal to 64 gives a linear equation.)1 mark
3. 2x = 64 − 10 = 54, so x = 54 ÷ 2 = 27.1 mark
4. Larger number = 27 + 10 = 37. The numbers are 27 and 37.½ mark

Check: 27 + 37 = 64 ✓ and 37 − 27 = 10 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing x + 10x = 64. “10 more than x” is x + 10, not 10x.
• Dividing 64 by 2 to get 32 and stopping. 32 and 32 have sum 64 but differ by 0, not 10.
• Finding x = 27 and giving only one number. The question asks for both numbers.

How this can come in the exam

Try one yourself

Two numbers add up to 100 and differ by 18. Find them.

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