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Difficulty: MediumDecimals and Scientific Notation

A manufacturing facility produces precision metal components. Each component has a mass of 8.4×10−48.4 \times 10^{-4} kilograms. If a shipment container holds a batch of these components with a total mass of 1.051.05 kilograms, how many components are in the container?

Answer: 1250 components

Answer

The container holds 1,250 components.
To find the number of components, divide the total batch mass by the mass per component: 1.05 kg8.4×10−4 kg\frac{1.05 \text{ kg}}{8.4 \times 10^{-4} \text{ kg}}. Expressing 110−4\frac{1}{10^{-4}} as 10410^4 transforms the expression into 1.058.4×104\frac{1.05}{8.4} \times 10^4. Dividing 1.051.05 by 8.48.4 yields 0.1250.125. Finally, 0.125×104=12500.125 \times 10^4 = 1250.

Step-by-Step Solution

1
Set up the ratio of total mass to single component mass
Number of components = 1.058.4×10−4\frac{1.05}{8.4 \times 10^{-4}}
Dividing total mass by individual component mass gives the total count.
2
Apply exponent rules to move the power of 10 to the numerator
1.058.4×104\frac{1.05}{8.4} \times 10^4
Since 110−4=104\frac{1}{10^{-4}} = 10^4, shifting the negative exponent to the numerator changes its sign.
3
Divide the decimal coefficients
1.058.4=0.125\frac{1.05}{8.4} = 0.125
Simplifying 105840\frac{105}{840} reduces to 18=0.125\frac{1}{8} = 0.125.
4
Evaluate the product with the place value shift
0.125×10,000=12500.125 \times 10,000 = 1250
Multiplying by 10410^4 shifts the decimal point 4 places to the right.

Key Concept

Division with scientific notation and decimal place value adjustment
Estimated Time:1m 30s
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Topics

Decimals and Scientific Notation

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