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

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

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

Cheers,
Brent

Originally posted by BrentGMATPrepNow on 13 Jan 2018, 09:44.
Last edited by BrentGMATPrepNow on 02 Feb 2021, 10:38, edited 1 time in total.
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Thanks Bunuel!! 13 is the answer
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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.

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.

Thanks
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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 $$x=k$$ --> $$7k+2k=117$$ --> $$k=13$$.

With the above approach it shouldn't take more than 30 secs.

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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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
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 $$x=k$$ --> $$7k+2k=117$$ --> $$k=13$$.

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

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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
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 $$x=k$$ --> $$7k+2k=117$$ --> $$k=13$$.

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

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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
thanks Bunuel! the way you solved this question is great
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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
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....

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

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

GMAT assassins aren't born, they're made,
Rich
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The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
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

Originally posted by gracie on 07 Dec 2017, 23:18.
Last edited by gracie on 13 Jan 2018, 12:11, edited 1 time in total.
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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
PareshGmat wrote:
One more method:

Just look at the ratio's provided

Earlier it was 3:4 & after addition of K to numerator & denominator, ratio went to 4:5

In such cases, we are just adding the same no. to numerator & denominator, so calculation can be done directly

4x + 5x = 117; 9x = 117; x = 13 = K

Bunuel - can you comment on the above method, for adding new ration directly and solve for one variable, when same number added in numerator and denominator ?
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Re: The ratio of two positive numbers is 3 to 4. If k is added [#permalink]
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

Solution:

We can let the original two numbers be 3x and 4x and create the equations:

(3x + k) / (4x + k) = 4/5

and

3x + k + 4k + k = 117

Simplifying the first equation, we have:

16x + 4k = 15x + 5k

x = k

Substituting k for x in the second equation, we have:

3k + k + 4k + k = 117

9k = 117

k = 13

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

A student ask me whether we're obligated to use the "multiplier strategy" (e.g., ratio = 3x/4x) for this question.
The answer is "No," and what follows is a solution that uses three variables.

Let A = the smaller of the two numbers
Let B = the bigger number

The ratio of two positive numbers is 3 to 4.
We can write: A/B = 3/4
Cross multiply: 4A = 3B
Rewrite as: 4A - 3B = 0

If k is added to each number the new ratio will be 4 to 5...
We can write: (A + k)/(B + k) = 4/5
Cross multiply: 5(A + k) = 4(B + k)
Expand: 5A + 5k = 4B + 4k
Rewrite as: 5A - 4B = -k

...the sum of the numbers will be 117
We can write: (A + k) + (B + k) = 117
Simplify: A + B + 2k = 117
Rewrite as: A + B = 117 - 2k

We now have the following system:
4A - 3B = 0
5A - 4B = -k
A + B = 117 - 2k

Solve, solve, solve....
A = 39, B = 52 and k = 13

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

(x + k) / (y + k) = 4 / 5

x + y + 2k = 117

Solve for x in the first:

x / y = 3 / 4

x = 3y / 4

Plug that into our second equation and solve for y

[(3y / 4) + k] / (y + k) = 4 / 5

5 * (3y / 4 + k) = 4(y + k)

15y + 20k = 16y + 16k

4k = y

if y = 4k and x = 3y / 4, then x = 3(4k) / 4 = 3k

Now plug that into our final equation

x + y + 2k = 117

3k + 4k + 2k = 117

9k = 117

k = 13
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