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Difficulty: Very hardDecimals and Scientific Notation

A high-precision instrument measures the mass of a single micro-particle PP as 4.8×10−84.8 \times 10^{-8} grams and a single micro-particle QQ as 8.0×10−98.0 \times 10^{-9} grams. A sample consists of a combination of PP and QQ particles in a ratio of 33 particles of PP for every 55 particles of QQ. If the total mass of the sample is 9.2×10−59.2 \times 10^{-5} grams, what is the total number of particles in the sample?

Answer: 4000

Answer

The total number of particles in the sample is 4,000.
Aligning powers of 10 shows that one P particle weighs 4.8×10−84.8 \times 10^{-8} g and one Q particle weighs 0.8×10−80.8 \times 10^{-8} g. A combined unit of 3 P particles and 5 Q particles has a mass of 3(4.8×10−8)+5(0.8×10−8)=18.4×10−8=1.84×10−73(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 18.4 \times 10^{-8} = 1.84 \times 10^{-7} g. Dividing the total sample mass of 9.2×10−59.2 \times 10^{-5} g by 1.84×10−71.84 \times 10^{-7} g yields 500 units. Since each unit contains 8 particles (3 + 5), the total number of particles is 500×8=4,000500 \times 8 = 4,000.

Step-by-Step Solution

1
Convert the mass of particle Q so that it shares the same exponent (10−810^{-8}) as particle P.
Mass of single particle Q = 0.8×10−80.8 \times 10^{-8} grams.
Aligning powers of 10 is necessary before performing addition of masses.
2
Find the combined mass of a fundamental ratio group consisting of 3 particles of P and 5 particles of Q.
Mass of one ratio group = 3(4.8×10−8)+5(0.8×10−8)=14.4×10−8+4.0×10−8=18.4×10−8=1.84×10−73(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 14.4 \times 10^{-8} + 4.0 \times 10^{-8} = 18.4 \times 10^{-8} = 1.84 \times 10^{-7} grams.
Determines the mass contributed by each set of 8 particles.
3
Divide the total mass of the sample by the mass of a single ratio group.
Number of groups = 9.2×10−51.84×10−7=9.21.84×102=5×102=500\frac{9.2 \times 10^{-5}}{1.84 \times 10^{-7}} = \frac{9.2}{1.84} \times 10^2 = 5 \times 10^2 = 500 groups.
Determines how many full ratio sets of particles make up the sample.
4
Multiply the number of groups by the total number of particles contained in each group (3+5=83 + 5 = 8).
Total particles = 500×8=4000500 \times 8 = 4000.
Yields the total count of individual particles in the sample.

Key Concept

Operations with scientific notation, decimal place value alignment, and weighted proportional sums.
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Topics

Decimals and Scientific Notation

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