2.7: Estimate Sums and Differences of Fractions and Mixed Numbers
Have you ever tried to estimate using fractions? Take a look at this dilemma.
Harriet and Matt have two identical jars. Harriet’s jar is \begin{align*}\frac{3}{10}\end{align*} full of pennies. Matt’s jar is \begin{align*}\frac{1}{4}\end{align*} full of pennies. If Matt and Harriet combine their pennies into one jar, how full will the jar be?
Do you know how to figure this out? We can use estimation to help us. Pay attention to this Concept and you will be able to figure out the fullness of the jar.
Guidance
When you estimate fraction sums and differences you will need to work with rounding fractions. You can work with rounding fractions in a couple of different ways. One way is to look at the relationship between the fraction and a whole.
Here are some guiding questions:
- Is this fraction close to one-half or one-whole?
- If I simplify the fractions that I am adding or subtracting, will they have a common denominator?
- Is this fraction so close to one-whole that it would make sense to round up to a whole?
Take a few minutes and write these guiding questions down in your notebook.
Now apply this information and estimate the following sum.
\begin{align*}\frac{4}{25}+\frac{11}{20}\end{align*}
The first fraction is close to \begin{align*}\frac{5}{25}\end{align*}, or \begin{align*}\frac{1}{5}\end{align*}.
The second fraction is close to \begin{align*}\frac{12}{20}\end{align*}, or \begin{align*}\frac{3}{5}\end{align*}.
Now you can easily add the rounded fractions.
\begin{align*}\frac{1}{5}+\frac{3}{5}=\frac{4}{5}\end{align*}
A good estimate for the sum is \begin{align*}\frac{4}{5}\end{align*}.
With this problem, it made sense to round the fractions so that we could simplify them. The simplified fractions had a common denominator. This makes our work easier.
Here is another one.
Estimate the difference: \begin{align*}\frac{24}{49}-\frac{7}{31}\end{align*}
This first fraction is about \begin{align*}\frac{1}{2}\end{align*}.
The second fraction is about \begin{align*}\frac{1}{4}\end{align*}.
\begin{align*}\frac{1}{2}-\frac{1}{4}=\frac{2}{4}-\frac{1}{4}=\frac{1}{4}\end{align*}
A good estimate for the difference is \begin{align*}\frac{1}{4}\end{align*}.
It seems tricky because there is not one set way of estimating sums and differences of fractions. You need to use your thinking skills and powers of reason to look at the relationship between the fraction and a whole or a half and so on. As you move into higher levels of mathematics, this will become necessary for many of your assignments. It is a good skill to begin practicing now.
Find a reasonable estimate for each sum or difference.
Example A
\begin{align*}\frac{1}{4} + \frac{6}{7}\end{align*}
Solution:\begin{align*}1 \frac{1}{4}\end{align*}
Example B
\begin{align*}\frac{8}{9} - \frac{1}{2}\end{align*}
Solution:\begin{align*}\frac{1}{2}\end{align*}
Example C
\begin{align*}\frac{4}{5} + \frac{9}{10}\end{align*}
Solution:\begin{align*}2\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
To find the how full the jar will be, write a simple equation to represent the problem. Let \begin{align*}x\end{align*} represent the amount of the jar that has been filled.
\begin{align*}x &= \frac{3}{10}+\frac{1}{4}\\ &=\left(\frac{3}{10} \cdot \frac{2}{2}\right)+\left(\frac{1}{4}\cdot\frac{5}{5}\right)\\ &=\frac{6}{20}+\frac{5}{20}\\ &=\frac{11}{20}\end{align*}
The jar will be \begin{align*}\frac{11}{20}\end{align*} full. Now let’s think about this logically. What information does this fraction tell you? Well, we can think in terms of halves or wholes. Since 10 is half of 20, we can say that this jar is a little more than half full.
This is our estimate of the sum of the jar.
Vocabulary
- Fraction
- represents a part of a whole.
- Improper fraction
- a fraction where the numerator is greater than the denominator. It means that we have more than one whole represented.
- Mixed Number
- a whole number and a fraction written together.
- Denominator
- the bottom number in a fraction tells you how many parts the whole has been divided into.
- Numerator
- the top number in a fraction. It tells you how many parts you have out of the whole.
Guided Practice
Here is one for you to try on your own.
Estimate the following difference.
\begin{align*}5 \frac{11}{12} - 2\end{align*}
Solution
To estimate this difference, first look at the mixed number.
\begin{align*}5 \frac{11}{12}\end{align*} is close to \begin{align*}6\end{align*}.
We can use \begin{align*}6 - 2\end{align*}.
\begin{align*}6 - 2 = 4\end{align*}
Our estimate is \begin{align*}4\end{align*}.
Video Review
Practice
Directions: Estimate each sum or difference.
1. \begin{align*}\frac{6}{7} + \frac{1}{22}\end{align*}
2. \begin{align*}\frac{1}{2} + \frac{9}{10}\end{align*}
3. \begin{align*}2 \frac{6}{7} + 4 \frac{1}{12}\end{align*}
4. \begin{align*}\frac{8}{9} + \frac{21}{22}\end{align*}
5. \begin{align*}\frac{16}{17} - \frac{1}{22}\end{align*}
6. \begin{align*}\frac{1}{2} + \frac{1}{9}\end{align*}
7. \begin{align*}\frac{11}{12} - \frac{21}{22}\end{align*}
8. \begin{align*}7 \frac{8}{10} - 3 \frac{1}{22}\end{align*}
9. \begin{align*}\frac{4}{6} + \frac{2}{3}\end{align*}
10. \begin{align*}11 \frac{6}{7} + 14 \frac{1}{22}\end{align*}
11. \begin{align*}9 \frac{1}{7} + 14 \frac{1}{22}\end{align*}
12. \begin{align*}\frac{18}{20} - \frac{1}{2}\end{align*}
13. \begin{align*}\frac{16}{32} - \frac{1}{2}\end{align*}
14. \begin{align*}5\frac{1}{2} - 2 \frac{1}{2}\end{align*}
15. \begin{align*}12 \frac{6}{12} + 15 \frac{1}{22}\end{align*}
Denominator
The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator .fraction
A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.improper fraction
An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.Numerator
The numerator is the number above the fraction bar in a fraction.Image Attributions
Here you'll learn to estimate sums and differences of fractions and mixed numbers.
Denominator
The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator .fraction
A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.improper fraction
An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.Numerator
The numerator is the number above the fraction bar in a fraction.