Two numbers are in the ratio 3 : 5. If each number is increased by 10, the new ratio becomes 5 : 7. What are the original two numbers?

Introduction / Context:
This ratio-and-proportion problem asks for two original numbers given their initial ratio and how that ratio changes after adding a constant to both numbers. It tests setting up a ratio equation and solving for the common multiplier.

Given Data / Assumptions:

• Initial ratio of the two numbers = 3 : 5.
• 10 is added to each number.
• New ratio becomes 5 : 7.

Concept / Approach:
If numbers are in the ratio a : b, we can write them as a*x and b*x . When the same number is added to both, form a new ratio and solve for x. Then compute actual numbers.

Step-by-Step Solution:
Let numbers be 3x and 5x. After adding 10 to each: (3x + 10) : (5x + 10) = 5 : 7. Cross-multiply: 7*(3x + 10) = 5*(5x + 10). 21x + 70 = 25x + 50. Rearrange: 70 - 50 = 25x - 21x ⇒ 20 = 4x ⇒ x = 5. Numbers = 3x = 15 and 5x = 25.

Verification / Alternative check:
Add 10: 15 + 10 = 25 and 25 + 10 = 35. Ratio 25 : 35 simplifies to 5 : 7, which matches the condition.

Why Other Options Are Wrong:
13, 22 does not give 5 : 7 after adding 10. 7, 9 and 3, 5 do not satisfy the new ratio when 10 is added.

Common Pitfalls:
Mixing up the order of terms, forgetting to cross-multiply correctly, or adding 10 to only one number. Always represent numbers as 3x and 5x before adding the constant to both.

Final Answer:
15, 25