Introduction / Context:
This question tests the concept of fourth proportional in a proportion relation. Proportional numbers are a common theme in quantitative aptitude, and knowing how to find the third or fourth proportional quickly is a useful exam skill. Here, you are given three numbers and asked to find the fourth proportional that completes the proportion based on the standard definition.
Given Data / Assumptions:
- First number a = 6.
- Second number b = 24.
- Third number c = 83.
- We are required to find the fourth proportional d.
Concept / Approach:
If a, b, c and d are in proportion, we write a : b = c : d. In such a case, d is called the fourth proportional to a, b and c. From a : b = c : d, we get a / b = c / d. Cross-multiplying gives a * d = b * c, so d = (b * c) / a. We can apply this formula directly to the given numbers once we identify a, b and c correctly.
Step-by-Step Solution:
1) We are given a = 6, b = 24 and c = 83.
2) For the fourth proportional d, we use a : b = c : d.
3) Therefore, 6 : 24 = 83 : d.
4) Write this in fraction form: 6 / 24 = 83 / d.
5) Simplify 6 / 24 to 1 / 4.
6) So 1 / 4 = 83 / d.
7) Cross-multiply to obtain d = 83 * 4.
8) Compute d = 332.
9) Therefore the fourth proportional to 6, 24 and 83 is 332.
Verification / Alternative check:
We verify by checking whether 6 : 24 equals 83 : 332. The ratio 6 : 24 simplifies to 1 : 4. The ratio 83 : 332 simplifies by dividing both terms by 83, giving 1 : 4. Since both ratios are identical, they confirm that 6 : 24 = 83 : 332, and therefore 332 is indeed the correct fourth proportional.
Why Other Options Are Wrong:
Option A (249) would give 83 : 249 which does not simplify to 1 : 4. Option C (166) gives 83 : 166 which is 1 : 2, not 1 : 4. Option D (498) would make 83 : 498 approximately 1 : 6, again incorrect. Option E (124.5) leads to a non-integer ratio. Only 332 maintains the correct proportional relationship with the given numbers.
Common Pitfalls:
Students sometimes mistakenly set up the proportion as 6 : 24 = d : 83, which would lead to the wrong formula. Another common issue is forgetting to simplify the initial ratio 6 : 24 and thus making the arithmetic unnecessarily complicated. Always carefully set the proportion as a : b = c : d when finding the fourth proportional and then cross-multiply correctly.
Final Answer:
The fourth proportional to 6, 24 and 83 is
332.
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