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2. Fractions, Decimals, & Percents
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Grade 6

2.01 Percents as fractions

Lesson
Practice

Percents as fractions

Percent means "one part out of every 100, “per 100” or how many “out of 100". In other words, 1\% is equal to one hundredth or \dfrac{1}{100}.

An image of 10 by 10 grid with 6 squares shaded.

Percents can be represented with a 10 \times 10 grid which has 100 total squares. Each square represents 1 \% or 1 square out of 100 total squares.

\text{percent of squares} = \text{number of shaded squares} \%

So 6 shaded squares represents 6 \% of the grid.

\text{fraction of squares}=\dfrac{\text{shaded squares}}{\text{total squares}}

So, the six shaded blue squares represent \dfrac{6}{100}.

The image shows with 2 rectangles. 1st rectangle divided into 4 equal parts and 1 part is shaded. 2nd rectangle is divided into 3 parts. Ask your teacher for more information.

Percents can represent amounts that are less than a whole or greater than a whole, with 100 \% representing one whole.

  • Percents less than a whole will be smaller than 100 \% and represent part of a whole. They can also be represented by fractions less than 1.

  • Percents greater than a whole will be larger than 100 \% and represent a quantitiy greater than the original. They can also be represented by fractions greater than 1.

Estimating percents can be helpful for quick calculations or when precise measurements are not necessary. When estimating a percent, it is helpful to be comfortable with calculating the benchmark percents of 0 \% \text{, } 25 \% \text{, } 50 \%, and 100 \%.

The image shows 4 rectangles. Showing fraction-decimal equivalence. Ask your teacher for more information.

25 \% = \dfrac{1}{4}

50 \% = \dfrac{1}{2}

75 \% = \dfrac{3}{4}

100 \% = 1

Double number lines can be used with benchmark percents to represent both percents less than 100 \% and greater than 100 \%. On this number line, 100 \% of the quantity is 24 minutes. And we can easily see other percents by looking at the values that line up.

A double numberline that shows time in minutes and its equivalent in percentage. Ask your teacher for more information.

Benchmark percents and their fraction equivalents will be helpful in converting between percents and fractions as well as estimating percents.

To convert any percent to a fraction, remember percent means "per 100". Create a fraction with the percent quantiity as the numerator and 100 as the denominator. This fraction may be reduced.

25 \% = \dfrac{25}{100} = \dfrac{1}{4}

To convert any fraction to a percent, there are two methods.

For the first method, create an equivalent fraction with 100 as the denominator. The numerator is the percent.

\dfrac{6}{4} = \dfrac{6 \cdot 25}{4 \cdot 25} = \dfrac{150}{100} = 150 \%

For the second method, divide the numerator by the denominator, and then multiply by 100\%.

\dfrac{6}{4} = 1.5

1.5 \cdot 100 \% = 150 \%

Examples

Example 1

Consider the grid below.

A 10 by 10 square grid with 50 squares shaded.
a

How many squares are shaded?

Worked Solution
Create a strategy

Count the shaded squares in the row and the number of rows.

Apply the idea

There are 5 rows of 10 squares shaded so there are 50 squares shaded.

5\cdot 10 = 50

b

What percent of the grid is shaded?

Worked Solution
Create a strategy

Find the fraction shaded then convert to a percent.

Apply the idea

We have 50 out of 100\ squares in the grid shaded. We can express this as fraction \dfrac{50}{100}.

Recall that percent means parts per hundred.

This means that \dfrac{50}{100}= 50\%

c

What fraction of the grid does this percent represent?

A
One quarter
B
One tenth
C
One half
D
One fifth
Worked Solution
Create a strategy

Observe the size of the shaded part of the grid model.

We can also find an equivalent fraction in (b) with a numerator of 1 (one).

Apply the idea

There are 50 shaded squares out of 100 squares. This can be written as \dfrac{50}{100} as a fraction. Looking at the model we can see that the shaded part is half the grid.

The numerator and denominator of \dfrac{50}{100} can also be divided by 50 to find an equivalent fraction with one in the numerator.

\displaystyle \dfrac{50 \div 50}{100\div 50}\displaystyle =\displaystyle \dfrac{1}{2}Divide by 50 to find equivalent fraction

The correct option is C.

Example 2

Use this model where one circle represents one whole:

5 circles all divided into 5 parts. Total of 22 parts shaded.
a

Write the fraction that represents the model.

Worked Solution
Create a strategy

Count the total parts in each circle of the model. This will represent one whole. Then count the number of shaded parts to form a fraction.

Apply the idea

Since each of the circles consists of 5 parts, 5 parts represents a whole and 5 should be the denominator of our fraction.

Counting the shaded parts, 22 parts are shaded, and 22 should be the numerator of our fraction. So, the fraction representing the model is \dfrac{22}{5} which can be reduced to 4 \dfrac{2}{5}.

Reflect and check

We could have also notice that 4 whole circles are filled in. So, we have 4 wholes and then whatever fraction is represented by the fifth circle that is partially shaded.

The fifth circle has 2 out of 5 parts shaded, or \dfrac{2}{5} of the circle. Therefore, the fraction represented by the model is 4 \dfrac{2}{5}.

b

Estimate the percent represented.

Worked Solution
Create a strategy

Convert the fraction to an approximate percent by considering the benchmark percent that is closest to our fraction.

Apply the idea

Given that our fraction that best represented the model was 4 \dfrac{2}{5}, we know the 4 whole can be written as 400 \%.

Visually looking at the last circle representing \dfrac{2}{5}, this is larger than 25 \% = \dfrac{1}{4} and smaller than 50 \% = \dfrac{1}{2}. Since the shaded part is near the middle of the two benchmarks, we can estimate this as about 38\%.

Therefore, the shaded portion is approximately 438 \%.

Reflect and check

When estimating, some variation is expected. Any estimates between 35 \% and 45 \% would be appropriate answers.

c

What exact percent of one whole circle is shaded?

Worked Solution
Create a strategy

Calculate the exact percent by dividing the number of shaded parts by the total parts in one whole circle and multiplying by 100 \%.

Apply the idea

Number of shaded parts: 22

Total parts: 25

\displaystyle 22 \div 5\displaystyle =\displaystyle 4.4Divide the number of shaded parts by the total parts in one circle
\displaystyle =\displaystyle 4.4 \cdot 100 \%Multiply 4.4 by 100 \%
\displaystyle =\displaystyle 440\%Evaluate
Reflect and check

Our exact percent is very close to our approximation. Consider ways to make an even better estimate.

Example 3

Write 24\% as a fraction.

Worked Solution
Create a strategy

To convert the percent into fraction, rewrite the percent as a fraction out of 100.

Apply the idea
\displaystyle 24\%\displaystyle =\displaystyle \dfrac{24 }{100 }Write the percent as a numerator over the denominator 100
Reflect and check

To simplify the fraction \dfrac{24}{100}, we find the greatest common divisor (GCD) of 24 and 100, which is 4. Dividing both the numerator and the denominator by 4, we can simplify our fraction:

\dfrac{24}{100} = \dfrac{24 \div 4}{100 \div 4} = \dfrac{6}{25}

A 10 by 10 square grid with 24 squares shaded.

We can model these fractions on a grid where the denominator is the total number of squares.

On a 10 by 10 grid representing 100\%, shading 24 squares shows the original percentage.

A 5 by 5 square grid with 6 squares shaded.

We can create a 5 by 5 grid to represent the simplified fraction \dfrac{6}{25}.

Each square in this grid represents 4 squares from the 10 \times 10 grid. Therefore, shading 6 squares corresponds to the same proportion of the whole.

This visualization helps confirm the equivalence of 24\% to \dfrac{6}{25}. Fractions can be simplified while representing the same value or percent of the whole.

Example 4

Write \dfrac{2}{5} as a percent.

Worked Solution
Create a strategy

We need to find an equivalent fraction with the denominator equal to 100. What number can we multiply 5 by to get 100? Then be sure to multiply the numerator and denominator by the same factor.

Apply the idea
\displaystyle \dfrac{2}{5} \displaystyle =\displaystyle ⬚What can we multiply 5 by to get a denominator of 100?
\displaystyle =\displaystyle \dfrac{2\cdot 20}{5\cdot 20}Multiply both the numerator and the denominator by 20
\displaystyle =\displaystyle \dfrac{40 }{100 }Evaluate
\displaystyle =\displaystyle 40\%40 for every 100 is 40\%
Reflect and check

When we are unsure how to get a denominator of 100 directly, we can divide 100 by the current denominator to find the multiplying factor.

100 \div 5 = 20

So, both numerator and denominator should be multiplied by 20 to create an equivalent fraction with a denominator of 100. This method is especially helpful for converting fractions where the denominator is not a simple factor of 100.

Example 5

Write 250\% as a mixed number in its simplest form.

Worked Solution
Create a strategy

To write a percent as a fraction divide by 100. Then convert the improper fraction into a mixed number.

Apply the idea
\displaystyle 250\%\displaystyle =\displaystyle \dfrac{250 }{100 }Divide by 100
\displaystyle =\displaystyle \dfrac{5 }{2 }Simplify the fraction
\displaystyle =\displaystyle 2\dfrac{1 }{2 }Convert to a mixed number
Reflect and check

A double number line is a powerful tool for understanding the conversion between percents and other amounts. Let's draw a double number line to represent this relationship:

A double number line: the top number line with marks from 1 to 3, and the bottom number line with marks 100% to 300%. Ask your teacher for more information.

The top number line represents the mixed number amounts. The bottom line of the double number line represents those amounts as percents. This model clearly shows that 250\% aligns with 2\dfrac{1}{2}, which matches our calculation.

Idea summary

Percent means “per 100” or how many “out of 100". 1\% is equal to one hundredth. 1\% = \dfrac{1}{100} = 0.01

We can convert any percent into a fraction by writing the percent value as the numerator and 100 as the denominator.

We can convert any fraction into a percent by finding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the \% symbol to represent the percent.

Outcomes

6.NS.1

The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

6.NS.1a

Estimate and determine the percent represented by a given model (e.g., number line, picture, verbal description), including percents greater than 100% and less than 1%.*

6.NS.1c

Represent and determine equivalencies among fractions (proper or improper) and mixed numbers that have denominators that are 12 or less or factors of 100 and percents incorporating the use of number lines, and concrete and pictorial models.*

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