Question

The ratio of two numbers is 3 : 5. If 39 is added to the first, and 14 is added to the second, then the ratio becomes 6 : 7. What will be the ratio if 11 is added to the first number and 6 is added to the second number?

• A. 7 : 9
• B. 5 : 9
• C. 5 : 7
• D. 2 : 3

Hint:

When simplifying ratios like 74:111, if you can't see the common factor immediately, try checking if they are multiples of prime numbers like 31, 37, or 41.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves ratios and algebraic equations. We represent the two numbers in terms of a common variable based on their initial ratio and then set up an equation based on the changes described.
Step 2: Key Formula or Approach:
Let the two numbers be $3x$ and $5x$. According to the problem: \[ \frac{3x + 39}{5x + 14} = \frac{6}{7} \]
Step 3: Detailed Explanation:
Cross-multiply the equation to solve for $x$: \[ 7(3x + 39) = 6(5x + 14) \] \[ 21x + 273 = 30x + 84 \] Rearrange the terms to isolate $x$: \[ 273 - 84 = 30x - 21x \] \[ 189 = 9x \] \[ x = \frac{189}{9} = 21 \] Now, find the actual numbers: First number = $3x = 3(21) = 63$
Second number = $5x = 5(21) = 105$
The question asks for the new ratio if 11 is added to the first and 6 to the second: New first number = $63 + 11 = 74$
New second number = $105 + 6 = 111$
Calculate the new ratio: \[ \text{Ratio} = \frac{74}{111} \] Both numbers are divisible by 37: \[ 74 = 37 \times 2, \quad 111 = 37 \times 3 \] \[ \text{Ratio} = \frac{2}{3} \]
Step 4: Final Answer:
The final ratio is 2 : 3.