If M = (3/7) ÷ (6/5) × (2/3) + (1/5) × (3/2) and N = (2/5) × (5/6) ÷ (1/3) + (3/5) × (2/3) ÷ (3/5), what is the exact value of the ratio M / N?

Introduction / Context:
This question is a classic example of simplifying expressions involving fractions, multiplication and division. Two expressions M and N are given in terms of rational numbers, and the goal is to find the exact value of M divided by N. This tests the ability to handle fractions systematically, to recall that division by a fraction is the same as multiplication by its reciprocal, and to reduce the final result to lowest terms.

Given Data / Assumptions:

• M = (3/7) ÷ (6/5) × (2/3) + (1/5) × (3/2).
• N = (2/5) × (5/6) ÷ (1/3) + (3/5) × (2/3) ÷ (3/5).
• All numbers are rational and standard arithmetic rules apply.
• We must compute the simplified fraction M / N.

Concept / Approach:
The idea is to simplify M and N separately and then form the ratio. For each division by a fraction, we multiply by the reciprocal. Next, we multiply and add fractions carefully, keeping numerators and denominators under control and simplifying whenever possible. Once M and N are in simplest fractional form, the ratio M / N is found by multiplying M by the reciprocal of N and simplifying the result.

Step-by-Step Solution:
Step 1: Simplify M. (3/7) ÷ (6/5) = (3/7) * (5/6) = 15 / 42 = 5 / 14 after simplification. Now multiply by (2/3): (5/14) * (2/3) = 10 / 42 = 5 / 21. Compute the second part: (1/5) * (3/2) = 3 / 10. So M = 5 / 21 + 3 / 10. Find a common denominator for 5/21 and 3/10, which is 210. Convert: 5/21 = 50 / 210, 3/10 = 63 / 210. Therefore M = (50 + 63) / 210 = 113 / 210. Step 2: Simplify N. First part: (2/5) * (5/6) ÷ (1/3) = (2/5) * (5/6) * (3/1) = (2 * 5 * 3) / (5 * 6) = 30 / 30 = 1. Second part: (3/5) * (2/3) ÷ (3/5) = (3/5) * (2/3) * (5/3) = (3 * 2 * 5) / (5 * 3 * 3) = 30 / 45 = 2 / 3. Thus N = 1 + 2 / 3 = 5 / 3. Step 3: Compute M / N. M / N = (113 / 210) ÷ (5 / 3) = (113 / 210) * (3 / 5) = 339 / 1050. Step 4: Simplify the fraction 339 / 1050 by dividing numerator and denominator by 3, yielding 113 / 350.

Verification / Alternative check:
As a check, one can use a calculator that handles fractions or decimal equivalents. Compute M as approximately 0.538095, N as approximately 1.66667, and then compute M / N ≈ 0.322857. Converting 113 / 350 to decimal gives 113 ÷ 350 ≈ 0.322857, which agrees with the previous value. This confirms that the fractional simplification is correct.

Why Other Options Are Wrong:
Options a, b and d represent other rational numbers that might result from partial simplification or arithmetic slips, such as forgetting to reduce a fraction to lowest terms or miscomputing one of the intermediate products. Option e, 1/2, is too large compared to the more accurate decimal value of about 0.323. Only 113/350 exactly matches the fully simplified and verified ratio M / N.

Common Pitfalls:
Typical mistakes include forgetting that division by a fraction means multiplying by its reciprocal, not simply dividing numerators or denominators. Another source of error is failing to simplify intermediate results, which can make later calculations more complicated and prone to mistakes. Students also sometimes add fractions without converting them to a common denominator first. Being systematic at each step and simplifying whenever possible helps avoid these problems.

Final Answer:
Therefore, the exact value of M / N is 113/350.