Definition

We combine the order operations PEMDAS with adding, subtracting, multiplying, and dividing fractions.

Rules for Order of Operations with Fractions

• First, we simplify any parentheses if any in the expression.
• Next, we simplify any exponents if present in the expression.
• We do multiplication and division before addition and subtraction.
• We do multiplication and division based on order of appearance from left to right in the problem.
• Next, we do addition and subtraction based on order of appearance from left to right in the problem.

Consider the following problems involving PEMDAS with adding, subtracting, multiplying, and dividing fractions.

Example 1

Evaluate 45[17−32(14)2]

Solution

Step 1:

As per the PEMDAS rule of operations on fractions we simplify the brackets or the parentheses first.

Step 2:

Within the brackets, the first we simplify the exponent as (14)2=116

Step 3:

Within the brackets, next we multiply as follows

17−32(14)2=17−32×116=17−2

Step 4:

Within the brackets, next we subtract as follows

17 – 2 So, [17−32(14)2]=15

Step 5:

45[17−32(14)2]=45[15]=45×15

So, simplifying we get

45×15=4×3=12

Step 6:

So, finally 45[17−32(14)2]=12

Example 2

Evaluate (367−117)×85−97

Solution

Step 1:

As per the PEMDAS rule of operations on fractions we simplify the brackets or the parentheses first.

Within the brackets, the first we subtract the fractions as follows

Step 2:

Next, we multiply as follows

(367−117)×85−97=257×85−97=407−97

Step 3:

We then subtract as follows

407−97=(40−9)7=317

Step 4:

So, finally (367−117)×85−97=317=437

Definition

Rules to convert a decimal to a mixed number and an improper fraction in simplest form.

• We read the decimal as whole number part, tenths, hundredths and so on and write it as a mixed number.
• Then we simplify the proper fraction of the mixed number and write it in lowest terms.
• Using algorithm, we convert the mixed number to an improper fraction.

Example 1

Convert 6.8 to a mixed number and an improper fraction in simplest form.

Solution

Step 1:

The decimal 6.8 is read as 6 and 8 tenths.

So, it can be written as a mixed number 6810.

Step 2:

The mixed number has a whole number part 6 and a fractional part 8/10 which can be reduced to lowest terms as 45. So, 6810=45.

Step 3:

The same mixed number can be converted into an improper fraction as follows. The denominator 5 is multiplied with whole number 4 and the product is added to the numerator 4 to give 6 × 5 + 4 = 34.

Step 4:

This becomes the numerator of the improper fraction and 5 is retained as the denominator of the improper fraction. We get 345

So, 6.8=645=345 in simplest form

Example 2

Convert 15.25 to a mixed number and an improper fraction in simplest form

Solution

Step 1:

The decimal 15.25 is read as 15 and 25 hundredths. So, it is written as a mixed number 1525100.

Step 2:

The mixed number has a whole number part 15 and a fractional part 25100 which is reduced to simplest form as 14. So, 1525100=1514.

Step 3:

The same mixed number can be converted into an improper fraction as follows. The denominator 4 is multiplied with whole number 15 and the product is added to the numerator 1 to give 15 × 4 + 1 = 61.

Step 4:

This becomes the numerator of the improper fraction and 4 is retained as the denominator of the improper fraction. We get 614

Step 5:

So, 1525100=1514=614 in simplest form

Definition

To convert a decimal into a mixed number and an improper fraction without simplifying.

We can follow two methods.

Method 1

• We read the decimal as whole number part, tenths, hundredths and so on and write it as a mixed number.
• Using algorithm, we convert the mixed number to an improper fraction.

We read the decimal and write it as a mixed number and an improper fraction as shown in this example.

Convert 13.06 into a mixed number and an improper fraction without simplifying.

• 13.06 can be read as thirteen and six hundredths.
• This can be written as mixed number 136100 without simplifying
• This can be further converted into improper fraction as follows. [13×100+6]100=1306100.
• So, 136100=1306100, an improper fraction. withoutsimplifying

Method 2

• Here, we drop the decimal point and write the number as the numerator of a fraction.
• We count the number of digits after the decimal point.
• Next, we write 1 followed by that many zeros as the denominator of same fraction.
• We get an improper fraction by writing both the numerator and denominator.
• This improper fraction can be converted into a mixed number by long division. withoutsimplifying

Consider the same decimal number 13.06.

• Here, we drop the decimal point and write the number 1306 as the numerator of a fraction.
• We count the number of digits after the decimal point. Here, it is 2 digits.
• Next, we write 1 followed by that many zeros as the denominator of same fraction.
• So, we get 13.06=1306100
• This improper fraction can be converted into a mixed number by long division as 1306100=136100 withoutsimplifying

Example 1

Convert 4.6 to a mixed number and improper fraction without simplifying.

Solution

Step 1:

The decimal 4.6 is read as 4 and 6 tenths.

Step 2:

So, it can be written as a mixed number 4610. The mixed number has a whole number part 4 and a fractional part 610 which can be reduced to lowest terms.

Step 3:

The same mixed number can be converted into an improper fraction as follows. The denominator 10 is multiplied with whole number 4 and the product is added to the numerator 6 to give 4 × 10 + 6 = 46.

Step 4:

This becomes the numerator of the improper fraction and 10 is retained as the denominator of the improper fraction.

Step 5:

So, 4.6=4610=4610 withoutsimplifying

Example 2

Convert 11.75 to a mixed number and improper fraction without simplifying

Solution

Step 1:

The decimal 11.75 is read as 11 and 75 hundredths.

Step 2:

So, it is written as a mixed number 1175100. The mixed number has a whole number part 11 and a fractional part 75100.

Step 3:

The same mixed number can be converted into an improper fraction as follows. The denominator 100 is multiplied with whole number 11 and the product is added to the numerator 75 to give 11 × 100 + 75 = 1175.

Step 4:

This becomes the numerator of the improper fraction and 100 is retained as the denominator of the improper fraction.

Step 5:

So, 1175100=1175100 withoutsimplifying.

Definition

Recall that a proper fraction is a fraction where the numerator is smaller than the denominator;

For example, 35.

Here numerator 3 is smaller than denominator 5.

Rules to convert a decimal to a proper fraction without simplifying.

• We drop the decimal point and write the number as the numerator of a fraction.
• We take the place value of the last digit of the decimal number and write it as the denominator of the fraction.

Example 1

Convert decimal 0.28 to a proper fraction without simplifying.

Solution

Step 1:

We drop the decimal and write the number 28 as the numerator of a fraction.

Step 2:

The place value of the last digit 8 is a hundredth. So, we write 100 as the denominator of the fraction to get

0.28=28100 withoutsimplifying

Example 2

Convert decimal 0.75 to a proper fraction without simplifying.

Solution

Step 1:

We drop the decimal and write the number 75 as the numerator of a fraction.

Step 2:

The place value of the last digit 5 is a hundredth. So we write 100 as the denominator of the fraction to get

0.75=75100 withoutsimplifying

This tutorial provides comprehensive coverage of converting decimals to fractions based on Common Core CCSS and State Standards and its prerequisites. Students can navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. This simple tutorial uses appropriate examples to help you understand converting decimals to fractions in a general and quick way.

Audience

This tutorial has been prepared for beginners to help them understand the basics of converting decimals to fractions. After completing this tutorial, you will find yourself at a moderate level of expertise in converting decimals to fractions, from where you can advance further.

Prerequisites

Before proceeding with this tutorial, you need a basic knowledge of elementary math concepts such as number sense, addition, subtraction, multiplication, division, whole numbers, fractions, types of fractions, decimals, comparing and ordering whole numbers and so on.