Difficulty: Medium
Correct Answer: 5/3
Explanation:
Introduction:
This question checks your ability to handle decimal arithmetic, bracket operations, and negative powers. The expression involves multiplication, addition, and an inverse (power of minus one), and finally asks for the answer in fractional form.
Given Data / Assumptions:
Concept / Approach:
We simplify inside each bracket first. Then we apply the negative power by taking the reciprocal of that result. Finally, we multiply the two parts. Working carefully with decimal multiplication and addition is crucial. Converting intermediate decimals to fractions can make simplification easier and help in obtaining the final answer as a reduced fraction.
Step-by-Step Solution:
Step 1: Compute 0.2 * 0.2 = 0.04.Step 2: Add 0.01: 0.04 + 0.01 = 0.05.Step 3: Compute 0.1 * 0.1 = 0.01.Step 4: Add 0.02: 0.01 + 0.02 = 0.03.Step 5: Raise the second bracket to the power -1: (0.03)^(-1) = 1 / 0.03.Step 6: Now the whole expression becomes 0.05 * (1 / 0.03) = 0.05 / 0.03.Step 7: Convert decimals to fractions: 0.05 = 5/100 and 0.03 = 3/100.Step 8: Compute (5/100) / (3/100) = (5/100) * (100/3) = 5/3.So the simplified result is 5/3.
Verification / Alternative check:
You can use decimal arithmetic: 0.05 ÷ 0.03 ≈ 1.6666..., and 5/3 as a decimal is 1.6666..., which matches. This confirms the correctness of the fractional result.
Why Other Options Are Wrong:
9/3 equals 3, 11/3 is approximately 3.67, 13/3 is about 4.33, and 7/3 is approximately 2.33. None of these match the decimal 1.6666..., so they cannot be correct values for the given expression.
Common Pitfalls:
Typical mistakes include misinterpreting the power -1 as multiplication by -1 instead of taking the reciprocal, or adding and multiplying decimals incorrectly. Another frequent error is to divide 3 by 5 instead of 5 by 3 when simplifying 0.05/0.03. Converting decimals to fractions and handling the arithmetic step by step helps avoid confusion.
Final Answer:
The value of the expression is 5/3.
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