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Before you get started, take this readiness quiz.
How many quarters are pictured? One quarter plus $2$ quarters equals $3$ quarters.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that
Let’s use fraction circles to model the same example, $\frac{1}{4}+\frac{2}{4}.$
Start with one $\frac{1}{4}$ piece. | ||
Add two more $\frac{1}{4}$ pieces. | ||
The result is $\frac{3}{4}$ . |
So again, we see that
Use a model to find the sum $\frac{3}{8}+\frac{2}{8}.$
Start with three $\frac{1}{8}$ pieces. | ||
Add two $\frac{1}{8}$ pieces. | ||
How many $\frac{1}{8}$ pieces are there? |
There are five $\frac{1}{8}$ pieces, or five-eighths. The model shows that $\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.$
Use a model to find each sum. Show a diagram to illustrate your model.
$\frac{1}{8}+\frac{4}{8}$
$\frac{5}{8}$
Use a model to find each sum. Show a diagram to illustrate your model.
$\frac{1}{6}+\frac{4}{6}$
$\frac{5}{6}$
[link] shows that to add the same-size pieces—meaning that the fractions have the same denominator —we just add the number of pieces.
If $a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0,$ then
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.
Find the sum: $\frac{3}{5}+\frac{1}{5}.$
$\frac{3}{5}+\frac{1}{5}$ | |
Add the numerators and place the sum over the common denominator. | $\frac{3+1}{5}$ |
Simplify. | $\frac{4}{5}$ |
Find the sum: $\frac{x}{3}+\frac{2}{3}.$
$\frac{x}{3}+\frac{2}{3}$ | |
Add the numerators and place the sum over the common denominator. | $\frac{x+2}{3}$ |
Note that we cannot simplify this fraction any more. Since $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}2$ are not like terms, we cannot combine them.
Find the sum: $-\frac{9}{d}+\frac{3}{d}.$
We will begin by rewriting the first fraction with the negative sign in the numerator.
$-\frac{a}{b}=\frac{-a}{b}$
$-\frac{9}{d}+\frac{3}{d}$ | |
Rewrite the first fraction with the negative in the numerator. | $\frac{\mathrm{-9}}{d}+\frac{9}{d}$ |
Add the numerators and place the sum over the common denominator. | $\frac{\mathrm{-9}+3}{d}$ |
Simplify the numerator. | $\frac{\mathrm{-6}}{d}$ |
Rewrite with negative sign in front of the fraction. | $-\frac{6}{d}$ |
Find the sum: $\frac{2n}{11}+\frac{5n}{11}.$
$\frac{2n}{11}+\frac{5n}{11}$ | |
Add the numerators and place the sum over the common denominator. | $\frac{2n+5n}{11}$ |
Combine like terms. | $\frac{7n}{11}$ |
Find the sum: $-\frac{3}{12}+\left(-\frac{5}{12}\right).$
$-\frac{3}{12}+\left(-\frac{5}{12}\right)$ | |
Add the numerators and place the sum over the common denominator. | $\frac{\mathrm{-3}+\left(\mathrm{-5}\right)}{12}$ |
Add. | $\frac{\mathrm{-8}}{12}$ |
Simplify the fraction. | $-\frac{2}{3}$ |
Find each sum: $-\frac{4}{15}+\left(-\frac{6}{15}\right).$
$-\frac{2}{3}$
Find each sum: $-\frac{5}{21}+\left(-\frac{9}{21}\right).$
$-\frac{2}{3}$
Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into $12$ slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or $\frac{7}{12}$ of the pizza) left in the box. If Leonardo eats $2$ of these remaining pieces (or $\frac{2}{12}$ of the pizza), how much is left? There would be $5$ pieces left (or $\frac{5}{12}$ of the pizza).
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