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?
Answer & explanation
Correct answer: option 4
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.