Prealgebra Section 5.5 – Fractions and Decimals
1 5.5 Fractions and Decimals
Steps for Changing a Fraction to a Decimal 1. Simplify the fraction, if possible.
2. Divide the numerator by the denominator. n d n d
How to know when to stop dividing:
1. You have a remainder of zero with no other digits in the dividend to drop down.
2. If the directions tell you to round your answer, you will stop at one place value after the given rounding place value.
3. You are able to recognize that the number will have a repeating decimal (ie. numbers behind the decimal point that repeat in a constant pattern).
Terminating Decimals
Terminating Decimals are decimal numbers that end. If you keep dividing, you will get a remainder of zero with no other digits in the dividend to drop down. They do not have repeating decimals and the directions usually will not tell you to round your answer.
Repeating Decimals
Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
We put a repeating bar over the digits that repeat in a pattern. (Make sure the bar only goes over the digits that are repeating.)
EXs: 1 0.33333333333 0.3
3 2 0.66666666666 0.6
3
1 0.0416666666 0.0416
24 7 0.636363636363 0.63
11
Example 1: Change 15
48 to a decimal.
Solution: Step 1: Simplify the fraction, if possible.
15 15 3 5 48 48 3 16
Step 2: Divide the numerator by the denominator. n d n d
Before we do the division we want to look at what kind of decimal we should be ending with.
Will it be a terminating decimal or a repeating decimal? (We already know we do not have to round since the directions do not tell us to.)
To see if we have a terminating decimal we will look at the prime factorization of the
denominator, 16. If the denominator has only factors of 2 and/or 5, then we know we have a terminating decimal.
16 2 2 2 2 Since we only have factors of 2, we know we have a terminating decimal.
So, let’s divide 16 5 16 5.0000 (Since we know we have a terminating decimal, we will add zeros to the back of the dividend until we get a remainder of zero.)
0.3125 16 5.0000
4 8 20 16 40 32 80 80 0
Hence, 15 0.3125 48
You Try It 1: Change 10
16 to a decimal.
Note:
You can tell whether a fraction will convert to a terminating decimal by looking at its denominator. If the denominator has a prime factorization consisting of only twos and/or fives, then the decimal representation of the fraction will terminate (end without rounding or having a repeating decimal). If the denominator has a prime factorization consisting of other factors than 2 and/or 5, then you will either need to round or have a repeating decimal.
Prealgebra Section 5.5 – Fractions and Decimals
3 Example 2: Change 3 7
20 to a decimal.
Solution: Step 1: Simplify the fraction, if possible. 7
20 is already simplified.
Step 2: Before we divide, first let’s look at the prime factorization of 20 to see if we have a terminating decimal.
20 2 2 5 Since we only have factors of 2 and 5, we know we have a terminating decimal.
So, let’s divide 20 7 20 7.00
0.35 20 7.00 6 0 1 00 1 00 0
Hence, 7 0.35
20 . But remember, we had 3 7
20. Therefore, 3 7 3.35 20 .
You Try It 2: Change 711
20 to a decimal.
Example 3: Change 1
12 to a decimal.
Solution: Step 1: Simplify the fraction, if possible. 1
12 is already simplified.
Step 2: Divide the numerator by the denominator. n d n d
First, let’s look at the prime factorization of 12 to see if we have a terminating decimal.
12 2 2 3 Since the prime factorization has more than just 2’s or 5’s (it has a 3), this will NOT be a terminating decimal. Also, since the directions did not tell us to round, we can assume this will be a repeating decimal.
So, let’s divide 12 1 12 1.0000
0.0833 12 1.0000
0 1 00
96 40 36 40 36 4
Hence, 1 0.0833 0.083
12 .
You can see here that you will just keep getting a 4 as a remainder.
This means the 3 in the quotient will just keep repeating.
You Try It 3: Change 5
12 to a decimal.
Example 4: Change 23
111 to a decimal.
Solution: Step 1: Simplify the fraction, if possible. 23
111 is already simplified.
Step 2: Divide the numerator by the denominator. n d n d
First, let’s look at the prime factorization of 111 to see if we have a terminating decimal.
111 3 37 Since the prime factorization has more than just 2’s or 5’s (it has a 3 and 37), this will NOT be a terminating decimal. Also, since the directions did not tell us to round, we can assume this will be a repeating decimal.
So, let’s divide 111 23 111 23.000000
0.207207 111 23.000000
22 2 80
0 800 777 230 222 80
0 800 777 23
Hence, 23 0.207207 0.207
111 .
You Try It 4: Change 5
33 to a decimal.
You can see here that the division is starting to repeat. You have 230 (which is what you started with), then 80, and then 800. So you know that the 0.207 will repeat.
Prealgebra Section 5.5 – Fractions and Decimals
5 Example 5: Simplify. 3 1.25
8
Solution: Let’s do this problem using fractions. This means we will need to change 1.25 into a fraction.
1.25 125 125 25 5 100 100 25 4
So now we have, 3 5
8 4
. Let’s get the LCD of 8 and simplify.
3 5 3 5 2 3 10 3 10 13
8 4 8 4 2 8 8 8 8
Hence, 3 1.25 13
8 8
. We can also do the problem with decimals. This means we will need to change 3
8 into a decimal.
You can use your calculator for this, 3 3 8 0.375
8
So now we have, 0.375 1.25 , again, you may use your calculator to get, 0.375 1.25 1.625 So, it is possible to get two different answers for this problem, 3 1.25 13
8 8
if you decide to work in fraction from, and 0.375 1.25 1.625if you decide to work in decimal form. Either answer is correct. You need to pick which route you would like to take at the beginning (fractions or decimals) and then you will only need to write one as your answer.
Hence, your answer is 3 1.25 13
8 8
OR 3 1.25 1.625
8 .
You Try It 5: Simplify. 7 6.5
8
Simplifying Expressions with Fractions and Decimals
To simplify expressions that contain both fractions and decimals, you need to make the numbers look the same. This means, you either need to change the fractions to decimals so that all your numbers are decimals. Or you need to change the decimals to fractions so that all your numbers are fractions.
Example 6: Simplify. 2 0.35
3
Solution: Let’s do this problem using fractions. This means we will need to change 0.35 into a fraction.
0.35 35 35 5 7 100 5 20
So now we have, 2 7
3 20
. Let’s get the LCD of 60 and simplify.
2 7 2 20 7 3 40 21 40 21 19
3 20 3 20 20 3 60 60 60 60
Hence, 2 0.35 19
3 60
.
You do NOT want to do this particular problem using decimals. The reason being is if you try to change 2
3 into a decimal you would get 0.6. You cannot work with a repeating decimal when you are computing. You also cannot round since the directions do not tell you to do so. Therefore, you can only do this problem using fractions. And therefore, there is only one possible correct answer.
Hence, 2 0.35 19
3 60
.
You Try It 6: Simplify. 4 0.25
9