Estimate and Find Fractional and Decimal Percents | Grade 6 Math | LevelKite

A percent tells you how many parts there are for every hundred. So $25\%$ means $25$ for every $100$ , or $\tfrac{25}{100}$ . You already know how to find a percent of a number: $25\%$ of $80$ is $0.25 \times 80 = 20$ . The number you take the percent of is the whole. It stands for $100\%$ . The result is the part.

This lesson adds two skills. First, you will estimate a percent of a number. To estimate is to find a value close to the exact answer by using easier numbers. Second, you will find exact answers when the percent is not a whole number, such as $9.5\%$ or $\tfrac{3}{4}\%$ .

Some percents are easy to work with because they match simple fractions. Take $10\%$ . It means $\tfrac{10}{100}$ . Dividing the top and the bottom by $10$ keeps the value the same, so $\tfrac{10}{100} = \tfrac{1}{10}$ . That makes $10\%$ of a number one tenth of it. You find one tenth by dividing by $10$ .

Easy percents like this are called benchmark percents. A benchmark is a familiar value that you compare other values with. To estimate a percent of a number:

Replace the percent with a nearby benchmark percent.
Replace the whole with a nearby number that is easy to divide.
Find the benchmark percent of that easier number.

Benchmark percents, tried on a whole of 200

Benchmark percent	Same value as a fraction	What to do to the whole	Result when the whole is 200
$1\%$	$\tfrac{1}{100}$	Divide by $100$	$200 \div 100 = 2$
$10\%$	$\tfrac{10}{100} = \tfrac{1}{10}$	Divide by $10$	$200 \div 10 = 20$
$20\%$	$\tfrac{20}{100} = \tfrac{1}{5}$	Divide by $5$	$200 \div 5 = 40$
$25\%$	$\tfrac{25}{100} = \tfrac{1}{4}$	Divide by $4$	$200 \div 4 = 50$
$50\%$	$\tfrac{50}{100} = \tfrac{1}{2}$	Divide by $2$	$200 \div 2 = 100$
$75\%$	$\tfrac{75}{100} = \tfrac{3}{4}$	Divide by $4$ , then multiply by $3$	$200 \div 4 = 50$ , then $50 \times 3 = 150$

Worked example

Estimate with a benchmark percent

Estimate $24\%$ of $83$ .

Replace the percent with a nearby benchmark. 24% is very close to 25%, and 25% equals one fourth.
$24\% \approx 25\% = \tfrac{1}{4}$
Replace the whole with a nearby number that is easy to divide by 4, because finding one fourth means dividing by 4. The number 80 is close to 83, and 80 divides by 4 with no remainder.
$83 \approx 80$
Find 25% of 80 by dividing 80 by 4, because 25% is one fourth. The result is your estimate.
$80 \div 4 = 20$
See how close the estimate is by finding the exact value. Write 24% as the decimal 0.24. Split 83 into 80 and 3 and multiply each piece, because the two products add up to the full product. 0.24 times 80 is 19.2, and 0.24 times 3 is 0.72.
$0.24 \times 83 = 19.2 + 0.72 = 19.92$

Answer

$24\%$ of $83$ is about $20$ . The exact value is $19.92$ , so the estimate is only $0.08$ away.

The number in front of the percent sign does not have to be a whole number. It can be a decimal, as in $9.5\%$ . It can be a fraction, as in $\tfrac{3}{4}\%$ . The meaning stays the same: for every hundred.

$9.5\%$ means $9.5$ for every $100$ . A whole of $200$ is two hundreds, so $9.5\%$ of $200$ is $9.5 + 9.5 = 19$ .
$\tfrac{3}{4}\%$ means three fourths of one for every $100$ . That is less than $1\%$ . For the same whole of $200$ , it is $\tfrac{3}{4} + \tfrac{3}{4} = 1\tfrac{1}{2}$ .

Because the meaning is the same, the method you already know still works. Only the number in front of the percent sign looks different.

Key concept

A percent is a factor that multiplies the whole

A factor is a number you multiply by. Let $p$ be the positive number in front of the percent sign. Let the whole be the number you take the percent of, and let the part be the result.

Percent means for every hundred, so $p\%$ is the factor $\tfrac{p}{100}$ . This is true whether $p$ is a whole number, a decimal or a fraction. To find the part, divide $p$ by $100$ and multiply the result by the whole.

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

Dividing by $100$ makes each digit worth one hundredth as much, so every digit moves two places to the right. That is how $9.5\%$ becomes the factor $0.095$ : $9.5 \div 100 = 0.095$ .

If the percent has a fraction such as a half or a fourth, write the fraction as a decimal first. The fraction $\tfrac{1}{2}$ and the decimal $0.5$ name the same number, so $\tfrac{1}{2}\%$ and $0.5\%$ are the same percent. Then divide by $100$ as before.

From a percent to a factor

Percent	Divide by 100	Factor
$9.5\%$	$9.5 \div 100$	$0.095$
$12.5\%$	$12.5 \div 100$	$0.125$
$0.4\%$	$0.4 \div 100$	$0.004$
$\tfrac{1}{2}\% = 0.5\%$	$0.5 \div 100$	$0.005$
$\tfrac{3}{4}\% = 0.75\%$	$0.75 \div 100$	$0.0075$
$2\tfrac{1}{2}\% = 2.5\%$	$2.5 \div 100$	$0.025$

Worked example

Find a decimal percent of a number

Find $9.5\%$ of $60$ .

Estimate first, so you know what size to expect. 9.5% is close to the benchmark 10%, and 10% of 60 is 60 divided by 10. The answer should be near 6.
$60 \div 10 = 6$
Turn 9.5% into a factor. Percent means for every hundred, so divide 9.5 by 100. Each digit moves two places to the right.
$9.5\% = \tfrac{9.5}{100} = 0.095$
Multiply the factor by the whole, because the part is the factor times the whole. Leave out the decimal point for a moment: 95 times 6 is 570, so 95 times 60 is 5700. The factor 0.095 has three decimal places and 60 has none, so the product needs three decimal places. That gives 5.700, which is 5.7.
$0.095 \times 60 = 5.7$
Check the size. 9.5% is a little less than 10%, and 60 is positive, so the part should be a little less than 6. The result 5.7 fits.
$5.7 < 6$

Answer

$9.5\%$ of $60$ is $5.7$ , which is close to the estimate of $6$ .

Worked example

Find a fractional percent of a number

Find $\tfrac{3}{4}\%$ of $400$ .

Predict the size with a benchmark. 1% of 400 is 400 divided by 100, which is 4. Three fourths of 1% is less than 1%, and 400 is positive, so the answer must be less than 4.
$400 \div 100 = 4$
Write the fraction as a decimal by dividing 3 by 4. Both forms name the same number, so the percent stays the same.
$\tfrac{3}{4}\% = (3 \div 4)\% = 0.75\%$
Divide by 100 to get the factor, because percent means for every hundred. Each digit moves two places to the right.
$0.75\% = \tfrac{0.75}{100} = 0.0075$
Multiply the factor by the whole to get the part. 75 times 4 is 300, so 75 times 400 is 30,000. The factor has four decimal places and 400 has none, so the product is 3.0000, which is 3.
$0.0075 \times 400 = 3$
Check the size and the meaning. 3 is less than 4, as predicted. Also, 400 is 4 hundreds, and three fourths for every hundred makes 4 times three fourths.
$4 \times \tfrac{3}{4} = 3$

Answer

$\tfrac{3}{4}\%$ of $400$ is $3$ .

Takeaway. A percent of a number is a factor times the whole. You get the factor by dividing the number in front of the percent sign by $100$ . This works the same way for whole-number, decimal and fractional percents. Benchmark percents such as $1\%$ , $10\%$ , $25\%$ and $50\%$ give you a quick estimate, and the estimate shows whether an exact answer is a sensible size.

Next, you will use these skills to solve percent word problems and to find mistakes in percent calculations.