If $a, b, c$ are three positive integers such that a and b are in the ratio $3 : 4$ while $b$ and $c$ are in the ratio $2:1$ , then which one of the following is a possible value of $(a + b + c)$ ?

$201$

$205$

$207$

$210$

✅ Correct Option: 3

We have three positive integers $a$ , $b$ , $c$ with ratios $a : b = 3 : 4$ and $b : c = 2 : 1$ .

From $a : b = 3 : 4$ , we get $a = 3k$ and $b = 4k$ for some positive integer $k$ .

From $b : c = 2 : 1$ , we get $b = 2m$ and $c = m$ for some positive integer $m$ .

Since both expressions represent $b$ :

$4k = 2m$

$m = 2k$

Substituting back:

$a = 3k$

$b = 4k$

$c = m = 2k$

Therefore:

$a + b + c = 3k + 4k + 2k = 9k$

Since $k$ must be a positive integer, possible values are $9, 18, 27, 36, ...$

For $207$ :

$9k = 207$

$k = 23$

This gives $a = 69$ , $b = 92$ , $c = 46$ .

Verification: $69:92 = 3:4$ and $92:46 = 2:1$ ✓

Therefore, $207$ is a possible value of $(a + b + c)$ .

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CAT Past Year Questions (Topic-Wise)

If $a, b, c$ are three positive integers such that a and b are in the ratio $3 : 4$ while $b$ and $c$ are in the ratio $2:1$ , then which one of the following is a possible value of $(a + b + c)$ ?