Percents Greater Than 100 and Less Than 1 | Grade 6 Math | LevelKite

You already know that percent means per 100. A percent compares a part with a whole. The whole is the reference amount: the amount that counts as $100\%$ . The part is the amount you compare with it.

The part does not have to fit inside the whole. You can read more pages than your goal, so the pages you read can be more than $100\%$ of the goal. The part can also be tiny, like a few pages of a very long book. Percents describe both cases.

Two benchmarks, or familiar reference points, help you make sense of them:

$100\%$ of an amount is the whole amount, because $\frac{100}{100} = 1$ .
$1\%$ of an amount is one hundredth of it, because $1\% = \frac{1}{100}$ .

Now think about a whole that is a positive amount. More than $100\%$ of it is more than the whole. A percent between $0\%$ and $1\%$ of it is less than one hundredth of the whole.

Different percents of the same whole, 200

Percent	Written per 100	As a decimal	That percent of 200	Compared with the whole
$0.5\%$	$\frac{0.5}{100}$	$0.005$	$1$	Less than one hundredth of 200
$1\%$	$\frac{1}{100}$	$0.01$	$2$	Exactly one hundredth of 200
$50\%$	$\frac{50}{100}$	$0.5$	$100$	Half of 200
$100\%$	$\frac{100}{100}$	$1$	$200$	Equal to 200
$150\%$	$\frac{150}{100}$	$1.5$	$300$	All of 200 plus half of 200
$200\%$	$\frac{200}{100}$	$2$	$400$	Twice 200

Key concept

Percents can go above 100 and below 1

A percent compares a part with a whole as a number per 100. Let $p$ stand for that number, so $p = 150$ means $150\%$ . To find $p\%$ of a whole, multiply the whole by $\frac{p}{100}$ . You can always divide by $100$ , because $100$ is not zero.

For a whole that is a positive amount:

If $p$ is greater than $100$ , then $\frac{p}{100}$ is greater than $1$ , so the part is greater than the whole.
If $p$ is between $0$ and $1$ , then $\frac{p}{100}$ is between $0$ and $0.01$ , so the part is less than one hundredth of the whole.

Either way, the whole is still the amount that counts as $100\%$ .

\text{part} = \frac{p}{100} \times \text{whole}

Look back at the table. Every row uses one idea: write the percent per 100, then multiply by the whole.

Dividing a number by $100$ makes each digit worth one hundredth as much, so each digit moves two places to the right:

$150\% = \frac{150}{100} = 1.5$
$0.5\% = \frac{0.5}{100} = 0.005$

You can also build an unusual percent from benchmarks:

$150\% = 100\% + 50\%$ , which is the whole plus half of the whole.
$0.5\%$ is half of $1\%$ , because $0.5$ is half of $1$ .

Both ways lead to the same part, because they describe the same percent in two forms. Solve with one way and check with the other.

Notice two rows that look alike but are far apart: $0.5\%$ of $200$ is $1$ , while $50\%$ of $200$ is $100$ .

Worked example

A percent greater than 100

A school sets a goal of collecting $240$ cans for a food drive. The students collect $125\%$ of the goal. How many cans do they collect?

Name the whole. The goal of 240 cans is the whole, so it counts as 100%. The answer will be in cans. Because 125% is more than 100% and the whole is positive, expect more than 240 cans.
Split 125% into two benchmarks you know. This is the same percent written as a sum, so it describes the same amount.
$125\% = 100\% + 25\%$
Find each piece. The 100% piece is all 240 cans. The 25% piece is one fourth of 240, because 25% is 25 hundredths, which equals one fourth. So divide 240 by 4. You divide by 4, which is not zero. The result 60 is exact (not rounded), so multiplying back gives the starting number: 60 times 4 is 240.
$100\% \text{ of } 240 = 240 \qquad 25\% \text{ of } 240 = 240 \div 4 = 60$
Add the two pieces, because together they make 125% of the goal.
$240 + 60 = 300$
Check with one multiplication. Write 125% per 100: 125 divided by 100 is 1.25. Multiplying 240 by 1.25 means taking 1 times 240 plus 0.25 times 240, so it must agree with the benchmark method.
$1.25 \times 240 = (1 \times 240) + (0.25 \times 240) = 240 + 60 = 300$

Answer

The students collect $300$ cans. That is $125\%$ of the $240$ -can goal, and it is more than the goal, as predicted.

Worked example

A percent less than 1

A tank holds $600$ liters of water. Overnight, $0.5\%$ of the water leaks out. How many liters leak out?

Name the whole. The amount of water in the tank, 600 liters, is the whole, so it counts as 100%. The answer will be in liters. Because 0.5% is between 0% and 1% and the whole is positive, expect less than 1% of 600 liters.
Find 1% first. Since 1% is one hundredth, divide the whole by 100. You divide by 100, which is not zero. The result 6 is exact, so multiplying back gives the starting number: 6 times 100 is 600.
$1\% \text{ of } 600 = 600 \div 100 = 6$
Use the fact that 0.5 is half of 1. So 0.5% is half of 1%, and you need half of 6 liters. You divide by 2, which is not zero. The result 3 is exact: 3 times 2 is 6.
$0.5\% \text{ of } 600 = 6 \div 2 = 3$
Check with one multiplication. Write 0.5% per 100: 0.5 divided by 100 is 0.005, or 5 thousandths. Multiplying 600 by 5 thousandths means multiplying by 5 and dividing by 1000. It must agree with the benchmark method.
$0.005 \times 600 = \frac{5 \times 600}{1000} = \frac{3000}{1000} = 3$

Answer

The tank loses $3$ liters. That is $0.5\%$ of the $600$ liters, and it is less than the $6$ liters that make up $1\%$ , as predicted.

Now turn the question around. Sometimes you know the part and the whole, and you want the percent. You already know how: divide the part by the whole, then write the result as a percent.

Three names make this easier to talk about. In $600 \div 100 = 6$ , the dividend is $600$ , the divisor is $100$ , and the quotient is $6$ . When you find a percent, the part is the dividend and the whole is the divisor. A divisor must never be zero, so the whole cannot be zero.

Nothing about the method changes for unusual percents. Only the size of the quotient changes. For a positive part and a positive whole:

If the part is greater than the whole, the quotient is greater than $1$ , so the percent is greater than $100\%$ .
If the part is less than one hundredth of the whole, the quotient is less than $0.01$ , so the percent is less than $1\%$ .

Make this prediction before you calculate. Then you will know right away whether your answer is the right size.

Worked example

Finding a large percent and a tiny percent

A class sets a goal of recycling $250$ kilograms of paper. By the end of the month, the class has recycled $400$ kilograms. On the first day, it recycled only $2$ kilograms. What percent of the goal is each amount?

Name the whole and predict. The goal of 250 kilograms is the whole, so it counts as 100%. It is also the divisor, and it is not zero. The month's 400 is greater than 250, so expect more than 100%. For the first day, 1% of 250 is 2.5, and 2 is less than 2.5, so expect less than 1%.
$1\% \text{ of } 250 = 250 \div 100 = 2.5$
Divide the month's amount by the whole. Write the division as a fraction, then multiply the top and the bottom by 4 so the bottom becomes 1000. The value stays the same, because 4 over 4 equals 1.
$\frac{400}{250} = \frac{400 \times 4}{250 \times 4} = \frac{1600}{1000} = 1.6$
Write the quotient as a percent. Percent means per 100, so rewrite 1.6 with 100 on the bottom. Multiply the top and the bottom by 100, which keeps the value the same because 100 over 100 equals 1.
$1.6 = \frac{1.6 \times 100}{1 \times 100} = \frac{160}{100} = 160\%$
Divide the first day's amount by the same whole. Use the same move: multiply the top and the bottom by 4.
$\frac{2}{250} = \frac{2 \times 4}{250 \times 4} = \frac{8}{1000} = 0.008$
Write this quotient as a percent the same way. The number over 100 is 0.8, which is less than 1, so the percent is less than 1%.
$0.008 = \frac{0.008 \times 100}{1 \times 100} = \frac{0.8}{100} = 0.8\%$
Check. Both quotients are exact and the divisor, 250, is not zero. So each quotient times 250 must equal its dividend.
$1.6 \times 250 = 400 \qquad 0.008 \times 250 = 2$

Answer

The month's total of $400$ kilograms is $160\%$ of the goal, and the first day's $2$ kilograms is $0.8\%$ of the goal. Both match the predictions: more than $100\%$ and less than $1\%$ .

Takeaway. The whole is the amount that counts as $100\%$ , and $1\%$ is one hundredth of it. For a positive whole, a percent greater than $100\%$ gives a part that is more than the whole, and a percent between $0\%$ and $1\%$ gives a part that is less than one hundredth of the whole. The percent sign always means per 100, even when the number in front of it is a decimal, so $160\% = 1.6$ and $0.8\% = 0.008$ . Predict the size of the answer before you calculate. When you divide, the divisor is not zero, and an exact quotient times the divisor gives back the dividend.