Percent Word Problems and Scale Errors | Grade 6 Math | LevelKite

A percent is a rate per hundred. So 35% means 35 for every 100.

In a word problem, look for what the percent is a percent of. That quantity is the whole. The amount the percent picks out is the part.

Try this situation: A school has 300 students. 35% of the students ride a bus. The whole is 300 students, because the percent is "of the students". That is 35 bus riders for every 100 students. There are 3 hundreds in 300, so the part is $3 \times 35 = 105$ students.

Most wholes are not a neat number of hundreds. In this lesson you will find the part for any whole a problem gives you. Then you will hunt for mistakes in percent work.

Before you calculate, read the problem for three things:

The whole. Ask: the percent is of what? Write the number and its unit.
The percent. This is the rate per hundred.
The question. Check what the answer should count or measure.

Next, turn the percent into a factor, which is a number you multiply by. 35% means 35 hundredths, so its factor is $\frac{35}{100}$ . As a decimal that is $0.35$ . Both forms name the same number, so they give the same product.

Why multiply? Taking 35 hundredths of a whole means cutting the whole into 100 equal pieces and taking 35 of them. That is dividing by 100, then multiplying by 35. Multiplying by $\frac{35}{100}$ does both jobs in one step, so the result is the same. Here it is for the 300 students:

\frac{35}{100} \times 300 = 35 \times \frac{300}{100} = 35 \times 3 = 105

Key concept

A percent of a quantity is a factor times the whole

Let $W$ be the whole, $p$ the number in front of the percent sign, and $P$ the part, in the same unit as $W$ . To find $p$ % of $W$ , multiply $W$ by the factor $\frac{p}{100}$ .

The factor also tells you what size to expect. When $W$ is positive (greater than 0):

a factor between 0 and 1 (a percent between 0% and 100%) gives a part smaller than $W$ ;
a factor of 1 (exactly 100%) gives a part equal to $W$ ;
a factor greater than 1 (a percent above 100%) gives a part larger than $W$ .

When $W$ is 0, the part is 0 for every percent.

P = \frac{p}{100} \times W

How the factor sets the size of the part

Whole	Percent	Factor	Calculation	Part compared with the whole
$40$	25%	$\frac{25}{100} = 0.25$ , between 0 and 1	$0.25 \times 40 = 10$	Smaller than 40
$40$	100%	$\frac{100}{100} = 1$	$1 \times 40 = 40$	Equal to 40
$40$	150%	$\frac{150}{100} = 1.5$ , greater than 1	$1.5 \times 40 = 60$	Larger than 40
$0$	25%	$\frac{25}{100} = 0.25$	$0.25 \times 0 = 0$	Equal to 0: zero stays zero

Worked example

Finding a part in a word problem

A class collects 80 cans for a food drive. 35% of the cans hold soup. How many cans hold soup?

Find the whole. The percent is "of the cans", so the whole is all 80 cans. Call the whole W, counted in cans.
$W = 80$
Write the percent as a factor. 35% means 35 hundredths, and 35 hundredths is the decimal 0.35.
$\frac{35}{100} = 0.35$
Multiply the whole by the factor to get the part, P, also counted in cans. Multiply 35 by 80 to get 2800, then divide by 100, because the factor counts hundredths.
$P = \frac{35}{100} \times 80 = \frac{2800}{100} = 28$
Check the size. The whole, 80, is positive and the factor 0.35 is between 0 and 1, so the part must be smaller than 80. Benchmarks are easy percents such as 25% and 50%. Here 25% of 80 is 20 and 50% of 80 is 40. The whole is positive, so a larger percent of it gives a larger part. That puts 35% of 80 between 20 and 40. The answer 28 fits.
$20 < 28 < 40$

Answer

Of the 80 cans, 28 hold soup.

Worked example

Choosing the whole with a decimal percent

A jar holds a sample of seawater. The seawater has a mass of 2000 grams, and the empty jar has a mass of 150 grams. Salt makes up 3.5% of the mass of the seawater. How many grams of salt are in the sample?

Find the whole. The problem gives two masses. The percent is "of the mass of the seawater", so the whole is the 2000 grams of seawater. The 150 grams of the jar is not part of the whole. Call the whole W, measured in grams.
$W = 2000$
Write the percent as a factor. 3.5% means 3.5 hundredths. Dividing by 100 shifts every digit two places to the right, so the 3 lands in the hundredths place and the 5 lands in the thousandths place.
$\frac{3.5}{100} = 0.035$
Multiply the whole by the factor to get the part, P, measured in grams. Multiply 3.5 by 2000 to get 7000, then divide by 100, because the factor counts hundredths.
$P = \frac{3.5}{100} \times 2000 = \frac{7000}{100} = 70$
Check the size. The whole, 2000, is positive and the factor 0.035 is between 0 and 1, so the part must be smaller than 2000. Get closer with 1% of 2000, which is 2000 divided by 100, or 20. Then 3% of 2000 is 3 times 20, or 60, and 4% is 4 times 20, or 80. The whole is positive, so 3.5% of it must fall between 60 and 80. The answer 70 fits.
$60 < 70 < 80$

Answer

The sample contains 70 grams of salt.

Now switch roles and become the checker. Many mistakes in percent work are scale errors. A scale error makes an answer the wrong size, often 10 or 100 times too large or too small. The digits can look right while the place value is wrong.

To find a mistake, predict before you recalculate:

Predict the size. Name the whole and the percent. For a positive whole, use the factor: between 0 and 1 means a smaller part, exactly 1 means an equal part, and greater than 1 means a larger part. A benchmark such as 1% or 10% of the whole gets you closer.
Compare. If the written answer does not fit your prediction, look for a mistake.
Trace the factor. Find the line where the percent became a number. Check that it shows the correct number of hundredths, and that the whole is multiplied by it.
Repair and recheck. Redo the work from the wrong line. Test the new answer against your prediction.

An answer that fits your prediction can still hide a small slip, so check the arithmetic too.

Worked example

Finding the mistake in a solution

A jar holds 800 beads. 0.5% of the beads are gold. Sam writes $0.5 \times 800 = 400$ and says that 400 beads are gold. Find Sam's mistake and fix it.

Predict the size. The whole is 800 beads, which is positive. The percent 0.5% is between 0% and 100%, so its factor is between 0 and 1 and the part must be smaller than 800. Get closer with a benchmark: 1% of 800 is 800 divided by 100, or 8 beads. Since 0.5% is less than 1% and the whole is positive, the part must be less than 8 beads.
$800 \div 100 = 8$
Compare. Sam's answer is 400 beads, which is far more than 8 beads. It does not fit the prediction, so look for a mistake.
$400 > 8$
Trace the factor. Sam multiplied by 0.5. But 0.5 is 50 hundredths, which is 50%. Sam dropped the percent sign without dividing by 100, so the work finds half of the beads.
$0.5 = \frac{50}{100}$
Repair the factor. 0.5% means 0.5 hundredths. Dividing by 100 shifts every digit two places to the right, so the 5 moves from the tenths place to the thousandths place.
$\frac{0.5}{100} = 0.005$
Multiply the whole by the correct factor. Multiply 0.5 by 800 to get 400, then divide by 100, because the factor counts hundredths.
$\frac{0.5}{100} \times 800 = \frac{400}{100} = 4$
Recheck. The new answer, 4 beads, is less than 8 beads, as predicted. It is also half of 8, which matches 0.5% being half of 1%. Sam skipped the division by 100, so Sam's answer was 100 times too large. That is a scale error.
$400 \div 100 = 4$

Answer

Sam used 0.5 as the factor, which finds 50% of the beads. The correct factor is 0.005, so 4 of the 800 beads are gold.

Three more mistakes and how the size check exposes them

Problem	Mistaken work	What the size check says	What went wrong	Correct work
25% of 60	$25 \times 60 = 1500$	60 is positive and the factor is between 0 and 1, so the part is smaller than 60. But 1500 is larger.	25 was used as the factor, so the division by 100 is missing. The answer is 100 times too large.	$0.25 \times 60 = 15$
20% of 45	$45 \div 0.2 = 225$	45 is positive and the factor is between 0 and 1, so the part is smaller than 45. But 225 is larger.	The whole was divided by the factor. Dividing a positive number by a positive number below 1 makes it larger.	$0.2 \times 45 = 9$
120% of 50	$0.12 \times 50 = 6$	50 is positive and the factor is greater than 1, so the part is larger than 50. But 6 is smaller.	12 hundredths was used, but 120% is 120 hundredths, or 1.2. The answer is 10 times too small.	$1.2 \times 50 = 60$

Here is the idea to keep. In a percent word problem, find the whole, write the percent as a number of hundredths, and multiply the whole by that factor. Then let the factor judge the answer. For a positive whole, a factor between 0 and 1 gives a smaller part, a factor of 1 gives a part equal to the whole, and a factor greater than 1 gives a larger part. A whole of 0 gives a part of 0. When an answer does not fit, trace the factor. That is where scale errors in percent work usually start.