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# [Smart Math] Ratio Proportion Problem 28

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

### Problem

The ratio of number of ladies to gents in a party was 3 : 2. When however, 20 more gentlemen joined the party, the ratio was reversed. How many ladies were there at the party?

1. 16
2. 24
3. 32
4. 36
5. 40

### The Usual Method

Let ‘x’ be the constant of proportionality. Hence, there will be 3x ladies and 2x gents in the party originally.

New total number of gents = 2x + 20

Hence, ratio of ladies to gents = $\frac{3x}{2x+20}=\frac{2}{3}$

$\therefore 9x=4x+40$

$\therefore 5x=40$

$\therefore x=8$

$\therefore$Original number of ladies = 3x = 3 x 8 = 24

(Ans: 2)

Estimated Time to arrive at the answer = 45 seconds.

### Using Technique

Since the total number of ladies at the party = 3x, the answer should be a multiple of 3. From the options, only option ‘2’ and ‘4’ are multiples of 3. Assuming option ‘2’ to be the right answer, the number of men becomes 2x = 16 (using 3x = 24 $\therefore x=8$) Hence, new total of gents = 20 + 16 = 36 and 24 : 36 = 2 : 3. This satisfies the conditions and so, ‘2’ is the right option.