LESSON: Fractions and Decimals
| Site: | MN Partnership for Collaborative Curriculum |
| Course: | Mathematics Essentials Q1 |
| Book: | LESSON: Fractions and Decimals |
| Printed by: | Guest user |
| Date: | Thursday, November 21, 2024, 2:31 PM |
Description
Decimals and Fractions
Table of contents
INTRODUCTION
READ: Write Fractions as Terminating Decimals
Write Fractions and Mixed Numbers as Terminating Decimals
Now that you have mastered fractions and their corresponding operations, it’s time to discover how they relate to decimals. You know that fractions and decimals are related because they are both ways of describing numbers that are not wholes. In essence, a fraction is simply another way of describing what a decimal describes. They both represents parts of a whole and both can show the same part in a different way. The fraction shows us the part using the fraction bar comparing part to whole and the decimal shows us the part using place value. To start off, we’ll see how fractions can be converted into decimals and how decimals can be converted into fractions.
How do we convert fractions to decimals and decimals to fractions?
First, remember that fractions and decimals are different ways of writing the same thing. Both show us how to represent a part of a whole. Think about how we talk about fractions and decimals because this will become useful as we convert them.
Say this value out loud 0.1. You can say “point one” or you can say “one tenth.” Does the second version sound a little bit familiar? It sounds like the fraction . It turns out that
.
How do we convert fractions into decimals?
We can easily convert fractions into decimals. You’ve probably noticed by now that a fraction is really a short way of writing a division expression. Writing is really like writing
. The way that we find out how to write
as a decimal is to go ahead and solve the division problem. Since 4 doesn’t go into 3, we have to expand the number over the decimal point.
How many times does 4 go into 3.0? Four goes into 3.0 .7 times.
Be sure when you are writing your quotient above the dividend to keep the original place of the decimal point. Since 4 does not divide evenly into 3.0 and we have a remainder of .2, we can go further to the other side of the decimal point by adding a 0 next to the remainder of .2.
4 goes evenly into .20 five times, so we have our final answer.
Example 1
Convert to a decimal.
We start this by changing it into a division problem. We will be dividing 1 by 4. You already know that 1 can’t be divided by four, so you will need to use a decimal point and add zeros as needed.
Our answer is .25.
How do we convert mixed numbers to decimals?
When you are working with mixed numbers like for example, it is easiest to simply set the whole number to the side and solve the division problem with the fraction. When you have completed the division problem with the fraction, make sure that you put the whole number back on the left side of the decimal point.
Example 2
Convert to a decimal.
First, set aside the 3. We will come back to that later.
Next, we divide 1 by 2. Use a decimal point and zeros as needed.
Now we add in the 3.
Our final answer is 3.5.
EXAMPLE 1
READ: Write Fractions and Mixed Numbers as Repeating Decimals
Write Fractions and Mixed Numbers as Repeating Decimals
By now you’ve gotten the hang of converting fractions to decimals. So far, we have been working with what are known as terminating decimals, or decimals that have an end like 0.75 or 0.5.
One reason that we sometimes use fractions instead of decimals is because some decimals are repeating decimals, or decimals that go on forever. If you try to find a decimal for by dividing, you can divide forever because
written as a decimal
It goes on and on. That’s why we usually just simply write a line above the number that repeats. For
, we write:
. Let’s check out some examples involving repeating decimals.
Example
Write using decimals
First, we rewrite as the division problem
. We already know that we will have to go on the right side of the decimal point, so we are going to begin by dividing 6 into 5.0.
Six goes into 5.0 .8 times, but we have the remainder of .2. Six goes into 0.2 .03 times and we have a remainder of .02. Since 6 always goes into 20 three times, and there will always be a remainder of 2, we can see that it will never evenly divide.
If you keep dividing, you will get 0.83333333333.... forever and ever.
Our final answer is .
What about mixed numbers?
Well, there are some mixed numbers where the fraction part is a repeating decimal. Let’s look at an example.
Example
Write using decimals.
Just as we did with the terminating decimals, we are going to leave the whole number, 2 to the side before we are ready to add it to the final answer. So, we are simply solving for the decimal equivalent of . We write the division problem
. How many times does 3 go into 2.0? It goes into 2.0 0.6 times.
We have 0.20 as the remainder. How many times does 3 go into 0.20? The answer is 0.06 times.
Are you noticing a pattern here? It is obvious that there will always be a remainder whether we divide 3 into 2.0, 0.2, 0.02, 0.002, or 0.0002 and on and on. Clearly is a repeating decimal.
For our final answer we write .
EXAMPLE 2
READ: Write Decimals as Fractions
Write Decimals as Fractions
Now that we have mastered writing fractions as decimals, it will be good to know how to go back the other way and write decimals as fractions.
Consider again the decimal 0.1. We already know that we can say that this number is “one-tenth.” It’s very easy to rewrite decimals as fractions because decimals are already expressed as fractions with a denominator that is a factor of ten.
We can also say that .
To convert decimals to fractions, we write the number to the right of the decimal place over a denominator equivalent to the last place value of the decimal number. So, if we have 0.877, we would write .
If we have simply 0.6, we can write , or in simplest terms,
. Always make sure to put your fraction in simplest terms.
Example 1
Convert 0.35 to a fraction.
Start by saying the decimal to yourself out loud. To say 0.35 out loud, we can say “35 hundredths,” so we can go ahead and write the fraction down.
That’s a big fraction. We want to make our lives a little bit easier, so we will reduce the fraction to simplest terms. This fraction expressed in simplest terms is .
Our final answer is .
Example 2
Convert 2.4 to a mixed number.
Just as we leave aside the whole number when converting mixed numbers to decimals, we will leave aside the numbers to the left of the decimal point when converting decimals to fractions. So, in this case, we just have to find out what 0.4 is expressed as a fraction.
Let’s write it directly as the fraction “four tenths” or . Can we simplify it? You bet.
.
2.4 expressed as a mixed number .
EXAMPLE 3
EXAMPLE 4
READ: Compare and Order Decimals and Fractions
Compare and Order Decimals and Fractions
Eventually, you will be able to convert common fractions to decimals and common decimals to fractions in your head. You already know some of the classics like . Knowing this off the top of your head will make it easy for you to compare and order between fractions and decimals. For now, we will use our expertise at converting to compare and order. It’s always helpful to check.
Example
Compare and 0.25 using <, > or =
To compare a fraction to a decimal or a decimal to a fraction, we will need to convert one of them, so that we can compare a fraction to a fraction or a decimal to a decimal. For this one, I will convert to a decimal. I divide 1.0 by 4. 4 goes into 1.0 .2 times. There is also a remainder of 0.20 and 4 goes into 0.20 0.05 times. Now we know that
written as a decimal is 0.25.
We compare it as .
Example
Compare and 1.30
Our work in estimating the value of fractions and rounding decimals can be helpful when comparing fractions and decimals because you can look at a fraction or a decimal and quickly have a sense of what the approximate value is. Take a look at the mixed number . Don’t be intimidated by the large denominator, it looks like we can simplify it. If we simplify it to
.
Now we can take 1.30 and make it a mixed number. 1.30 becomes .
Our final answer is that .
We can use all of these strategies when ordering fractions and decimals too. Be sure that they are in the same form and then order them from least to greatest or from greatest to least.