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[Smart Math] Ratio Proportion Problem 50

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

Problem

Two vessels contain spirit and water mixed in the ratio 5 : 3 and 3 : 1 respectively. In what ratio should they be mixed together to get the ratio of spirit and water in the final mixture as 2 : 1?

1. 1 : 2
2. 5 : 4
3. 4 : 5
4. 2 : 1
5. 3 :1

The Usual Method

The ratio of spirit to total volume in the first vessel = $\frac{5}{5+3}=\frac{5}{8}$

Similarly that in the second vessel = $\frac{3}{3+1}=\frac{3}{4}$

Required spirit to total volume = $\frac{2}{2+1}=\frac{2}{3}$

Hence using Alligation technique, we get:

$\frac{5}{8}$                                       $\frac{3}{4}$

$\frac{2}{3}$

$\frac{2}{24}$                                     $\frac{1}{24}$

i.e. $\frac{2}{24}:\frac{1}{24}$ or 2 : 1

(Ans: 4)

Estimated Time to arrive at the answer = 60 seconds.

Using Technique

Understand that by mixing $\frac{5}{8}$ of spirit to $\frac{3}{4}$ of another spirit; we need $\frac{2}{3}$ of spirit in the mixture. Also know that $\frac{3}{4}$ > $\frac{2}{3}$ > $\frac{5}{8}$. Hence we would need less of $\frac{3}{4}$ strength spirit than $\frac{5}{8}$ strength spirit to get the mixture of intermediate type. Hence options ‘1’ and ‘3’ are eliminated since in both these cases the spirit of strength $\frac{3}{4}$ is more than that of strength $\frac{5}{8}$.

Also, it can be visually noted that by simply adding the two mixtures in equal quantities, i.e. $\frac{6}{8}+\frac{5}{8}$, we get: $\frac{11}{16}$ (Note: Here $\frac{6}{8}+\frac{5}{8}$ = $\frac{11}{16}$ and not = $\frac{11}{8}$ as the numerator as well as the denominator are quantities. Visualize that 6 liters of spirit in 8 liters of mixture and 5 liters of spirit in 8 liters of mixture will give a total of 6 + 5 = 11 liters of spirit in a total of 8 + 8 = 16 liters of mixture).

What we need actually is $\frac{2}{3}$ or $\frac{6}{9}$ or $\frac{8}{12}$.

$\frac{8}{12}$ can be written as $\frac{8}{12}$ = $\frac{3}{4}+\frac{5}{8}$ (Note: Same logic as explained in the previous note.)

i.e. the mixture of strength $\frac{5}{8}$ is double that of $\frac{3}{4}$. Hence the ratio of 2 : 1.

(Note: In this particular case the time needed using technique may be marginally less than that taken by the usual method. The idea is to show the various ways to attempt such problems.)

(Ans: 4)

Estimated Time to arrive at the answer = 45 seconds.
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