Question

The ratio of two numbers is \(3:4\). If 8 is added to each number, the ratio becomes \(4:5\). If 10 is subtracted from each of the original numbers, what will be their new ratio?

• A. \(2:3\)
• B. \(8:7\)
• C. \(5:9\)
• D. \(7:11\)

Hint:

For ratio problems:
• Represent the numbers as \(ax\) and \(bx\).
• Form an equation using the changed ratio.
• Find \(x\), substitute it back, and calculate the required ratio.

Answer 1

The Correct Option is D

Concept: When the ratio of two numbers is given, represent them as multiples of a common variable. Use the changed ratio to determine the variable, and then compute the required ratio.

Step 1: Represent the numbers.
Since the ratio is \(3:4\), let the numbers be \[ 3x \quad \text{and} \quad 4x. \]

Step 2: Use the given condition after adding 8.
According to the question, \[ \frac{3x+8}{4x+8}=\frac45. \] Cross-multiplying, \[ 5(3x+8)=4(4x+8) \] \[ 15x+40=16x+32 \] \[ x=8. \] Hence, the original numbers are \[ 3x=24,\qquad 4x=32. \]

Step 3: Subtract 10 from each original number.
\[ 24-10=14, \] \[ 32-10=22. \] Thus, the new ratio is \[ 14:22. \]

Step 4: Simplify the ratio.
Divide both terms by their common factor \(2\): \[ 14:22=7:11. \]

Step 5: Final conclusion.
Therefore, the required ratio is \[ \boxed{7:11} \]