Two-person age ratio with known difference: The ratio of the ages of two persons is 4 : 7. The older person is 30 years more than the younger. What is the sum of their present ages (in years)?

Introduction / Context:
When an age ratio and an absolute difference are both given, the scaling factor for the ratio can be found directly from the difference of the ratio parts. Then compute individual ages and their sum.

Given Data / Assumptions:

• Present ratio = 4 : 7.
• Difference between actual ages = 30 years (older minus younger).
• Let ages be 4x and 7x (older = 7x).

Concept / Approach:
The difference in actual ages equals the difference in ratio parts times the scale x: (7 − 4) x = 3x. Set 3x = 30 to find x, then calculate the sum.

Step-by-Step Solution:
1) 3x = 30 ⇒ x = 10.2) Ages are 4x = 40 and 7x = 70.3) Sum = 40 + 70 = 110.

Verification / Alternative check:
Ratio check: 40 : 70 simplifies by 10 to 4 : 7. Difference is 30, matching the statement.

Why Other Options Are Wrong:

• 100/120/130/90 do not satisfy both the required ratio and the 30-year difference simultaneously.

Common Pitfalls:
Confusing which is older/younger or adding the difference to the ratio numbers instead of to actual ages can derail the calculation.

Final Answer:
110