Converting fractions, decimals and percentages

Here you will learn about comparing fractions, decimals, and percentages, including how to define a fraction, a decimal, and a percentage, convert between them, and compare and order them.

Students first learn about comparing fractions and decimals in the fifth grade and expand this knowledge to percents in the sixth and seventh grades when working with ratio and proportional relationships and the number system.

What are fractions, decimals, and percentages?

Fractions, decimals, and percentages are different ways of expressing the same value.

• Fractions

Fractions equal parts of a whole. 

The denominator of a fraction (number below the line) shows how many equal parts the whole has been divided into. The numerator of a fraction (number above the line) shows how many of the equal parts we have.

3 parts shaded out of 5 equal parts.

Read as three-fifths

Step-by-step guide: Fractions

• Decimals

Decimals are numbers that have a whole part and a fractional part that are separated by a decimal point. Our decimal system splits whole numbers into tenths, hundredths, thousandths, and so on.

0.15 is represented on the place value chart.

Read as fifteen-hundredths.

Step-by-step guide: Decimals

• Percentages

Percentages are numbers which are expressed as a fraction of 100. Percent means “number of parts per hundred” and the symbol used for percent is, \% (percent sign).

42 shaded parts out of 100 equal parts is \, \cfrac{42}{100} \, which is 42\% .

Step-by-step guide: Percents

Let’s look at how you can express twenty-hundredths, \, \cfrac{20}{100} \, , as a fraction, decimal and percent.

FRACTION	DECIMAL	PERCENT
As a fraction: \cfrac{20}{100}=\cfrac{20\div 20}{100\div 20}= \cfrac{1}{5} 20 shaded pieces out of 100 equal pieces. OR 1 shaded piece out of 5 equal pieces.	As a decimal: \cfrac{20}{100}=0.20 20 shaded pieces where each piece represents 0.01.	As a percent: \cfrac{20}{100}=20 \% 20 shaded pieces out of 100 pieces.

What is converting fractions, decimals, and percentages?

In order to compare fractions, decimals and percentages, you will need to know how to convert them to the same form.

Converting Fractions, Decimals, and Percents \hspace{1.2cm}

to Fractions with a Denominator of \textbf{100} \hspace{1.2cm}

Step-by-step guide: Decimal to a fraction

Step-by-step guide: Percent to a fraction

Converting Fractions & Decimals \hspace{1.2cm}

to Percents \hspace{1.2cm}

Step-by-step guide: Decimal to a percent

Step-by-step guide: Fraction to a percent

Step-by-step guide: Powers of 10

Converting Fractions and Percents \hspace{1.2cm}

to Decimals \hspace{1.2cm}

Fractions (Use long division)	Decimals (Change to a fraction and then decimal in the hundredths place)
	50\%=\cfrac{50}{100}=0.50=0.5 Notice how 50 becomes 0.50 which is the same as dividing 50 by 100
	75\%=\cfrac{75}{100}=0.75 Notice how 75 becomes 0.75 which is the same as dividing 75 by 100
	33\%=\cfrac{33}{100} =0.33 Notice how 33 becomes 0.33 which is the same as dividing 33 by 100

Step-by-step guide: Fraction to a decimal

Step-by-step guide: Percent to a decimal

Comparing fractions, decimals, and percentages

Now that you know how to convert fractions, decimals, and percentages to the same form, let’s compare them.

For example, list the following numbers in order from least to greatest.

\cfrac{1}{4} \hspace{0.5cm} 0.19 \hspace{0.5cm} 0.3 \hspace{0.5cm} 26 \% \hspace{0.5cm} \cfrac{1}{5}

Before comparing the numbers, convert them all to fractions with a denominator of 100.

\cfrac{1}{4}=\cfrac{1 \, \times \, 25}{4 \, \times \, 25}=\cfrac{25}{100}

0.19=\cfrac{19}{100}

0.3=\cfrac{3}{10}=\cfrac{3 \, \times \, 10}{10 \, \times \, 10}=\cfrac{30}{100}

26 \%=\cfrac{26}{100}

\cfrac{1}{5}=\cfrac{1 \, \times \, 20}{5 \, \times \, 20}=\cfrac{20}{100}

The numbers in order from least to greatest:

0.19=\cfrac{19}{100} \, , \; \cfrac{1}{5}=\cfrac{20}{100} \, , \; \cfrac{1}{4}=\cfrac{25}{100} \, , \; 26 \%=\cfrac{26}{100} \, , \; 0.3=\cfrac{30}{100}

0.19, \; \cfrac{1}{5} \, , \; \cfrac{1}{4} \, , \; 26 \%, \; 0.3

Another way you could have done the comparison is to convert all the numbers to percents.

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

0.19 \, → multiply by 100 to change a decimal to a percent.

0.19 \times 100=19. This is the same as moving the decimal point 2 decimal places to the right.

0.19=19 \%

0.3 \, → multiplied by 100 changes a decimal to a percent.

0.3 \times 100=30. This is the same as moving the decimal point 2 decimal places to the right.

0.3=30 \%

\cfrac{1}{5}=\cfrac{20}{100}=20 \%

The numbers from least to greatest is:

0.19=19 \%, \; \cfrac{1}{5}=20 \% \, , \; \cfrac{1}{4}=25 \%, \; 26 \%, \; 0.3=30 \%

0.19, \; \cfrac{1}{5} \, , \; \cfrac{1}{4} \, , \; 26 \%, \; 0.3

What are fractions, decimals and percents?

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5 – Number and Operations – Base Ten (5.NBT.A.3.a)
  Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, for example,
  347.392 = 3 \times 100 + 4 \times 10 + 7 \times 1 + 3 \times \bigl(\cfrac{1}{10}\bigr) + 9 \times \bigl(\cfrac{1}{100}\bigr) + 2 \times \bigl(\cfrac{1}{1000}\bigr).

• Grade 6 – Ratios and Proportional Relationships (6.RP.A.3.c)
  Find a percent of a quantity as a rate per 100 (for example, 30\% of a quantity means \; \cfrac{30}{100} \; times the quantity); solve problems involving finding the whole, given a part and the percent.

• Grade 7 – The Number System (7.NS.2d)
  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

How do you convert fractions, decimals, and percentages?

In order to compare fractions, decimals, and percentages, convert them to the same form.

1. Choose a form (fraction, decimal, percentage).
2. Convert the numbers to the form you chose.
3. Write the numbers in the new form. If you’re comparing the numbers, go to Step 4.
4. Write the numbers in the correct order.

Converting fractions, decimals, and percentages examples

Example 1: convert fraction, decimal, and percentage to the same form

Convert the numbers below to fractions with a denominator of 100.

\cfrac{3}{4} \, , \; 75 \%, \; 0.75

1. Choose a form (fraction, decimal, percentage).

Convert the numbers to fractions.

2Convert the numbers to the form you chose.

Since you are converting the numbers to fractions, use equivalent fractions to convert the denominator of the fraction to 100.

\cfrac{3}{4}= \cfrac{3 \, \times \, 25}{4 \, \times \, 25}= \cfrac{75}{100}

\cfrac{3}{4}= \cfrac{75}{100}

Write the decimal as a fraction.

0.75 is seventy-five hundredths, which is \, \cfrac{75}{100} .

Write the percent as a fraction.

75 \% is \, \cfrac{75}{100} \, .

3Write the numbers in the new form. If you’re comparing the numbers, go to Step 4.

\cfrac{3}{4}= \cfrac{75}{100}

0.75= \cfrac{75}{100}

75 \% =\cfrac{75}{100}

You are not comparing the numbers, so you do not have to go to Step 4.

Example 2: convert fraction, decimal and percentage to a decimal

Convert the numbers below to decimals.

\cfrac{2}{5} \, , \; 40 \%

Convert the numbers to the form you chose.

Rewrite the fraction as a division problem, then use long division to divide the numerator by the denominator.

2 \div 5

2 \div 5=0.4

Divide the percent number by 100.

40 \% =\cfrac{40}{100} = {40}\div 100

Recall from dividing by powers of 10, you can simply move the decimal point two places to the left when dividing by 100.

Example 3: convert fraction, decimal, and percentage to decimals

Convert the numbers below to decimals.

\cfrac{2}{9} \, , \; 67 \%

Convert the numbers to the form you chose.

Divide the numerator by the denominator using long division \, \cfrac{2}{9} = 2\div 9

2 \div 9=0 . \overline{2}

Make the percent a decimal by dividing the percent by 100 which is the same as moving the decimal point two places to the left.

67\%=\cfrac{67}{100} = 67\div 100

Using the rules of dividing by powers of ten 67 \div 100=0.67

Example 4: compare fractions, decimals, and percentage by converting them to fractions

Write the numbers in order from least to greatest.

\cfrac{1}{2} \, , \; 0.24,33 \%, \; \cfrac{7}{20}

Example 5: compare fractions, decimals, and percentage by changing them all to percentages

Write the numbers in order from greatest to least.

45\% , \; 0.32, \; \cfrac{1}{4} \, , \; \cfrac{2}{5} \, , \; 0.18

Example 6: compare fractions, decimals, and percentage by changing them all to decimals

Write the numbers in order from least to greatest.

78 \%, \; 0.9, \; \cfrac{2}{3} \, , \; 61 \%

Convert the numbers to the form you chose.

Divide the numerator by the denominator using long division, \, \cfrac{2}{3} \, is 2 \div 3

2 \div 3=0 . \overline{6} which is a repeating decimal.

0 . \overline{6}=0.66666\ldots

0.66666\ldots Rounded to the nearest hundredth is 0.67

To change the percents to decimals, divide the percents by 100 or move the decimal point two places to the left.

78 \%=\cfrac{78}{100}=78 \div 100

Using the rules of dividing by powers of ten, 78 \div 100 = 0.78

61 \%=\cfrac{61}{100}=61 \div 100

Using the rules of dividing by powers of ten, 61 \div 100 = 0.61

Teaching tips for comparing fractions, decimals, and percentages

• Use visual models so students can see the connections between fractions, decimals, and percentages.

• When first comparing fractions, decimals, and percentages, start by first converting the numbers into either fractions with a denominator of 100 or percentages. It is usually the easiest way for students to compare the numbers.

• Long division is useful when converting fractions and percentages to decimals, be sure to review the process with students.

• Fraction worksheets, decimal worksheets, and percent worksheets have their place, but using strategies to develop conceptual understanding is more meaningful to your learners.

Easy mistakes to make

• Not remembering the definition of a percent
  Not remembering the definition of a percent which is a fraction of 100. All percents can be written as a fraction over 100, which is useful when comparing fractions, decimals, and percents.

• Not remembering to make the denominator \bf{100} when comparing fractions, decimals, and percents
  For example, when converting 0.7 to a fraction, leave it as is \, \cfrac{7}{10} instead of changing it to \, \cfrac{70}{100}.

• Writing the answer in the wrong order
  For example, writing the numbers in ascending order means writing them from least to greatest. Writing the numbers in descending order means writing them from greatest to least.

• Original Values
  Not giving the final answer in the value’s original form.
  For example, after converting the numbers to the same form to compare them, you forget to write the final answer using the original numbers and not the converted numbers.

The next lessons are

• Arithmetic
• Properties of equality
• Addition and subtraction
• Percent to fraction
• Decimal to fraction

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