The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
Answer: The integers are 42 and 105.
Step-by-step solution
Given: Difference = 63; Ratio = 2 : 5
To find: The two integers
To find: The two integers
Idea: Numbers in the ratio 2 : 5 can be written 2k and 5k for some number k. Their difference is 3k.
- Let the integers be 2k and 5k (so their ratio is 2 : 5).1 mark
- Difference: 5k − 2k = 63, so 3k = 63.1 mark
- k = 21. The integers are 2 × 21 = 42 and 5 × 21 = 105.1 mark
The two integers are 42 and 105.
Check: 105 − 42 = 63 ✓ and 42 : 105 = (42 ÷ 21) : (105 ÷ 21) = 2 : 5 ✓.
Answer to write in the exam
Let the integers be 2k and 5k
5k − 2k = 63 ⇒ 3k = 63 ⇒ k = 21
∴ The integers are 42 and 105
Common mistakes that cost marks
- Taking the integers as 2 and 5 themselves, or as x and x + 63 without using the ratio.
- Writing 5k + 2k = 63. The question gives the difference, so subtract.
- Stopping at k = 21. k is only the multiplier; the integers are 2k and 5k.
How this can come in the exam
MCQ (1 mark)
Two numbers are in the ratio 3 : 7 and their difference is 48. The larger number is
- 84
- 36
- 112
- 70
Show answer
(A) 84
7k − 3k = 4k = 48, so k = 12; the larger is 7 × 12 = 84.
Short answer (3 marks)
Two positive integers are in the ratio 4 : 9 and their sum is 117. Find them.
Show answer
Let them be 4k, 9k (1 mark). 13k = 117 (1 mark), k = 9: the integers are 36 and 81 (1 mark).Try one yourself
The difference between two positive integers is 45 and their ratio is 1 : 4. Find them.
Show answer
4k − k = 3k = 45, so k = 15. The integers are 15 and 60.
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