Difficulty: Easy
Correct Answer: 81
Explanation:
Introduction / Context:
This problem again uses the concept of third proportional, similar to earlier questions but with different numbers. It reinforces the formula and the understanding that third proportional is defined through a specific proportional relationship between three numbers.
Given Data / Assumptions:
Concept / Approach:
If a, b and c are in proportion such that a : b = b : c, then c is called the third proportional to a and b. In equation form this is a / b = b / c leading to a * c = b^2 and therefore c = b^2 / a. We will substitute a = 16 and b = 36 into this formula.
Step-by-Step Solution:
Step 1: Let the third proportional be x.
Step 2: By definition, 16 : 36 = 36 : x.
Step 3: Convert this to a fraction equation: 16 / 36 = 36 / x.
Step 4: Cross multiply: 16x = 36 * 36.
Step 5: Compute 36 * 36 = 1296.
Step 6: So 16x = 1296, which gives x = 1296 / 16.
Step 7: Simplify 1296 / 16 by dividing numerator and denominator by 16: 1296 ÷ 16 = 81.
Step 8: Hence the third proportional is 81.
Verification / Alternative check:
Check if 16 : 36 equals 36 : 81. Compute 16 / 36 = 4 / 9, and 36 / 81 = 4 / 9 after simplification. Since both ratios are equal, the proportional relation holds and confirms that 81 is the correct third proportional to 16 and 36.
Why Other Options Are Wrong:
Options such as 79, 84 or 88 do not satisfy the equality 16 : 36 = 36 : x. For these values, the fraction 36 / x will not simplify to 4 / 9. For instance, 36 / 84 simplifies to 3 / 7, which is not equal to 4 / 9, so 84 cannot be the third proportional.
Common Pitfalls:
A common mistake is to mix up third proportional with mean proportional and wrongly set 16 : x = x : 36. Another error is to perform the square on the wrong number or to miscalculate 36 squared. Remember that the formula c = b^2 / a comes from the proportionality a / b = b / c, which is the defining property of third proportional.
Final Answer:
The third proportional to 16 and 36 is 81.
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