Two numbers are in the ratio 4 : 7. If 30 is added to each number, the ratio becomes 5 : 8. What is the average (arithmetic mean) of the original two numbers?

Introduction:
Ratios with additive changes are solved by expressing numbers as multiples of a common factor and applying the new ratio after the addition. Once the base numbers are found, compute the mean.

Given Data / Assumptions:

• Original numbers: 4x and 7x.
• After adding 30: (4x + 30) : (7x + 30) = 5 : 8.
• We seek average = (4x + 7x)/2.

Concept / Approach:
Translate the ratio equality into a proportion and solve for x. Then compute the original numbers and their mean.

Step-by-Step Solution:

Verification / Alternative check:
Add 30: 150 and 240 → ratio 5 : 8, correct.

Why Other Options Are Wrong:
135, 145, 155, 175 are not the mean of the solved pair or stem from algebraic errors.

Common Pitfalls:
Misapplying the ratio or forgetting both numbers receive the +30 increment.

Final Answer:
165