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The ratio of two positive numbers is 3 to 4. If k is added [#permalink]

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03 Jun 2010, 07:17

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The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

Kindly share the time in which you have answered the question.

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.
_________________

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

Kindly share the time in which you have answered the question.

I think I approached this in a slightly different manner than Bunuel:

9x = 117 7x = 117 - 2k

Combine the equations:

2x = 2k

k = x = 117 / 9 = 13

Took about the same time as Bunuel but I found it easier.

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

Kindly share the time in which you have answered the question.

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.

We have \(\frac{3x}{4x} = \frac{3}{4}\)................ (1)

Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]

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06 Oct 2015, 09:05

Bunuel wrote:

Hussain15 wrote:

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

Kindly share the time in which you have answered the question.

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.

Some general question to the number properties in this case. I thought when we have a ratio of two numbers, like 3 to 4, this means --> 3:4 = 3/7 and 4/7 Why can we here express this in the fraction 3x/4x and not 3x/(4x+3x) and 4x/(3x+4x) ???

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."

Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]

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25 Oct 2015, 03:29

Bunuel wrote:

Hussain15 wrote:

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

Kindly share the time in which you have answered the question.

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.

Bunuel,

I couldnt understand why you took a/b = 3x / 4x I started with a/b = 3/4 and completely lost the plot. Is this wrong approach?
_________________

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.

The ratio of two positive numbers is 3 to 4. If k is added [#permalink]

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07 Dec 2017, 22:18

Hussain15 wrote:

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

9x=117 x=13 7x=91 91+2k=117 k=13 B

Last edited by gracie on 13 Jan 2018, 11:11, edited 1 time in total.

The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?

A. 1 B. 13 C. 14 D. 18 E. 21

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