Two numbers are in the ratio 7 : 5. If the first number is increased by 14 and the second number is decreased by 5, then the ratio becomes 4 : 2. What is the sum of the original two numbers?

Correct option is A

Given:

Two numbers are in the ratio $7:5$.

The first number is increased by 14 and the second number is decreased by 5

Solution: 

First number = 7x,

Second number = 5x

The first number is increased by 14 and the second number is decreased by 5,

New first number = $$7x + 14$$ 

New second number = $$5x - 5$$ 

The new ratio is given as 4 : 2 

$$\frac{7x + 14}{5x - 5} = \frac{4}{2}$$ 

$$\frac{7x + 14}{5x -5} = 2$$ 

$$7x + 14 = 2 (5x - 5)$$  

$$7x + 14 = 10x - 10$$ 

$$7x - 10x = -10 -14$$ 

$$-3x = -24$$ 

$$x = \frac{-24}{-3} = 8$$ 

First number = $$7 \times 8 = 56$$ 

Second number = $$5 \times 8 = 40$$ 

The sum of the original two numbers = $$56 + 40 = 96$$ 

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