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Board of Directors D
Joined: 01 Sep 2010
Posts: 3401
In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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15 00:00

Difficulty:   85% (hard)

Question Stats: 54% (02:30) correct 46% (02:44) wrong based on 201 sessions

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x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

_________________

Originally posted by carcass on 12 Sep 2013, 09:32.
Last edited by Bunuel on 12 Sep 2013, 09:37, edited 1 time in total.
Moved to PS forum.
Math Expert V
Joined: 02 Sep 2009
Posts: 58116
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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3
7
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

$$\frac{8a}{c} + \frac{16b}{c}=\frac{8a+16b}{c}=\frac{8a+16b}{a+b}$$.

You can notice that it's a weighted average formula for $$a+b$$ items, where the weight of each of the items of $$a$$ is 8 and the weight of each of the items of $$b$$ is 16. Now, since b > a, then the average must be closer to 16, then to 8. Only D fits.

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##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 58116
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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2
Bunuel wrote:
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

$$\frac{8a}{c} + \frac{16b}{c}=\frac{8a+16b}{c}=\frac{8a+16b}{a+b}$$.

You can notice that it's a weighted average formula for $$a+b$$ items, where the weight of each of the items of $$a$$ is 8 and the weight of each of the items of $$b$$ is 16. Now, since b > a, then the average must be closer to 16, then to 8. Only D fits.

Similar question to practice from OG:
Quote:
If x, y, and k are positive numbers such that (x/(x+y))(10) + (y/(x+y))(20) = k and if x < y, which of the following could be the value of k?

A. 10
B. 12
C. 15
D. 18
E. 30

Discussed here: if-x-y-and-k-are-positive-numbers-such-that-x-x-y-128231.html
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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1
1
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

x = 8a/c + 16b/c
x= 8(a+b)/c + 8b/c
now substituting c = a + b in the above equation
x = 8 + 8b/(a+b)
The ratio b/(a+b) is smaller than 1 thus answer choice E is eliminated... and as a < b ... thus the ratio b/(a+b) is larger than 1/2 or 0.5... thus
8*b/(a+b) is larger than 4... thus the only answer choice left is D
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In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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Can anyone explain this in detail?
Math Expert V
Joined: 02 Sep 2009
Posts: 58116
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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vijayann wrote:
Can anyone explain this in detail?

Please specify what part in the solutions above is unclear.
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In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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x = 8(a/c) + 16(b/c)

just by looking at the options .. we should be able to understand that 8 or its multiple i.e 16 should be divisible by C..why ? Because none of the options are in p/q form.

let us say c =4 which gives us a=1 and b=3

the equation thus becomes x=8(1/4) + 16(3/4) =2+12=14 --close enough the other value c can take is 8
which gives us possible values for (a,b) as (1,7) (2,6) (3,5)

take 1,7

x=8(1/8)+16(7/8)=15

That's how I managed to do.
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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oh man..I did the long way..
8a+8b/a+b = 8
now, 8b/a+b =??
we can eliminate A right away..
tried several numbers..then got to:
a=1
b=7
a+b=8
8b=56.
56/8=7.
7+8=15.

D
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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CounterSniper wrote:
x = 8(a/c) + 16(b/c)

just by looking at the options .. we should be able to understand that 8 or its multiple i.e 16 should be divisible by C..why ? Because none of the options are in p/q form.

let us say c =4 which gives us a=1 and b=3

the equation thus becomes x=8(1/4) + 16(3/4) =2+12=14 --close enough the other value c can take is 8
which gives us possible values for (a,b) as (1,7) (2,6) (3,5)

take 1,7

x=8(1/8)+16(7/8)=15

That's how I managed to do.

I did exactly the same way! Can someone clarify if this method is OK to use for future such questions? Feels like guess work and lucked out with this one.
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

Simplifying, we have:

x = (8a + 16b)/c

x = (8a + 16b)/(a + b)

We see that we have a weighted average, and since b is greater than a, then x must be closer to 16 and also between 8 and 16, so x could be 15.

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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: In the formula above, a, b, c, and d are positive numbers.   [#permalink] 28 May 2019, 18:06
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# In the formula above, a, b, c, and d are positive numbers.

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