The ratio of two positive numbers is 3 to 4 --> \(\frac{a}{b}=\frac{3x}{4x}\), for some positive integer \(x\).

If k is added to each number the new ratio will be 4 to 5 --> \(\frac{3x+k}{4x+k}=\frac{4}{5}\) --> \(15x+5k=16x+4k\) --> \(x=k\).

The sum of the numbers will be 117 (I believe it means that the sum of the numbers after we add k to each) --> \(3x+k+4x+k=117\) --> \(7x+2k=117\) --> as from above \(x=k\) --> \(7k+2k=117\) --> \(k=13\).

Answer: B.

With the above approach it shouldn't take more than 30 secs.
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Thanks Bunuel!! 13 is the answer
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We have \(\frac{3x}{4x} = \frac{3}{4}\)................ (1)

Also, \(\frac{3x+k}{4x+k} = \frac{4}{5}\) .............. (2)

Looking at (1) & (2), we know that denominator of (1) = numerator of (2)

So we can directly equate:

4x = 3x + k

x = k = 13 = Answer
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Hi LaxAvenger,

You could refer to both parts of the ratio as you've described, but it would create more work for you overall. In these types of ratio question, when taking a 'math' approach, it's almost always easiest to do the work in 'part to part' format and not "part-to-whole TO "part-to-whole."

GMAT assassins aren't born, they're made,
Rich
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Hi sagar2911,

When we're told that the ratio of A:B is 3 to 4, that means that A = some multiple of 3 and B = the equivalent multiple of 4.

For example:
A = 3 and B = 4
A = 6 and B = 8
A = 9 and B = 12
Etc.

Mathematically, since we don't know WHICH multiple we're dealing with, we have to assign a variable to it. Thus...

A/B = 3X/4X

This allows us to construct more complex equations that are based on that original ratio.

GMAT assassins aren't born, they're made,
Rich
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Hi All,

Many Test Takers would approach this question algebraically, but you can also approach this question by working back from the final piece of information and TESTing THE ANSWERS.

We're told that two numbers will end up in the ratio of 4:5 and their sum will be 117

4:5 is the same as saying 4X:5X

4X + 5X = 117
9X = 117
X = 13

Thus, the two "end" numbers are 4(13) and 5(13): 52 and 65

The question asks for the value of K - the value that was added to the ORIGINAL 2 numbers to get us to 52 and 65.

We know from the original ratio of 3:4 that the first number is a multiple of 3 and the second number is the equivalent multiple of 4. So we're looking for whichever answer, when subtracted from 52 and 65, gives us a number that's a multiple of 3 and another that's the SAME multiple of 4....

Answer A:
52-1 = 51 = 3(17)
65-1 = 64 = 4(16)
NOT the same multiple.

Answer B:
52-13 = 39 = 3(13)
65-13 = 52 = 4(13)
This IS the SAME multiple, so this must be the answer.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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The ratio of two positive numbers is 3 to 4.
Let 3x = the smaller number
Let 4x = the larger number

Aside: Notice that this ensures that their ratio will equal 3/4, since 3x/4x = 3/4

If k is added to each number the new ratio will be 4 to 5 . . .
When k is added to each value, the NEW values are 3x + k and 4x + k
So, (3x + k)/(4x + k) = 4/5
Cross multiply to get: 16x + 4k = 15x + 5k
Rearrange terms to get: x = k


. . . and the sum of the numbers will be 117
The two NEW values are 3x + k and 4x + k
So, we can write: (3x + k) + (4x + k) = 117
Simplify to get 7x + 2k = 117

Since x = k , we can take the above equation and replace x with k to get...
7k + 2k = 117
Simplify: 9k = 117
Solve: k = 13

Answer: B

Cheers,
Brent
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