If the wealth of A is 4/9 times the wealth of B and the wealth of C is 7/6 times the wealth of B, what is the ratio of the wealth of C to the wealth of A?

Introduction / Context:
This ratio problem involves three people A, B and C, whose wealths are related to B by given fractional multipliers. We must find the ratio of the wealth of C to the wealth of A. It tests understanding of proportional relationships when multiple quantities are expressed in terms of a common reference quantity.

Given Data / Assumptions:

• Wealth of A is 4/9 times the wealth of B.• Wealth of C is 7/6 times the wealth of B.• All wealth values are positive and measured in the same monetary units.• We must determine the ratio C : A.

Concept / Approach:
When multiple quantities are compared to a common base, it is convenient to represent the base using a variable. Let the wealth of B be some amount, say b. Then A and C can be written in terms of b using the given fractional relationships. The ratio of C to A is then obtained by forming a fraction of C over A, substituting the expressions in terms of b, and simplifying. Because b appears in both numerator and denominator, it cancels out, leaving a ratio in simplest terms.

Step-by-Step Solution:
Step 1: Let the wealth of B be b (in suitable units).Step 2: A's wealth is given as 4/9 of B's wealth, so A = (4 / 9) * b.Step 3: C's wealth is given as 7/6 of B's wealth, so C = (7 / 6) * b.Step 4: We are required to find the ratio C : A.Step 5: Express C : A as a fraction and simplify: C / A = ((7 / 6) * b) / ((4 / 9) * b).Step 6: Cancel b from numerator and denominator, since b is nonzero: C / A = (7 / 6) / (4 / 9).Step 7: Division of fractions is done by multiplying by the reciprocal, so C / A = (7 / 6) * (9 / 4).Step 8: Multiply numerators and denominators: (7 * 9) / (6 * 4) = 63 / 24.Step 9: Simplify 63 / 24 by dividing numerator and denominator by their greatest common divisor, which is 3. This gives 21 / 8.Step 10: Therefore, the ratio C : A is 21 : 8.

Verification / Alternative check:
To verify, we can assign a convenient value to b, such as b = 72, which is divisible by both 9 and 6. Then A = (4 / 9) * 72 = 4 * 8 = 32. C = (7 / 6) * 72 = 7 * 12 = 84. The ratio C : A is then 84 : 32. Divide both numbers by 4 to get 21 : 8. This matches the simplified ratio obtained by algebra, confirming that the result is correct and independent of the particular value chosen for b.

Why Other Options Are Wrong:
• 8 : 21: This is simply the inverse of the correct ratio and would correspond to A : C, not C : A.• 27 : 14 and 14 : 27: These ratios do not arise from the given fractional relationships with B and do not simplify to 21 : 8 when checked with actual values.

Common Pitfalls:
Some learners misread the question and compute the ratio A : C instead of C : A. Others might incorrectly handle the fractional multipliers and attempt to work with percentages instead, leading to confusion. Mistakes can also occur during the simplification of 63 / 24 if the greatest common divisor is not correctly identified. Using the method of representing B's wealth as a variable, carefully canceling common factors, and methodically simplifying fractions helps avoid these issues.

Final Answer:
The ratio of the wealth of C to the wealth of A is 21 : 8.