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Two positive integers that have a ratio of 1 : 2

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by ErikaPrepScholar » Tue Jan 16, 2018 6:41 am
Ratios can be kind of confusing, so let's express them as a fraction instead:

We have a ratio of 1:2, so we can express that as
$$\frac{1}{2}$$

We know that to keep our same ratio, we can do things like $$\frac{1}{2}=\frac{1+1}{2+2}=\frac{2}{4}$$ $$\frac{1}{2}=\frac{1+2}{2+4}=\frac{3}{6}$$ $$\frac{1}{2}=\frac{1+3}{2+6}=\frac{4}{8}$$ and so on. Notice that we increase the fraction (our original ratio) in a 1:2 ratio - we add twice as much to the denominator as we add to the numerator.

So what if we increased the fraction in a 1:1 ratio - we add the same amount to the denominator as we add to the numerator. If adding twice as much to the denominator as the numerator gave us the same fraction before, then adding as much to the numerator should give us a larger fraction - in other words, the original ratio should increase. We can try this out:
$$\frac{1}{2}=\frac{1+1}{2+1}=\frac{2}{3}$$ $$\frac{1}{2}=\frac{1+2}{2+2}=\frac{3}{4}$$ $$\frac{1}{2}=\frac{1+3}{2+3}=\frac{4}{5}$$

So the increasing our original ratio in a 1:1 ratio will increase the original ratio from 1/2 to something greater than 1/2.

Looking at the answer options, the only one that is greater than 1/2 is C, so that's our answer.

We could also solve this by building equations, but logic is always a better first effort - even if we can't use it to fully solve, it generally helps us to eliminate answer options and better understand what the question is asking.
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by GMATGuruNY » Tue Jan 16, 2018 7:28 am
lheiannie07 wrote:Two positive integers that have a ratio of 1 : 2 are increased in a ratio of 1 : 1. Which of the following could be the resulting integers?

(A) 2 and 4
(B) 5 and 13
(C) 15 and 28
(D) 33 and 68
(E) 71 and 143
The original integers are in a 1:2 ratio.
They are increased in a 1:1 ratio, implying that each is increased by the SAME VALUE.
Implication:
When the same value is SUBTRACTED from the correct answer choice, the result must be a 1:2 ratio for the original integers.
Strategy:
In each answer choice, reduce the two given integers by the same amount.
The correct answer choice will yield a 1:2 ratio for the original integers.

A: 2 and 4
If we reduce 2 and 4 by the same amount, the following option is yielded for the original integers.
1 and 3.
The result in red does not constitute a 1:2 ratio.
Eliminate A.

B: 5 and 13
If we keep reducing 5 and 13 by the same amount, the following options are yielded for the original integers.
4 and 12
3 and 11
2 and 10
1 and 9.
None of the results in red constitutes a 1:2 ratio.
Eliminate B.

C: 15 and 28
14 and 27
13 and 26
Success!
The result in green constitute a 1:2 ratio.

The correct answer is C.
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by Scott@TargetTestPrep » Thu Jan 18, 2018 1:33 pm
lheiannie07 wrote:Two positive integers that have a ratio of 1 : 2 are increased in a ratio of 1 : 1. Which of the following could be the resulting integers?

(A) 2 and 4
(B) 5 and 13
(C) 15 and 28
(D) 33 and 68
(E) 71 and 143
Let's the original numbers be x and 2x and let each number increased by y (where both x and y are positive integers). Thus the new numbers are x + y and 2x + y. Now let's go through each answer choice and see which pair could be the resulting integers.

(A) 2 and 4

x + y = 2
2x + y = 4

Subtracting the first equation from the second we have:

x = 2

However, that means y = 0, which contradicts our assumption. A can't be the answer.

(B) 5 and 13

x + y = 5
2x + y = 13

Subtracting the first equation from the second, we have:

x = 8

However, that means y = -3, which contradicts our assumption. B can't be the answer.

(C) 15 and 28

x + y = 15
2x + y = 28

Subtracting the first equation from the second, we have:

x = 13

That means y = 2, which fits our assumption. Thus, C is the answer.

(We will leave the readers, as an exercise, to show that neither D nore E can be the answer.)

Answer: C

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