The integers a, b, and c are positive a/b = 5/2, and a/c

Question

The integers a, b, and c are positive, a/b = 5/2, and a/c = 7/5. What is the smallest possible value of 2a + b?

A. 63
B. 70
C. 84
D. 95
E. 105

Bunuel · Accepted Answer

Since  \frac {a}{b}=\frac {5}{2} , and  \frac {a}{c}=\frac {7}{5}  then  a  must be a multiple of both 5 and 7, so the lowest value of  a  is  5*7=35  (note that  a  is a positive integer). Then the lowest value of  b  would be  2*7=14 , as  \frac {a}{b}=\frac {5}{2}=\frac {5*7}{2*7}=\frac {35}{14} , so the lowest value of  2a+b=2*35+14=84 .

Answer: C.

Or: since  \frac {a}{b}=\frac {5}{2}  then  a=\frac{5b}{2}  and  2a + b=2*\frac{5b}{2}+b=6b , so it's a multiple of 6. The only multiple of 6 among the answer choices is 84 (C).

Answer: C.

pharm · Answer

why is the value of '2' for b have '7' multiplying to it , 7 is the the value for the fraction a/c . I understand how you are getting the lowest value for 'a' but not for 'b' .

Thanks

Bunuel · Answer

The lowest value of  a  is 35. Now, if a=35, then from a/b = 5/2 we'll have that b=14.

Hope it's clear.

pharm · Answer

Ok so since a/b = 5/2 and a/c = 7/5 . You are multiple across the '7' to 5 & 2 in order to get their LCM. Correct?

Bunuel · Answer

I don't understand what you mean. Please, elaborate your question.

pharm · Answer

you said that from a/b = 5/2 , b=14 . Since in a/b , 'b' was = 2 . the common number is '7' that is being multipled to 5 and 2 in order to reach the lowest possible values correct?
so to get " b's " lowest possible value you multiplied = '2 * 7= 14' . Does '7' hold any significance that it was used for the values in 'a' & 'b' to reach there lowest possible values ?

Kris01 · Answer

It is mentioned that a/c=7/5. So from this ratio we know that 7 needs to be a factor of a. From a/b=5/2 we again know that a must contain 5. Hence, the minimum value of a must be 35.

Using the ratio a/b=5/2, we know that b must have 2 and must also contain 7 as a also contained 7 which was cancelled while calculating he simplest form of ratio.

PareshGmat · Answer

\frac{a}{b} = \frac{5}{2}

\frac{2a}{b} = \frac{10}{2}

Using componendo / dividendo

\frac{2a + b}{b} = \frac{10+2}{2}

2a +b = 6b

Looking at the options available, only 84 is divisible by 6

Answer = C

Bunuel, can you kindly tell-

Q1: Is this method correct to solve?

Q2: a/c = 7/5 >> Is this term given just to confuse? As I never required that info in solving this problem.

ScottTargetTestPrep · Answer

We are given that:

a/b = 5/2 or a : b = 5 : 2

and

a/c = 7/5 or a : c = 7 : 5

We need to determine the smallest value of 2a + b.

Since a is a multiple of 5 and of 7, we see it’s a multiple of the LCM of 5 and 7, which is 35.

Thus we now have:

a : b = 5 x 7 : 2 x 7 = 35 : 14

a : c = 7 x 5 : 5 x 5 = 35 : 25

So the minimum value of a is 35 and the minimum value of b is 14 (since a, b, and c must be integers), and thus the minimum value of 2a + b is 2(35) + 14 = 70 = 14 = 84.

Answer: C

Abhishek009 · Answer

\frac{a}{b} = \frac{5}{2} , &  \frac{a}{c} = \frac{7}{5}

Make numerators same base

So,  \frac{a}{b} = \frac{35}{14} , &  \frac{a}{c} = \frac{35}{25}

So We have  a = 35 ,  b = 14  &  c = 25

Thus, The value of  2a + b = 2*35 + 14 = 84 , Answer must be (C)

Mansoor50 · Answer

JUst in case its not obvious that a is a multiple of 35:

b/a + c/a = 2/5 + 5/7

(b+c)/a = 39/35

this clearly shows that a is a multiple of 35

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The integers a, b, and c are positive a/b = 5/2, and a/c

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