Scientific Notation: Word Problems and Error Checks | Grade 8 Math | LevelKite

In this lesson you put scientific notation to work. You will solve word problems about huge quantities, such as distances in space and computer storage, and you will learn to find and fix mistakes in scientific notation work.

Recall that a nonzero number is in scientific notation when it is written as $a \times 10^n$ , where the exponent $n$ is an integer and the coefficient $a$ satisfies $1 \le |a| < 10$ . For example, $340{,}000 = 3.4 \times 10^5$ .

Start with meaning. A word problem gives you quantities, not just numbers. Before you compute, name what each number measures and its unit, such as kilometers or bytes. A rate tells how much of one quantity goes with one unit of another, such as kilometers per second. Next, decide what the question asks for. Its meaning, not a single key word, tells you which operation to use. It also helps to write every number in scientific notation before you compute.

Let the meaning of the question choose the operation

What the question asks for	Operation	What happens to the units
A steady rate over an amount of time, or a total of equal groups	Multiply	Units multiply: km per second × seconds gives km
How many equal parts fit, or how many times as large one amount is	Divide by a nonzero amount	Units divide: bytes ÷ bytes per song gives songs, and km ÷ km gives a plain number
The combined amount of two quantities of the same kind	Add	Both amounts need the same unit, and the sum keeps it
How much more one amount is than another	Subtract	Both amounts need the same unit, and the difference keeps it

Worked example

Example 1: How far sunlight travels

Light travels about $3 \times 10^5$ kilometers each second. Sunlight takes about 500 seconds to reach Earth. About how far does sunlight travel to reach Earth?

Let $d$ be the distance in kilometers. The light covers $3 \times 10^5$ km in each of the 500 seconds, which is a steady rate over an amount of time, so multiply. Write 500 as $5 \times 10^2$ , because $5 \times 100 = 500$ .
$d = (3 \times 10^5\ \text{km/s}) \times (5 \times 10^2\ \text{s})$
You may multiply factors in any order, so group the coefficients together and the powers of ten together. The seconds cancel, which leaves kilometers.
$d = (3 \times 5) \times (10^5 \times 10^2)\ \text{km}$
Multiply the coefficients: $3 \times 5 = 15$ . Multiply the powers of ten by adding the exponents, because five factors of 10 times two more factors of 10 make seven factors of 10.
$d = 15 \times 10^{5+2}\ \text{km} = 15 \times 10^{7}\ \text{km}$
The distance is positive, so in scientific notation its coefficient must be at least 1 and less than 10. Since 15 is too large, write $15 = 1.5 \times 10^1$ and add the exponents again. This changes the form but not the value.
$d = 1.5 \times 10^1 \times 10^7\ \text{km} = 1.5 \times 10^{1+7}\ \text{km} = 1.5 \times 10^{8}\ \text{km}$
Check: the answer is a distance in kilometers, as the question asks. In decimal form, $1.5 \times 10^8$ and $15 \times 10^7$ both equal 150,000,000, so rewriting kept the value.
$1.5 \times 10^8 = 150{,}000{,}000 = 15 \times 10^7$

Answer

Sunlight travels about $1.5 \times 10^8$ kilometers, or about 150,000,000 km, to reach Earth.

Worked example

Example 2: Songs on a memory card

A memory card holds $1.28 \times 10^{11}$ bytes of data. A song file uses about 4,000,000 bytes. About how many song files fit on the card?

Let $s$ be the number of song files, a whole number. You want how many 4,000,000-byte parts fit into the card, so divide. First write $4{,}000{,}000 = 4 \times 10^6$ so both numbers are in scientific notation. Bytes divided by bytes per song leaves songs.
$s = \frac{1.28 \times 10^{11}}{4 \times 10^{6}}$
Because neither 4 nor $10^6$ is zero, you can divide the coefficients and the powers of ten separately. Dividing $10^{11}$ by $10^6$ cancels six of the eleven factors of 10, so subtract the exponents; this is valid because the base 10 is not zero.
$s = \frac{1.28}{4} \times \frac{10^{11}}{10^{6}} = 0.32 \times 10^{11-6} = 0.32 \times 10^{5}$
The result is positive, so its coefficient must be at least 1 and less than 10, but 0.32 is less than 1. Multiply the coefficient by 10 and subtract 1 from the exponent, so the value stays the same.
$s = 0.32 \times 10^{5} = 3.2 \times 10^{5-1} = 3.2 \times 10^{4}$
Check by undoing the division: the number of songs times the bytes per song should give back the size of the card. Multiply, then rewrite $12.8 \times 10^{10}$ in scientific notation.
$(3.2 \times 10^4)(4 \times 10^6) = (3.2 \times 4) \times 10^{4+6} = 12.8 \times 10^{10} = 1.28 \times 10^{11}$

Answer

About $3.2 \times 10^4$ , or about 32,000, song files fit on the card.

Finding errors. A mistake in scientific notation work often leaves a clue in the answer, so test an answer before you trust it. The table lists four checks. The undo check uses an inverse operation, one that reverses another: multiplying by a nonzero number reverses dividing by it, and adding a number reverses subtracting it. If a check fails, reread the setup, then look for the first line of work that is not equal to the line before it. Explain why that line fails, and redo the work from there.

Four checks for a scientific notation answer

Check	Ask yourself	It can catch
Form	For a nonzero answer, is the coefficient at least 1 and less than 10 in size, and is the exponent an integer?	A coefficient left as 15 or 0.32
Size	Is the answer a sensible size? Compare it with a quick estimate or with the numbers you started with.	A wrong exponent, or coefficients combined without matching powers of ten
Units and sign	Is the unit what the question asks for, and is the sign possible?	A wrong operation, a dropped unit or a negative distance
Undo	Does the inverse operation lead back to a number you started with?	An arithmetic slip in any step

Key concept

Solve from meaning, then check

Let the meaning of the quantities and the question choose the operation, and keep each unit with its number. Write a nonzero result as $a \times 10^n$ , where $n$ is an integer and $1 \le |a| < 10$ . Then test the answer before you trust it by checking its form, its size and its unit. A correct-looking form alone does not make an answer right.

Worked example

Example 3: Find and fix the error

Earth's average distance from the Sun is about $1.5 \times 10^8$ km, and Mercury's is about $5.8 \times 10^7$ km. A student found how much farther from the Sun Earth is and wrote: $5.8 - 1.5 = 4.3$ , so the answer is $4.3 \times 10^8$ km. Find the error and correct it.

Check the student's answer first. The form check passes: the answer is nonzero, its coefficient 4.3 is at least 1 and less than 10, and the unit is km. The size check fails. Both distances are positive, so their difference must be less than the larger distance, yet the student's answer is larger than Earth's whole distance.
$4.3 \times 10^8 > 1.5 \times 10^8$
Now find the step that broke. The student treated the difference as $(5.8 - 1.5) \times 10^8$ , which is not equal to it, because the coefficients count different powers of ten: 1.5 counts hundred-millions, but 5.8 counts ten-millions. That is like saying 3 hundreds minus 2 tens is 1 hundred, when it is really 280. Ignoring the powers of ten also made Mercury look farther away.
$1.5 \times 10^8 - 5.8 \times 10^7 \neq (5.8 - 1.5) \times 10^8$
To add or subtract coefficients, first write both numbers with the same power of ten. Rewrite Earth's distance with $10^7$ : multiply the coefficient by 10 and subtract 1 from the exponent, which keeps the value the same.
$1.5 \times 10^8 = 15 \times 10^7$
Now both coefficients count ten-millions, so the distributive property lets you subtract them and keep the common factor $10^7$ . The coefficient 9.2 is at least 1 and less than 10, so this nonzero result is already in scientific notation.
$15 \times 10^7 - 5.8 \times 10^7 = (15 - 5.8) \times 10^7 = 9.2 \times 10^7$
Check by undoing the subtraction: adding Mercury's distance back should give Earth's distance. The corrected answer is also less than $1.5 \times 10^8$ , as a difference of these positive distances must be.
$9.2 \times 10^7 + 5.8 \times 10^7 = (9.2 + 5.8) \times 10^7 = 15 \times 10^7 = 1.5 \times 10^8$

Answer

The student subtracted coefficients whose powers of ten did not match. Earth is about $9.2 \times 10^7$ km, or about 92,000,000 km, farther from the Sun than Mercury.

Takeaway: Scientific notation makes very large and very small quantities easier to work with, but the meaning of the problem still decides the operation. Carry the units through to your answer, and say about when the data are rounded. Then check size and units, not just form. When a check fails, look for a wrong setup or for the first line that is not equal to the line before it, and fix the work there.