Difficulty: Easy
Correct Answer: 0.00243
Explanation:
Introduction / Context:
This problem checks your understanding of multiplying decimal numbers and correctly placing the decimal point in the final answer. When several decimals are multiplied, it is easy to miscount the total number of decimal places. A systematic approach avoids such mistakes and leads to the right product.
Given Data / Assumptions:
Concept / Approach:
One way is to multiply step by step while paying attention to decimal places. Another way is to write each decimal as an integer divided by a power of 10, then combine the powers. Both methods will yield the same result. We will use stepwise multiplication for clarity and then cross-check with the decimal-place method.
Step-by-Step Solution:
Step 1: Multiply 3 × 0.3 = 0.9. Step 2: Multiply the result by 0.03: 0.9 × 0.03 = 0.027. Step 3: Multiply by 0.003: 0.027 × 0.003 = 0.000081. Step 4: Finally, multiply by 30: 0.000081 × 30 = 0.00243. Step 5: Therefore, the exact product is 0.00243.
Verification / Alternative check:
Write each factor as a fraction: 3 = 3, 0.3 = 3 / 10, 0.03 = 3 / 100, 0.003 = 3 / 1000, and 30 = 30.
The product of integers is 3 × 3 × 3 × 3 × 30 = 2430.
The product of denominators is 10 × 100 × 1000 = 1000000.
So the product is 2430 / 1000000 = 0.00243, confirming the result.
Why Other Options Are Wrong:
0.0000243 and 0.000243: These are smaller by factors of 10 or 100, caused by misplacing the decimal point.
0.0243: This is larger by a factor of 10, also due to decimal miscounting.
0.243: This is larger by a factor of 100, clearly not matching the careful calculation.
Common Pitfalls:
Typical mistakes include multiplying the numbers but not correctly tracking the total number of decimal places, or rounding too early. Writing numbers as fractions over powers of 10 gives a clear view of how many zeros appear in the denominator and prevents such errors.
Final Answer:
The exact value of the product is 0.00243.
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