A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?
Step-by-step solution
To find: The two numbers
Idea: Let the smaller number be x, so the larger is 5x. After adding 21, the larger new number (5x + 21) must be twice the smaller (x + 21).
- Let the numbers be x and 5x. After adding 21: x + 21 and 5x + 21.½ mark
- The larger new number is twice the smaller: 5x + 21 = 2(x + 21).1 mark
- 5x + 21 = 2x + 42 ⇒ 3x = 21 ⇒ x = 7.1 mark
- The numbers are 7 and 5 × 7 = 35. (The other way round, x + 21 = 2(5x + 21), gives x = −73, which makes 5x negative, so it is rejected because the number must be positive.)½ mark
Check: 35 = 5 × 7 ✓. New numbers 28 and 56, and 56 = 2 × 28 ✓.
Answer to write in the exam
Let the numbers be x and 5x
5x + 21 = 2(x + 21)
5x + 21 = 2x + 42 ⇒ 3x = 21 ⇒ x = 7
5x = 35
∴ The numbers are 7 and 35
Common mistakes that cost marks
- Adding 21 to only one of the numbers. The question adds it to both.
- Writing 2(5x + 21) = x + 21 (twice the larger equals the smaller), which gives a negative answer.
- Writing 2x + 21 instead of 2(x + 21): “twice the new number” doubles the whole of x + 21.
How this can come in the exam
One number is 3 times another. When 10 is added to each, the larger becomes twice the smaller. The smaller number is
- 5
- 10
- 15
- 20
Show answer
(B) 10
3x + 10 = 2(x + 10) ⇒ x = 10.
One number is 4 times another. If 6 is subtracted from each, the larger becomes 7 times the smaller. Find the numbers.
Show answer
4x − 6 = 7(x − 6) (1 mark) ⇒ 4x − 6 = 7x − 42 ⇒ 3x = 36 (1 mark) ⇒ x = 12; the numbers are 12 and 48 (1 mark).Try one yourself
One number is 6 times another. If 15 is added to each, the larger becomes 3 times the smaller. Find the numbers.
Show answer
6x + 15 = 3(x + 15) ⇒ 3x = 30 ⇒ x = 10. The numbers are 10 and 60.
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