Difficulty: Medium
Correct Answer: 3.5
Explanation:
Introduction / Context:
This is an algebraic problem involving squares and fractions. We are told that when 89² is added to the square of an unknown number, the sum is 16202. We need to determine one twenty-sixth of that unknown number. The question tests equation formation and manipulation with square numbers.
Given Data / Assumptions:
- Let the unknown number be n.
- n² + 89² = 16202.
- We need n / 26 (one twenty-sixth of n).
Concept / Approach:
We first compute 89², subtract it from 16202 to find n², and then take the positive square root to get n. Once n is known, we divide it by 26 to get n / 26, which should match one of the options.
Step-by-Step Solution:
Step 1: Let the required number be n. Then n² + 89² = 16202.Step 2: Compute 89² = 89 × 89 = 7921.Step 3: Substitute into the equation: n² + 7921 = 16202.Step 4: Subtract 7921 from both sides: n² = 16202 − 7921.Step 5: Calculate the difference: 16202 − 7921 = 8281.Step 6: So n² = 8281. Take the square root: n = √8281.Step 7: Recognize that 91² = 8281, so n = 91.Step 8: We are asked for n / 26, so compute 91 / 26.Step 9: 91 divided by 26 = 3.5.
Verification / Alternative check:
Check the original equation: 91² = 8281, 89² = 7921, and 8281 + 7921 = 16202. This confirms that n = 91. Dividing 91 by 26 gives 3.5, which matches the derived result and one of the answer choices.
Why Other Options Are Wrong:
Values like 5.65, 2.7 and 6.66 would correspond to different values of n and therefore would not satisfy the original equation n² + 89² = 16202. Testing them by multiplying by 26 and then squaring shows that none produce n² = 8281.
Common Pitfalls:
Errors often occur in computing 89² or in subtracting 7921 from 16202. Some students also misidentify the square root of 8281. Knowing that 90² = 8100 and 95² = 9025 helps estimate that 8281 lies between them, leading to the correct value n = 91 by checking 91² explicitly.
Final Answer:
One twenty-sixth of the number is 3.5.
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