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# [Smart Math] Ratio Proportion Problem 39

Here’s and example of a SMART MATH problem for RATIO PROPORTION.

### Problem

In what proportion must a number be divided so that 1/3rd of the first part and 1/8th of the second part are together equal to 1/4th of the number?

1. 6 : 8
2. 6 : 4
3. 2 : 3
4. 1 : 4
5. 5 : 4

### The Usual Method

Let the two parts be ‘x’ and ‘y’.

Hence, $\frac{1}{3}x+\frac{1}{8}y=\frac{1}{4}(x+y)$

$\frac{8x+3y}{24}=\frac{1}{4}(x+y)$

$\therefore 8x+3y=6(x+y)$

$\therefore 8x+3y=6x+6y$

$\therefore 2x=3y$

$\therefore \frac{x}{y}=\frac{3}{2}=\frac{6}{4}$

(Ans: 2)

Estimated Time to arrive at the answer = 45 seconds.

### Using Technique

Since, the first part has to be divided by 3 (1/3rd of first part), it should be either 3 or a multiple of 3.

Similarly the second part has to be divided by 8 (1/8th of second part), so it should be some multiple of 2 (prime factor of 8).

You will observe that only options ‘1’ and ‘2’ satisfy these conditions. Now taking option ‘1’, we verify it as follows:

Reduce 6 : 8 to 3 : 4 first

$\frac{1}{3}\times 3\times k+\frac{1}{8}\times 4\times k=k+\frac{k}{2}=1.5k$ where ‘k’ is the constant of proportionality

But 1.5k is not 1/4th of 7k. Hence this option does not satisfy.

Similarly taking option ‘2’ we verify it as follows:

Reduce 6 : 4 to 3 : 2 first

$\frac{1}{3}\times 3\times k+\frac{1}{8}\times 2\times k=k+\frac{k}{4}=1.25k$

Total number is = 3k + 2k = 5k

1.25k = 1/4th of 5k. Hence it satisfies.

(Ans: 2)

Estimated Time to arrive at the answer = 15 seconds.
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