Model and Calculate a Percent of a Number | Grade 6 Math | LevelKite

Percent means per hundred. So $30\%$ is the same as $\frac{30}{100}$ or $0.30$ . You already know how to move between these forms.

In this lesson you find a percent of a number. The number you start with is the whole. The whole is $100\%$ , because it is all $100$ hundredths of itself.

So $30\%$ of $50$ means $30$ hundredths of $50$ . Here the whole is $50$ . Your job is to find how much $30$ hundredths of it is.

A bar model is a picture of this idea. One bar stands for the whole, so the full bar is $100\%$ . When you cut the bar into equal parts, each part gets an equal share of the $100\%$ and an equal share of the number.

In the bar below, the full bar stands for $50$ . It has $10$ equal parts, and $3$ of them are shaded. The shaded parts show $30\%$ of $50$ . The worked example after the bar shows why, and finds the amount.

The full bar stands for

50

, which is

100\%

. Three of its ten equal parts are shaded.

Worked example

Find 30% of 50 with a bar model

Use the bar model to find $30\%$ of $50$ .

The full bar is 100%, and it has 10 equal parts. Share the 100% equally among the 10 parts to find the percent that one part shows.
$100\% \div 10 = 10\%$
The full bar also stands for the number 50. The parts are equal, so each part holds an equal share of 50. Divide 50 by 10 to find the amount in one part.
$50 \div 10 = 5$
Find how many parts make 30%. Each part is 10%, so 3 parts make 30%. These are the 3 shaded parts in the bar.
$3 \times 10\% = 30\%$
Each shaded part holds 5. Multiply by the 3 shaded parts to find the amount they hold together.
$3 \times 5 = 15$

Answer

$30\%$ of $50$ is $15$ .

Look back at what you did with the bar. You divided $50$ by $10$ , then multiplied by $3$ . That is the same as multiplying $50$ by the fraction $\frac{3}{10}$ :

$\frac{3}{10} \times 50 = \frac{150}{10} = 15$

It works because $3$ of the $10$ equal parts is $\frac{3}{10}$ of the bar, and finding a fraction of a number means multiplying by that fraction.

Now connect this to the percent. The fraction $\frac{3}{10}$ is equivalent to $\frac{30}{100}$ , which is $30\%$ , and both equal $0.30$ . All of these forms name the same share of the whole, so multiplying by any of them gives the same result. This means you do not always need a drawing. You can write the percent as a fraction or a decimal and multiply.

Key concept

To find a percent of a number, multiply

A percent of a number is that many hundredths of the number. To find it, write the percent as a fraction over $100$ or as a decimal. Then multiply by the number.

In the formula, $n$ is the number you start with, which is the whole, or $100\%$ . The letter $p$ is the number in front of the percent sign. For $30\%$ of $50$ , $p = 30$ and $n = 50$ , so you get $\frac{30}{100} \times 50 = \frac{1500}{100} = 15$ . This is the same amount the bar model gave.

p\% \text{ of } n = \frac{p}{100} \times n

Worked example

Find 35% of 80 by multiplying

Find $35\%$ of $80$ without a model.

Name the whole and the percent. The whole is 80, and you want 35 hundredths of it. Write 35% as a fraction over 100, because percent means per hundred.
$35\% = \frac{35}{100}$
Multiply the fraction by the whole. Multiply 35 by 80 on top and keep 100 on the bottom.
$\frac{35}{100} \times 80 = \frac{35 \times 80}{100} = \frac{2800}{100}$
Divide 2800 by 100 to write the result as one number.
$\frac{2800}{100} = 28$
Check with the decimal form. 35/100 equals 0.35, which names the same share of the whole, so the result must match. 35 times 80 is 2800, and 0.35 has two decimal places, so the result is 28.00, or 28.
$0.35 \times 80 = 28.00 = 28$

Answer

$35\%$ of $80$ is $28$ .

A percent can be greater than $100\%$ . For example, $150\%$ means $150$ hundredths. That is all of the whole and $50\%$ more, so one bar is not enough to show it.

To model $150\%$ of $60$ , use two bars. The two bars are the same size, and each one stands for the same whole, $60$ . So each bar is $100\%$ .

This time each bar has $4$ equal parts instead of $10$ . A model does not need $10$ parts every time. Pick a number of equal parts that lets you shade the percent with whole parts. In the figure below, $6$ parts are shaded: all $4$ parts of the first bar and $2$ parts of the second bar.

Two same-sized bars, each standing for the same whole,

60

. Six parts are shaded: one full bar and two parts of the next.

Worked example

Find 150% of 60 with two bars

Find $150\%$ of $60$ . Use the two-bar model, then check by multiplying.

Each bar is the same size and stands for the same whole, 60, so each bar is 100%. One bar has 4 equal parts. Share the 100% equally among the 4 parts.
$100\% \div 4 = 25\%$
Share the whole, 60, among the 4 parts of one bar. Both bars stand for 60, so every part in both bars holds this amount.
$60 \div 4 = 15$
Find how many 25% parts make 150%. Six parts do. They are the 6 shaded parts: one full bar and 2 parts of the second bar.
$6 \times 25\% = 150\%$
Multiply the amount in one part by the 6 shaded parts.
$6 \times 15 = 90$
Check by multiplying. 150% means 150 per hundred, so write it as 150 over 100 and multiply by the whole, 60.
$\frac{150}{100} \times 60 = \frac{9000}{100} = 90$
Compare the result with the whole. The whole, 60, is positive, and you multiplied it by 150/100, which is 1.5, a number greater than 1. So the result should be larger than 60, and it is.
$90 > 60$

Answer

$150\%$ of $60$ is $90$ .

The big idea: a percent of a number is that many hundredths of the number. In a bar model, each same-sized bar is the whole, or $100\%$ . Equal parts split the percent and the number in the same way, so you can count parts to find the amount. Multiplying gives the same amount: write the percent as a fraction over $100$ or as a decimal, then multiply by the number.

Next, you will estimate percents of numbers and work with percents that include a fraction or a decimal.