Question

the ratio of number of ladies to that of gents at the party was 3:2. When 30 more gents joined the party, the ratio was reversed. The number of ladies present in the party was
(a)36
(b)32
(c)42
(d)16

Answer 1

Hint: Consider the number of gents as x, number of the ladies as y. Given that the ratio of number of ladies to number of gents 3:2. The condition is given by if there are 30 more gents in the party then the ratio reversed. Which means the ratio of ladies to gents becomes 2:3. Use these conditions to form linear equations. Solve them to get the values of x and y.

Complete step-by-step answer:
Let consider the number of gents as x, number of the ladies as y. Given that the ratio of number of ladies to number of gents 3:2...... Condition (1)

So, mathematically, condition (1) can be represented as y : x = 3 : 2 .

\[\Rightarrow \text{ }\dfrac{3}{2}=\dfrac{y}{x}\]

Now cross multiply the above equation which gives us 3x = 2y........ (i)

The condition is given by if there are 30 more gents in the party then the ratio reversed. Which means the ratio of ladies to gents becomes 2:3.....condition (2)

On adding 20 more gents, the number of gents becomes (x + 30). So, mathematically, condition (2) can be represented as y : (x + 30) = 2 : 3 .

\[\Rightarrow \text{ }\dfrac{2}{3}=\dfrac{y}{x+30}\]

Now, we will cross multiply the above equation, which gives us 2x + 60 = 3y ........ (ii)

Now, we will solve equation (i) and equation (ii) to find the number of gents and the number of ladies in the party.

From equation (i), we have 3x = 2y.

$\Rightarrow x=\dfrac{2y}{3}$

Substituting in equation (ii), we get: $\left( 2\times \dfrac{2y}{3} \right)+60=3y$ .

$\Rightarrow \dfrac{4y}{3}+60=3y$

$\Rightarrow 4y+180=9y$

$\Rightarrow 5y=180$

$\Rightarrow y=36$

Therefore, the ladies in the party are 36.

Now, we substitute the value of y in equation (i) to get the number of gents in the party.

3x = 2y

\[\Rightarrow \] 3x = 72

$\Rightarrow $ x = 24

Therefore, there are 24 gents in the party.

Therefore, there are 36 ladies and 24 gents in the party. Hence, the correct option is option(a).

Note: While making equations, make sure that equations are framed carefully according to the data given. Do not form incorrect equations, as the answer depends upon these equations. Incorrect equations will lead to wrong answers.