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# UNDERSTANDING FRACTIONS

Let’s Review Fractions

A fraction is part of a whole or part of a group.

Part of a whole

3/4 of the circle is blue

Part of a group

3/4 of the group is purple

Let’s Review Fractions

• The line that separates the numerator (3) from the denominator (4) is a division         symbol. It tells us that we can divide these two numbers.

•  The top number 3 is the numerator.

•  The bottom number 4 is the denominator.

Let’s Review Fractions

Let's review some division before we go on. We’ll have to understand division well to divide the numerator by the denominator in a fraction.

Reading and Writing Fractions as Decimals

Fractions and Decimals

•    A decimal is another way of representing a fraction.

•    A decimal has no numerator or denominator.

•    For example, the fraction 5/10 can be expressed as a decimal, 0.5

•    The period ( . ) represents the decimal point.

•    We can read five-tenths as 0.5.

Writing Fractions as Decimals

•    Fractions with 10, 100, or 1000 in the denominator can be represented as decimals.

1/10=0.1           1/100=0.01          1/1000=0.001

•    The fraction 1/10 can be expressed as 0.1 or one-tenth.

•    The fraction 1/100 can be expressed as 0.01 or one-hundredth.

•    The fraction 1/1000 can be expressed as 0.001 or one-thousandth.

Reading and Writing Fractions as Decimals

Writing Fractions as Decimals

• For fractions with a denominator of 10, the decimal is expressed as tenths.

1/10= 0.1

• For 1/10 we write the decimal point (.) and then the numerator. In this example, the numerator is 1. It takes the tenths position, so the decimal is written as one-tenth and read as 0.1.

The places after the decimal point are called decimal places.

Reading and Writing Fractions as Decimals

Take a look at the chart above:

• The decimal 0.9 has 1 decimal place.

• The decimal 0.23 has 2 decimal places.

• The decimal 0.765 has 3 decimal places.

Writing Fractions as Decimals

Example 1: Write the fraction 1/10 as a decimal.

Let’s find the decimal that is equal to 1/10. We can start with the denominator.

•    The denominator is 10. So, we know that the new decimal will be in the tenths place.

•    The numerator is 1, so we can place the 1 in the tenths place.

Example 2: Write the fraction 7/10 as a decimal.

Let’s find the decimal that is equal to 7/10. We can start with the denominator.

• The denominator is 10. So, we know that the new decimal will be in the tenths place.

• The numerator is 7, so we can place the 7 in the tenths place.

Example 3: Write the fraction 21/100 as a decimal.

Let’s find the decimal that is equal to 21/100. We can start with the denominator.

• The denominator is 100. So, we know that the new decimal will have two decimal places because it is in the hundredths place.

• The numerator is 21, so we can place the 2 in the tenths place and the 1 in the hundredths place

• Therefore,  the new decimal is 0.21.

Note: If a fraction has 100 in the denominator, the equivalent decimal will have two decimal places.

Example 4: Write the fraction 153/1000 as a decimal.

Let’s find the decimal that is equal to 153/1000. We can start with the denominator.

• The denominator is 1000. So, we know that the new decimal will have three decimal places because it is in the thousandths place.

• The numerator is 153, so we can place 1 in the tenths place, 5 in the hundredths place, and 3 in the thousandths place. Therefore, the new decimal is 0.153.

Note : If a fraction has 1000 in the denominator, the equivalent decimal will have three decimal places.

Converting Fractions to Decimals

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When the fraction has 10, 100, or 1000 in the denominator, we can rewrite that fraction as a decimal, as we’ve done in the previous sections.

But with other fractions, we may need to use division to convert them to decimals.

• A fraction is made up of two parts: a numerator and a denominator. It is used to represent the number of parts selected out of the total number of parts.

• As we saw earlier, the line in a fraction that separates the numerator and denominator can be rewritten using the division symbol.

• So, to convert a fraction to a decimal, divide the numerator by the denominator. This will give us our answer as a decimal.

It is important to master expressing fractions in different ways.

• When given a fraction, place the numerator inside the division symbol. This is called the dividend.

• When given a fraction, place the denominator on the outside of the division symbol. This is called the divisor.

Example 1: Convert the fraction  2/5 to a decimal.

Goal: Long division to get to no remainders.

Step 1: We know that 5 does not go into 2 at all.

However, 5 does go into 20. We know this because you can multiply 5 x 4 = 20 (this means that 5 goes into 20, 4 times).

Step 2: Next, you subtract 20 from 20 like this 20 – 20 = 0. Now you have no remainders left over.

Example 1: Convert the fraction 2/5 to a decimal.

Let’s look at how the steps on the previous page are carried out.

So, the fraction  is 0.4 as a decimal.

Look at the chart below. The fractions on the left are represented as decimals on the right.

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Example 1: Convert the fraction 4/5 to a decimal.

Let’s take a closer look at converting the fraction 4/5 into a decimal. We can use long division with Expression 3 in the table.

### Example 1: Convert the fraction to a decimal.

Goal: Long division to get no remainders.

Step 1: We know that 5 does not go into 4 at all. However, 5 does go into 40. We know this because you can multiply 5 x 8 = 40 (this means that 5 goes into 40, 8 times).

Step 2: Next, you subtract 40 from 40 like this 40 - 40 = 0.

Now, you have nothing left over.

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Example 1: Convert the fraction 4/5 to a decimal.

Let’s look at how the steps on the previous section are carried out.

So, the fraction 4/5 is 0.8 as a decimal.

Example 2: Convert the fraction 2/5 to a decimal.

Goal: Long division to get no remainders.

Step 1: We know that 5 does not go into 2 at all. However, 5 does go into 20. We know this because you can multiply 5 x 4 = 20 (this means that 5 goes into 20, 4 times).

Step 2: Next, you subtract 20 from 20 like this 20 – 20 = 0.

Now, you have nothing left over.

Example 2: Convert the fraction to a decimal.

Let’s look at how the steps are carried out.

So, the fraction 2/5 is 0.4 as a decimal.

Converting Fractions to Decimals with Long Division (Hundredths)

Example 3: Convert the fraction 3/4 to a decimal.

Goal: Long division to get no remainders.

Step 1: We know that 4 does not go into 3 at all. However, 4 does goes into 30. We know this because you can multiply 4 x 7 = 28 (this means that 4 goes into 30, 4 times).

Step 2: Next, you subtract 28 from 30 like this 30 - 28 = 2. Now, you bring down the 0. Remember our goal is to get to the point that we have no remainders.

Step 3: So to get no remainders we need to complete the last step. This time, 4 goes into 20 exactly 5 times and we will have nothing left over.

Example 3: Convert the fraction 3/4 to a decimal.

Let’s look at how the steps on the previous section are carried out.

So, the fraction 3/4 is 0.75 as a decimal.

Example 4: Convert the fraction 1/4 to decimal.

Goal: Long division to get no remainders.

Step 1: We know that 4 does not go into 1 at all. However, 4 does go into 10. We know this because you can multiply 4 x 2 = 8 (this means that 4 goes into 10, 2 times).

Step 2: Next, you subtract 8 from 10 like this 10 – 8 = 2. Now, you can bring down the 0. Remember our goal is to get to the point that we have no remainders.

Step 3: So to get no remainders, we need to complete the last step. This time, 4 goes into 20 exactly 5 times and we will have nothing left over.

Example 4: Convert the fraction 1/4 to a decimal.

Let’s look at how the steps on the previous section are carried out.

So, the fraction 1/4 is 0.25 as a decimal

### Writing Decimals as Fractions

Example 1: Write the decimal 0.08 as a fraction.

Step 1:

Let’s find the numerator of the new fraction.

• The number after the decimal point is 8, so we place 8 in the numerator.

Step 2:

Let’s find the denominator of the new fraction.

• We start by writing 1 in the denominator when converting this decimal to a fraction.

• Find the number of decimal places after the decimal point. For example, in 0.08, there are two decimal places. So, we add two zeros next to the 1. The denominator is 100.

Example 1: Write the decimal 0.08 as a fraction.

Step 3:

• With 8 in the numerator and 100 in the denominator, the new fraction is 8/100

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You may be asked to write the decimal as a fraction in its simplest form.

Example 2: Write the decimal 0.30 as a fraction in its simplest form.

Step 1:

Let’s find the numerator of the new fraction.

• The number after the decimal point is 30, so we place 30 in the numerator.

Step 2:

Let’s find the denominator of the new fraction.

• We start by writing 1 in the denominator when converting this decimal to a fraction.

• Find the number of decimal places after the decimal point. In For example, in 0.30, there are two decimal places. So, we add two zeros next to the 1. The denominator is 100.

Example 2: Write the decimal 0.30 as a fraction in its simplest form.

Step 3:

• With 3 in the numerator and 100 in the denominator, the new fraction is 30/100.

• Now, we are ready to simplify this fraction.

Step 4:

• Find the Greatest Common Factor (GCF) of 30 and 100. The GCF is a factor you can divide both numbers by.

Example 2: Write the decimal 0.30 as a fraction in its simplest form.

Step 5:

• Divide the numerator and denominator by the GCF of 10 to solve for the simplest form.

Example 3: Write the decimal 0.4 as a fraction.

Step 1:

Let’s find the numerator of the new fraction.

• The number after the decimal point is 4, so we place 4 in the numerator.

Step 2:

Let’s find the denominator of the new fraction.

• We start by writing 1 in the denominator when converting this decimal to a fraction.

• Find the number of decimal places after the decimal point. For example, in 0.4, there is one decimal place. So, we place one zero next to the one. The denominator is 10.

Example 3: Write the decimal 0.4 as a fraction.

Step 3:

• With 4 in the numerator and 10 in the denominator, the new fraction is 4/10

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Example 4: Write the decimal 0.35 as a fraction.

Step 1:

Let’s find the numerator of the new fraction.

• The number after the decimal point is 35, so we place 35 in the numerator.

Step 2:

Let’s find the denominator of the new fraction.

• We start by writing 1 in the denominator when converting this decimal to a fraction.

• Find the number of decimal places after the decimal point. For example, in 0.35, there are two decimal places. So, we place two zeros next to the one. The denominator is 100.

Example 4: Write the decimal 0.35 as a fraction.

Step 3:

• With 35 in the numerator and 100 in the denominator, the new fraction is 35/100

Example 5: Write the decimal 0.002 as a fraction.

Step 1:

Let’s find the numerator of the new fraction.

• The number after the decimal point is 2, so we place 2 in the numerator.

Step 2:

Let’s find the denominator of the new fraction.

• We start by writing 1 in the denominator when converting this decimal to a fraction.

• Find the number of decimal places after the decimal point. For example, in 0.002, there are three decimal places. So, we place three zeros next to the one. The denominator is 1000.

### Example 5: Write the decimal 0.002 as a fraction.

Step 3:

• With 2 in the numerator and 1000 in the denominator, the new fraction is 2/1000

Converting Decimals to Fractions

We know that we can convert decimals to fractions and vice versa. So, let’s take a closer look at this example below and convert the decimal to the fraction.

Example 1: 0.3 (three-tenths) = 3/10

Step 1:

• Write down 0.3 divided by 1 like this 0.3/1

Note: Placing a decimal over 1 does not change the value.

Step 2:

• Multiply both the numerator and the denominator by a whole number.

• Use 10 (because there is 1 digit after the decimal place)

Note: Do you see how it changes the numerator to a whole number?

The decimal 0.3 is the fraction 3/10

Example 2: 0.9 (nine-tenths) =9/10

Step 1:

• Write down 0.9 divided by 1 like this 0.9/1

Note: Placing a decimal over 1 does not change the value.

Step 2:

• Multiply both the numerator and the denominator by a whole number to get a fraction.

• Use 10 (because there is 1 digit after the decimal place).

Note: Do you see how it changes the numerator to a whole number?

The decimal 0.9 is the fraction 9/10

Example 3: 0.04 (four-hundredths) = 4/100

Step 1:

• Write down 0.04 divided by 1 like this 0.04/1

Note: Placing a decimal over 1 does not change the value.

Step 2:

• Multiply both the numerator and the denominator by a whole number to get a fraction.

• Use 100 (because there are 2 digits after the decimal place).

Note: Do you see how it changes the numerator to a whole number?

The decimal 0.04 is the fraction 4/100

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