For two real numbers m and n, the three-part ratio (m + n) : (m − n) : (m · n) is given as 7 : 1 : 60. What is the ratio 1/m : 1/n for these numbers?

Introduction / Context:
This problem connects algebraic expressions and ratios. We are told that three expressions involving two numbers, namely m + n, m − n and m·n, are in a particular ratio. From this information, we need to deduce the ratio of the reciprocals 1/m and 1/n. Questions like this test algebraic manipulation skills and understanding of how ratios behave under substitution and simplification.

Given Data / Assumptions:
• (m + n) : (m − n) : (m · n) = 7 : 1 : 60. • m and n are real numbers and non-zero (so that 1/m and 1/n are defined). • We must find 1/m : 1/n.

Concept / Approach:
We interpret the triple ratio by introducing a scaling constant k such that m + n = 7k, m − n = 1k and m·n = 60k. The first two equations allow us to solve for m and n in terms of k. Once m and n are known, we impose the product condition m·n = 60k to find k. Finally, we form the ratio 1/m : 1/n, which simplifies to n : m, and then we express it in the simplest integer form.

Step-by-Step Solution:
Step 1: Let m + n = 7k and m − n = k. Step 2: Add the two equations: (m + n) + (m − n) = 7k + k ⇒ 2m = 8k ⇒ m = 4k. Step 3: Subtract the second from the first: (m + n) − (m − n) = 7k − k ⇒ 2n = 6k ⇒ n = 3k. Step 4: Now use the third part of the ratio: m·n = 60k. Step 5: Substitute m and n: m·n = (4k) * (3k) = 12k^2. Step 6: Set 12k^2 = 60k. Since k is not zero, divide both sides by k to get 12k = 60 ⇒ k = 5. Step 7: Therefore m = 4k = 20 and n = 3k = 15. Step 8: Now find 1/m : 1/n. This is (1/20) : (1/15). Step 9: Multiply both terms by the common denominator 60 to clear the fractions: (1/20) * 60 : (1/15) * 60 = 3 : 4.

Verification / Alternative check:
We can verify by checking the original given ratio: m + n = 20 + 15 = 35, m − n = 20 − 15 = 5, and m·n = 300. Now form the ratio 35 : 5 : 300. Divide each term by 5 to get 7 : 1 : 60, which matches exactly. This confirms the correctness of m and n and therefore the reciprocal ratio 3 : 4.

Why Other Options Are Wrong:
• Ratios like 4 : 3, 8 : 7 and 7 : 8 would correspond to different values of m and n and would not reproduce the given 7 : 1 : 60 relationship.

Common Pitfalls:
A common mistake is to forget to check the third ratio condition m·n = 60k after solving the first two equations, which could lead to an incorrect k. Another error is to invert the ratio incorrectly when moving from m : n to 1/m : 1/n. Remember that 1/m : 1/n simplifies to n : m, not m : n.

Final Answer:
Thus, the required ratio 1/m : 1/n is 3 : 4.