GMAT Club Olympics 2024 (Day 2): The ratio of two positive integers

Question

The ratio of two positive integers, m and n, is 4 to 3. If the values of m and n are increased in a ratio of 1 to 1, which of the following cannot be the resulting integers?

I. 8 to 5
II. 6 and 5
III. 15 and 14

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Bunuel · Accepted Answer

GMAT Club Official Explanation:

When two numbers are  increased  in a ratio of 1:1, it implies that the  same positive number  was added to both of them.

If a fraction is between 0 and 1, adding the same positive number to both the numerator and denominator increases the value of the fraction. For instance, if we add 2 to the numerator and denominator of 1/3, we get (1 + 2)/(3 + 2) = 3/5, which is greater than the initial fraction of 1/3.

If a fraction is more than 1, adding the same positive number to both the numerator and denominator decreases the value of the fraction. For instance, if we add 2 to the numerator and denominator of 3/2, we get (3 + 2)/(2 + 2) = 5/4 = 1.25, which is less than the initial fraction of 3/2 = 1.5.

For the question at hand, the original ratio was 4:3, which is more than 1 (≈1.3), thus adding the same positive number to both the numerator and the denominator will decrease the value of the new ratio, bringing it closer to, but still above 1. Therefore, both 6 and 5 (6/5 = 1.2), and 15 and 14 (15/14 ≈ 1.07), are possible values for the resulting integers since their ratio is less than 1.3, while 8 to 5 cannot be the resulting integers, as their ratio is 1.6, which is greater than 1.3.

Answer: A.

jack5397 · Answer

The ratio of two positive integers, m and n, is 4 to 3. If the values of m and n are increased in a ratio of 1 to 1, which of the following cannot be the resulting integers?

I. 8 to 5
II. 6 and 5
III. 15 and 14

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

IMO - A

Only critical part to understand in this question is increased in 1:1 ratio. It means increased by same qty so difference between m:n will always remain 1. So only option A does not have difference of 1. Hence option A.

Riddhig · Answer

The ratio is m/n = 4/3. 
Ratio added has to be 1/1 so anything that can be broken down to a 1:1 can be added, for example: 2/2, 3/3 and so on.

Now let's look at the options:
1. 8 to 5: if I add a 4 to numerator and denominator, the resulting ratio will be 8:7 (cannot be the resulting integer)
2. 6 to 5: can be obtained by adding 2 to numerator and denominator
3. 15 to 14: can be obtained by adding 11 to numerator and denominator

Hence, the answer should be A: I only

vanitha777 · Answer

m/n = 4/3

m+1/n+1 = 4x+1/3x+1 = odd/ (odd or even)

By this logic option 2 is also not possible since numerator must be odd . PLease let me know where Iam wrong
 Bunuel

Bunuel · Answer

Why are you assuming m and n increase by 1? The question says they increase in a ratio of 1 to 1, meaning the same number is added to both, not necessarily 1. For example, if m = 4 and n = 3, adding 2 to each gives 6 and 5, which is valid. Also, it looks like you missed the full explanation already provided above.

GMAT Club Olympics 2024 (Day 2): The ratio of two positive integers

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GMAT Club Olympics 2024 (Day 2): The ratio of two positive integers