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Ratio Problem

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Ratio Problem

by HPengineer » Wed Dec 15, 2010 8:18 pm

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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?

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

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by 4GMAT_Mumbai » Wed Dec 15, 2010 8:32 pm

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Hi,

Let us start from the reverse ...

If the ratio of numbers is 4:5 and their total is 117; then the numbers are 52 and 65.

Now, (52 - k) / (65 - k) = 3:4

208 - 4k = 195 - 3k

k = 13. Hence, B
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by HPengineer » Wed Dec 15, 2010 9:23 pm

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Thanks 4gmat but how did you come up with those number 52 and 65 so easily? I could not conjure up those numbers is there an easier algebraic approach?
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by classyjay » Wed Dec 15, 2010 10:28 pm

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HPengineer wrote:Thanks 4gmat but how did you come up with those number 52 and 65 so easily? I could not conjure up those numbers is there an easier algebraic approach?

Let's suppose each number is a and b.
a+b = 117 --> a = 117-b
a/117:b/117 = 4:5
4b/117 = 5a/117
4b = 5a
4b = 5(117-b)
9b = 585
Therefore, b = 65/a =52
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by 4GMAT_Mumbai » Wed Dec 15, 2010 10:55 pm

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HPengineer wrote:Thanks 4gmat but how did you come up with those number 52 and 65 so easily? I could not conjure up those numbers is there an easier algebraic approach?
Hi,

If you know two numbers are in the ratio 4:5; then you could assume that the numbers are 4a and 5a (where a is any real number)

Now, we know that the sum of the numbers is 117.

So, 4a + 5a = 117; 9a = 117; a = 13.

Now that a is 13, the numbers are 4 *13 and 5 * 13.

I hope this helps. Thanks.
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by GMATGuruNY » Thu Dec 16, 2010 4:28 am

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HPengineer 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

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A great way to handle ratio problems is to use a ratio box. Let's call the two unknown values X and Y and make a box for the second ratio of 4 to 5. Here's what the box looks like:

__________X____Y____Total

Ratio:_____4____5_____9

Multiplier:

Actual:_______________117

In the top row, we put the ratio of X:Y = 4:5.
The last column is the total: 4+5 = 9.
In the bottom row, we put any known values. The sum of X and Y is 117, so this number is put under "Total" in the bottom row.

To determine the middle row -- the multiplier -- we divide the known actual value (total = 117) by its corresponding value in the top row (9): 117/9 = 13. The multiplier is placed in every column along the middle row:

____________X____Y____Total

Ratio:_______4____5______9

Multiplier:__13__ 13_____13

Actual:__________________117



Now we can complete the box by multiplying every value in the top row by the multiplier:

____________X_____Y______Total

Ratio:_______4_____5_______9

Multiplier:__13____13_____13

Actual:____52____65______117


So after k has been added to both values, X=52 and Y=65. Now we can plug in the answer choices, which represent the value of k. When the correct answer choice is subtracted from 52 and 65, the ratio will be 3:4.

Only answer choice B works: k=13.
52-13 = 39.
65-13 = 52.
39/52 = 3/4. Success!

The correct answer is B.
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by Brent@GMATPrepNow » Mon Sep 09, 2019 2:08 pm

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HPengineer 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
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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