Lesson Objectives

- Demonstrate an understanding of fractions
- Demonstrate an understanding of multiples and powers
- Learn how to convert a decimal fraction into a decimal
- Learn how to convert a decimal into a decimal fraction

## How to Convert a Decimal Fraction (Base 10 Fraction) into a Decimal

A decimal fraction (base 10 fraction) is a fraction whose denominator is a power of ten. Specifically, we would say the denominator is a multiple of 10: 10, 100, 1000, 10,000,…

Some examples of decimal fractions:

$$\frac{6}{10}, \frac{13}{100}, \frac{7293}{100,\hspace{-.1em}000}$$ We have all seen decimal numbers when shopping or making financial transactions. It is not too often that the price for an item with tax comes out to a whole number. Suppose the price of a can of tuna with tax was: $$\$1\frac{38}{100}$$ Decimals make our life much easier, we can write the same price for our can of tuna as: $$\$1.38$$ When we work with a decimal number, we will see the "." or decimal point as part of the number. The numbers after a decimal point represent part of a whole. In our example above, we priced our tuna can at $1.38. The part after the decimal point (38) represents 38 parts out of 100 parts. It is identical to the fraction 38/100: $$.38=\frac{38}{100}$$

In the first step, we found that our denominator of 10 had 1 zero. This means we will move our decimal point one place to the left: $$.3$$ When working with decimals, we can place a 0 in front (to the left) of the decimal point if no whole number is present. This will not change the value of the number and makes it easier to identify the decimal point: $$\frac{3}{10}=0.3$$ Example 2: Convert each into a decimal. $$\frac{52}{1000}$$ Count the number of zeros in the denominator (3), and then delete the denominator: $$\frac{52}{\cancel{1000}}$$ Place a decimal point after the final digit (rightmost) of the number: $$52.$$ Move the decimal point to the left by the same number of places as the number of zeros counted from the first step:

In the first step, we found that our denominator of 1000 had 3 zeros. This means we will move our decimal point three places to the left: $$.052$$ Our number 52 has only two digits. We need to move our decimal point three places to the left. In order to accomplish this, we will add one zero in front of our 5. This will give us a three-digit number 052 and allow us to move by the appropriate number of places left. Again, we can place a 0 in front (to the left) of the decimal point for clarity: $$\frac{52}{1000}=0.052$$ We may also encounter a mixed number where the fraction part is a decimal fraction. In this case, we just write the whole number in front or to the left of the decimal point. Let's look at an example.

Example 3: Convert each into a decimal. $$13\frac{231}{10,\hspace{-.1em}000}$$ We will deal with our whole number (13) last. Let's begin with our fraction part.

Count the number of zeros in the denominator (4), and then delete the denominator: $$\frac{231}{\cancel{10,\hspace{-.1em}000}}$$ Place a decimal point after the final digit (rightmost) of the number: $$231.$$ Move the decimal point to the left by the same number of places as the number of zeros counted from the first step:

In the first step, we found that our denominator of 10,000 had 4 zeros. This means we will move our decimal point four places to the left: $$.0231$$ Our number 231 has only three digits. We need to move our decimal point four places to the left. In order to accomplish this, we will add one zero in front of our 2. This will give us a four-digit number 0231 and allow us to move by the appropriate number of places left. Again, we can place a 0 in front (to the left) of the decimal point for clarity: $$\frac{231}{10,\hspace{-.1em}000}=0.0231$$ In this case, we have a whole number part. So we can just replace our 0 in front with our whole number part 13: $$13\frac{231}{10,\hspace{-.1em}000}=13.0231$$

In this case, we have a zero to the left of the decimal point.

Count the number of decimal places:

We have 4 decimal places.

Write the decimal part (29) over a denominator that begins with 1 and is followed by 4 zeros: $$0.0029=\frac{29}{10,\hspace{-.1em}000}$$ Notice how we wrote 29 instead of 0029. After being moved into the numerator of the fraction, the two zeros in front of 29 do not add any value and are excluded.

Example 5: Convert each into a decimal fraction (do not simplify the fraction). $$17.091$$ Write any whole number:

17

Count the number of decimal places:

We have 3 decimal places.

Write the decimal part (91) over a denominator that begins with 1 and is followed by 3 zeros: $$17.091=17\frac{91}{1000}$$

Some examples of decimal fractions:

$$\frac{6}{10}, \frac{13}{100}, \frac{7293}{100,\hspace{-.1em}000}$$ We have all seen decimal numbers when shopping or making financial transactions. It is not too often that the price for an item with tax comes out to a whole number. Suppose the price of a can of tuna with tax was: $$\$1\frac{38}{100}$$ Decimals make our life much easier, we can write the same price for our can of tuna as: $$\$1.38$$ When we work with a decimal number, we will see the "." or decimal point as part of the number. The numbers after a decimal point represent part of a whole. In our example above, we priced our tuna can at $1.38. The part after the decimal point (38) represents 38 parts out of 100 parts. It is identical to the fraction 38/100: $$.38=\frac{38}{100}$$

### Convert a Decimal Fraction into a Decimal

- Count the number of zeros in the denominator, and then delete the denominator
- Place a decimal point after the final digit (rightmost) of the number
- Move the decimal point to the left by the same number of places as the number of zeros counted from the first step
- When there are not enough places, we can add zeros as needed

In the first step, we found that our denominator of 10 had 1 zero. This means we will move our decimal point one place to the left: $$.3$$ When working with decimals, we can place a 0 in front (to the left) of the decimal point if no whole number is present. This will not change the value of the number and makes it easier to identify the decimal point: $$\frac{3}{10}=0.3$$ Example 2: Convert each into a decimal. $$\frac{52}{1000}$$ Count the number of zeros in the denominator (3), and then delete the denominator: $$\frac{52}{\cancel{1000}}$$ Place a decimal point after the final digit (rightmost) of the number: $$52.$$ Move the decimal point to the left by the same number of places as the number of zeros counted from the first step:

In the first step, we found that our denominator of 1000 had 3 zeros. This means we will move our decimal point three places to the left: $$.052$$ Our number 52 has only two digits. We need to move our decimal point three places to the left. In order to accomplish this, we will add one zero in front of our 5. This will give us a three-digit number 052 and allow us to move by the appropriate number of places left. Again, we can place a 0 in front (to the left) of the decimal point for clarity: $$\frac{52}{1000}=0.052$$ We may also encounter a mixed number where the fraction part is a decimal fraction. In this case, we just write the whole number in front or to the left of the decimal point. Let's look at an example.

Example 3: Convert each into a decimal. $$13\frac{231}{10,\hspace{-.1em}000}$$ We will deal with our whole number (13) last. Let's begin with our fraction part.

Count the number of zeros in the denominator (4), and then delete the denominator: $$\frac{231}{\cancel{10,\hspace{-.1em}000}}$$ Place a decimal point after the final digit (rightmost) of the number: $$231.$$ Move the decimal point to the left by the same number of places as the number of zeros counted from the first step:

In the first step, we found that our denominator of 10,000 had 4 zeros. This means we will move our decimal point four places to the left: $$.0231$$ Our number 231 has only three digits. We need to move our decimal point four places to the left. In order to accomplish this, we will add one zero in front of our 2. This will give us a four-digit number 0231 and allow us to move by the appropriate number of places left. Again, we can place a 0 in front (to the left) of the decimal point for clarity: $$\frac{231}{10,\hspace{-.1em}000}=0.0231$$ In this case, we have a whole number part. So we can just replace our 0 in front with our whole number part 13: $$13\frac{231}{10,\hspace{-.1em}000}=13.0231$$

### Convert a Decimal into a Decimal Fraction

- Write any whole number
- Count the number of decimal places - the number of places that occur after (to the right of) the decimal point
- Write the decimal part (the part that occurs after the decimal point) over a denominator that begins with 1 and is followed by the same number of zeros as decimal places counted in our previous step

In this case, we have a zero to the left of the decimal point.

Count the number of decimal places:

We have 4 decimal places.

Write the decimal part (29) over a denominator that begins with 1 and is followed by 4 zeros: $$0.0029=\frac{29}{10,\hspace{-.1em}000}$$ Notice how we wrote 29 instead of 0029. After being moved into the numerator of the fraction, the two zeros in front of 29 do not add any value and are excluded.

Example 5: Convert each into a decimal fraction (do not simplify the fraction). $$17.091$$ Write any whole number:

17

Count the number of decimal places:

We have 3 decimal places.

Write the decimal part (91) over a denominator that begins with 1 and is followed by 3 zeros: $$17.091=17\frac{91}{1000}$$

#### Skills Check:

Example #1

Convert into a decimal number. $$\frac{756}{1000}$$

Please choose the best answer.

A

75.6

B

7560

C

7.56

D

0.756

E

0.0756

Example #2

Convert into a decimal number. $$43\frac{43}{1000}$$

Please choose the best answer.

A

43.43

B

43.043

C

43.0043

D

4,343,000

E

0.4343

Example #3

Convert into a decimal number. $$\frac{420}{10{,}000}$$

Please choose the best answer.

A

4,200,000

B

0.042

C

0.0042

D

0.00042

E

42.042

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