Split 94 into two parts such that one-fifth of the first part and one-eighth of the second part are in the ratio 3 : 4. What is the first part?

Introduction / Context:This problem turns a verbal ratio condition on fractional parts into equations in two variables. Once the ratio is translated, cross-multiplication and the sum constraint lead directly to the required parts.

Given Data / Assumptions:

• Total = 94 split into x (first part) and y (second part).
• (x/5) : (y/8) = 3 : 4.
• x + y = 94.

Concept / Approach:From (x/5)/(y/8) = 3/4 ⇒ (8x)/(5y) = 3/4 ⇒ 32x = 15y. Express y in terms of x and use x + y = 94 to solve for x.

Step-by-Step Solution:32x = 15y ⇒ y = (32/15)xx + (32/15)x = 94 ⇒ (47/15)x = 94x = 94 * 15 / 47 = 30Hence, first part = 30

Verification / Alternative check:Compute parts: First = 30, Second = 64. Then x/5 = 6 and y/8 = 8; ratio 6 : 8 reduces to 3 : 4, as required.

Why Other Options Are Wrong:27, 36, 48, and 24 do not satisfy both the ratio condition and the sum constraint simultaneously.

Common Pitfalls:Setting 1/5 of the first equal to 1/8 of the second (equality) instead of the specified 3 : 4 ratio; forgetting to apply the total-sum constraint after forming the ratio equation.

Final Answer:30