If a and b are positive integers, is a/b = 5/8 ?

A really tricky question indeed. It requires solid concept on algebraic manipulation to crack.

The question asks whether \(\frac{a}{b}\) = \(\frac{5}{8}\)?

Statement 1 says \(\frac{1}{2}\) < \(\frac{a}{b}\) < \(\frac{2}{3}\) OR .5 <\(\frac{a}{b}\) <.66

Here the inequality satisfy multiple values for \(\frac{a}{b}\) between .5 & .66 as for example \(\frac{a}{b}\) = \(\frac{3}{5}\) , \(\frac{5}{9}\) etc. NOT sufficient.

Statement 2 says b=8. However there is no mention of a. Hence NOT sufficient.

Combining 1 & 2, we get \(\frac{1}{2}\) < \(\frac{a}{8}\) < \(\frac{2}{3}\)
Now trail out a with multiple values. a = 4 gives .5 , a = 5 gives .62 , a=6 gives .75.

Only a=5 satisfy the range. [0.5 to 0.66]

Thus Option (C) wins.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a and b are positive integers, is a/b = 5/8 ?

(1) 1/2 < a/b < 2/3

(2) b = 8

In the original condition, there are 2 variables (a,b) and 2 equations from the 2 conditions, making (C) our likely answer.
If we combine the 2 conditions, 1/2=12/24<a/b<16/24
When b-8, a/8=3a/24, from 12/24<3a/24<16/24, there is only a=5 that satisfies the inequality, so the condition is sufficient as unique answer is given.
The answer becomes (C).

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

Hi,
I am confused with the option 2. My aproach is the following: if a/b = 5/8 and b=8, then we can substitute 8 for b and have an equality a/8 = 5/8, which can be solved for a. In this case we find that a = 5 and 5/8 = 5/8.
Appreciate your feedback.

Hi Rashad046,
In the beginning I thought the same way you did, but then found out that "a/b = 5/8" is not given, it is the target question, so we don't know yet if a/b = 5/8, we need to find that out from the info in the 2 statements.
I hope I am not mistaken.
Thanks
AladdinT

GMATPrepNow

Bunuel

FROM QUES. STEM IS CLEAR THAT 5/8 = 0.625 THEN IT IS DIRECTLY SHOWS IN THE STATEMENT 1. THAT VALUE F A/B LIES BETWEEN 0.5 TO 66.6.

IMO - ANSWER IS A

The question asks: IS a/b = 5/8?

For (1) a/b can certainly be 5/8, because 5/8 is between 1/2 and 2/3 but a/b can also be any other number between 1/2 and 2/3, for example, 0.6. So, (1) is not sufficient.

In this question, the fact that a and b are positive integers is what makes the question easier to deal with. Had this constraint not been given, this question would have been much more difficult to solve.

We are to find out if \(\frac{a}{b}\) = \(\frac{5}{8}\) i.e. if a and b are in the ratio of 5:8.

From statement I, we know that \(\frac{a}{b}\) is greater than 0.5 but lesser than 0.67. Clearly, one possible value that \(\frac{a}{b}\) can take is \(\frac{5}{8}\) i.e. 0.625.
But, unfortunately, this is not the only value that is in between 0.5 and 0.67. \(\frac{5}{9}\) is also in the same range.

Therefore, statement I is insufficient to answer the question. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II, we know nothing about a. We only know that b = 8. This is surely insufficient to answer the question.
Answer option B can be eliminated.

Combining both statements I and II, with b = 8, the only way in which \(\frac{a}{b}\) can be greater than 0.5 but lesser than 0.67 is by having a = 5.

If a = 4, b = 8, a/b = ½ which violates the condition given in statement I. Any value of a<4 will bring the value of a/b below ½, which is not acceptable.

If a = 6, b = 8, a/b = ¾ which violates the condition given in statement I. Any value of a>6 will increase the value of a/b beyond 0.67, which is not acceptable again.

Therefore, the only possible value for a is 5. When a = 5 and b = 8, \(\frac{a}{b}\) = \(\frac{5}{8}\). The combination of statements is sufficient.
The correct answer option is C.

Hope this helps!

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

(DS) This is a 2 variables question with Condition(1) with 1 equation and Condition (2) with 1 equation

We are aware that each condition would usually give us an equation, resulting in a total of 2 equations, one each from condition (1), and condition (2). Since we need 2 equations to match the number of variables and equations in the original condition, an equal number of equations and variables should logically give us an answer C.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

=> We have to find whether \(\frac{a }{ b}\) = \(\frac{5 }{ 8}\)

Second and the third step of Variable Approach: From the original condition, we have 2 variables (f and m). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

The answer is C because the number of variables and equations is 2. You should never get this question wrong as it is a basic, 30-39 level question.

Let’s take a look at both conditions together

Condition 1: \(\frac{1 }{2} < \frac{a }{ b} < \frac{2 }{3}\) and Condition 2: b = 8

=> \(\frac{1 }{ 2} < \frac{a }{ 8} < \frac{2 }{ 3}\)

=> 4 < a < \(\frac{16 }{ 3}\)

=> 4 < a < 5.33

a being integer will have value as 5. Hence, a = 5 and b = 8.

Is \(\frac{a }{ b} = \frac{5 }{ 8}\) - YES

The answer is unique YES according to CMT(1), so the conditions combined are sufficient.

Both conditions (1) and (2) together are sufficient.

So, C is the correct answer.

Answer: C

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