What is the fourth proportional to the numbers 336, 288 and 161 (that is, 336 : 288 = 161 : x)?

Introduction / Context:
This question tests the concept of fourth proportional in ratio and proportion. When three numbers are given, the fourth proportional is defined using a proportional relationship, similar to how we define the third proportional. The numbers 336, 288 and 161 are given, and we must find the value x that completes the proportion 336 : 288 = 161 : x. This is a standard and frequently tested idea in quantitative reasoning.

Given Data / Assumptions:
• First number a = 336. • Second number b = 288. • Third number c = 161. • Fourth proportional d = x must satisfy 336 : 288 = 161 : x.

Concept / Approach:
If a, b, c and d are in proportion, then a : b = c : d. In terms of fractions, a / b = c / d. From here, we can solve for d in terms of a, b and c. Specifically, d = (b * c) / a. So this is a straightforward computation problem where we plug the numbers into this formula. We need to be careful with arithmetic but no advanced algebra is required.

Step-by-Step Solution:
Step 1: Start from the proportion: 336 : 288 = 161 : x. Step 2: Express it as a fraction equation: 336 / 288 = 161 / x. Step 3: Cross multiply: 336 * x = 288 * 161. Step 4: Compute the right side: 288 * 161. Step 5: 288 * 161 = 288 * (160 + 1) = 288 * 160 + 288 * 1 = 46,080 + 288 = 46,368. Step 6: So 336 * x = 46,368. Now solve for x: x = 46,368 / 336. Step 7: Divide 46,368 by 336. This gives x = 138.

Verification / Alternative Check:
We can verify by checking whether 336 : 288 is equal to 161 : 138. Simplify 336 : 288 by dividing both numbers by 48. We get 336 / 48 = 7 and 288 / 48 = 6, so 336 : 288 = 7 : 6. Now simplify 161 : 138 by checking their common factors. 161 = 7 * 23 and 138 = 6 * 23. So 161 : 138 = 7 * 23 : 6 * 23 = 7 : 6. Since both ratios simplify to 7 : 6, the proportion holds and x = 138 is indeed correct.

Why Other Options Are Wrong:
• 184, 115 and 134 do not satisfy the proportion 336 : 288 = 161 : x. If you substitute any of them in place of x and cross multiply, the equality fails. • These numbers are included as plausible looking distractors but lack the precise proportional relationship with the given three numbers.

Common Pitfalls:
Common mistakes include inverting the proportion (writing 288 : 336 = 161 : x) or using a wrong formula such as x = (a * c) / b. Another source of errors is in doing the multiplication and division inaccurately, especially with larger numbers. Breaking the multiplication into simpler steps, as we did, helps maintain accuracy. Simplifying ratios early can also act as a quick check on the final answer.

Final Answer:
The fourth proportional to 336, 288 and 161 is 138.