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Difficulty: MediumDecimals and Scientific Notation

An automated micro-dispenser delivers liquid reagents in precise doses. Each dose has a volume of 3.6×10−53.6 \times 10^{-5} liters. If a reservoir containing 0.01620.0162 liters of reagent is emptied completely by delivering these equal doses, how many doses were delivered?

Answer: 450

Answer

450
Dividing the total volume of 1.62×10−21.62 \times 10^{-2} liters by the single dose volume of 3.6×10−53.6 \times 10^{-5} liters yields 1.623.6×10−2−(−5)=0.45×103=450\frac{1.62}{3.6} \times 10^{-2 - (-5)} = 0.45 \times 10^3 = 450 doses.

Step-by-Step Solution

1
Express the total reservoir volume in scientific notation
0.0162=1.62×10−20.0162 = 1.62 \times 10^{-2} liters
Converting decimals into standard scientific notation simplifies multiplication and division operations.
2
Set up the division for the number of doses
Number of doses=1.62×10−23.6×10−5\text{Number of doses} = \frac{1.62 \times 10^{-2}}{3.6 \times 10^{-5}}
The total volume divided by the volume per single dose yields the total dose count.
3
Compute the division of coefficients and exponent terms independently
1.623.6=0.45\frac{1.62}{3.6} = 0.45 and 10−210−5=10−2−(−5)=103\frac{10^{-2}}{10^{-5}} = 10^{-2 - (-5)} = 10^3
Applying the exponent quotient rule 10a/10b=10a−b10^a / 10^b = 10^{a-b} yields −2−(−5)=3-2 - (-5) = 3.
4
Convert from scientific notation to a standard integer
0.45×103=4500.45 \times 10^3 = 450
Multiplying 0.450.45 by 1,0001,000 shifts the decimal point 3 places to the right.

Key Concept

Division of numbers in scientific notation and place value manipulation
Estimated Time:1m 30s
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Topics

Decimals and Scientific Notation

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