New York 7 - 2020 Edition

2.05 Converting fractions to decimals

Lesson

We've already looked at how to change decimals to fractions. Now let's look at how to go the other way and change fractions to decimals.

Think about the names of the columns in our place value table: tenths, hundredths, thousandths and so on. So to change fractions to decimals, it's easiest if we have a denominator of $10,100,1000$10,100,1000 and so on (so it matches the numbers in the place value table).

Convert $\frac{3}{10}$310 to a decimal.

This fraction means "three tenths," so to write it in the place value table, it would be

Let's look at another example.

Convert $\frac{47}{100}$47100 to a decimal.

This time, we need to make sure our decimal finishes in the hundredths column, so we would write this as $0.47$0.47

If we have a fraction with a denominator that is not a power of $10$10, the we want to convert it so it does.

So the process goes:

1) Find a number you can multiply the denominator by to make it a power of $10$10.

2) Multiply the numerator and denominator by that number.

3) Then write down just the top number, putting the decimal point in the correct spot.

**Question**: Convert $\frac{4}{5}$45 to a decimal.

**Think:** To change the denominator to $10$10, we need to know how many tenths are in $1$1 fifth.

**Do**: From our work with fractions, we know that $2$2 tenths = $1$1 fifth, so for every fifth we have (and we have $4$4 of them) that is $2$2 tenths. So $4$4 fifths, is the same as $8$8 tenths or $0.8$0.8.

Let's look at another example.

**Question**: Convert $\frac{159}{300}$159300 to a decimal.

**Think**: To change the denominator $100$100, we ask ourselves, how many three-hundredths are in a hundredth. The answer is $3$3. So for every $3$3 three-hundredths we have (and we have $159$159 of them) we have $1$1 hundredth. This means that $159$159 three-hundredths is the same as $53$53 hundredths. What we are doing here is dividing $300$300 by $3$3. Whatever we do to the denominator, we have to do to the numerator.

**Do**: $\frac{159}{300}$159300 = $\frac{53}{100}=0.53$53100=0.53

We use the same process to convert mixed numbers.

**Question**: Convert $4\frac{567}{1000}$45671000 to a decimal.

**Think:** The $4$4 belongs in the units column, then the $567$567 will come after the decimal point.

**Do:** $4.567$4.567

**Question**: Write $15+\frac{4}{10}+\frac{5}{1000}$15+410+51000

**Think:** Where these numbers belong in the place value table (look- we have no hundredths).

**Do: **$15.405$15.405

We may need to do some conversion or even addition before we write a decimal.

**Question**: Write $4+\frac{4}{5}+\frac{15}{100}$4+45+15100

**Think:** This would be $4+\frac{8}{10}+\frac{15}{100}$4+810+15100. In $\frac{15}{100}$15100, the one belongs in the tenths column, so we will have to add this on to our other tenths. In other words, we could think of it as $4+\frac{8}{10}+\frac{1}{10}+\frac{5}{100}$4+810+110+5100

**Do:** $4.95$4.95

Remember!

Notice that with all of these examples, the decimal stopped or terminated, meaning it didn't continue on forever. We could say that all of these decimals end in $0$0. For example, looking at question $7$7 we can say that the answer is $4.950$4.950. We can add a zero at the end of any of the decimals above, that is because they are rational numbers.

When rational numbers are converted to decimals, one of two things happens:

- The decimal terminates (ends in $0$0)
- The decimal repeats (for example $\frac{1}{3}=0.33333$13=0.33333... repeating forever)

Write the fraction $\frac{1}{2}$12 as a decimal.

Write the fraction $\frac{349}{500}$349500 as a decimal.

Evaluate $8+\frac{8}{10}+\frac{1}{100}+\frac{1}{1000}$8+810+1100+11000, expressing your answer as a decimal.

Evaluate $\frac{3}{10}+\frac{3}{50}$310+350, expressing your answer as a decimal.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

(d) Convert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.