Two numbers are in the ratio 3 : 5. If 3 is subtracted from the first number and 3 is added to the second number, the ratio becomes 1 : 3. The sum of the numbers is?

24

The correct answer is Option (4) → 24

1. Define the Numbers

Since the numbers are in the ratio 3 : 5, let the numbers be:

• First number $= 3x$
• Second number $= 5x$

2. Set Up the Equation

According to the problem, if we subtract 3 from the first and add 3 to the second, the new ratio becomes 1 : 3:

$\frac{3x - 3}{5x + 3} = \frac{1}{3}$

3. Solve for $x$

Cross-multiply the fractions to solve for $x$:

$3(3x - 3) = 1(5x + 3)$

$9x - 9 = 5x + 3$

Rearrange the terms to group the $x$ variables on one side:

$9x - 5x = 3 + 9$

$4x = 12$

$x = 3$

4. Calculate the Sum of the Numbers

The sum of the original numbers is $3x + 5x = 8x$.

$\text{Sum} = 8 \times 3$

$\text{Sum} = 24$

(Verification: The numbers are $9$ and $15$. Subtracting 3 from 9 gives 6, and adding 3 to 15 gives 18. The ratio $6:18$ is indeed $1:3$.)

Conclusion

The sum of the two numbers is 24.