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Let's examine the following diagram.
2 one-fifths and 1 one fifth is shaded.
It is shown in the shaded regions of the diagram that
(2 one-fifths) + (1 one-fifth) = (3 one-fifths)
That is,
$\frac{2}{5}+\frac{1}{5}=\frac{3}{5}$
From this observation, we can suggest the following rule.
Find the following sums.
$\frac{3}{7}+\frac{2}{7}$ . The denominators are the same. Add the numerators and place that sum over 7.
$\frac{3}{7}+\frac{2}{7}=\frac{3+2}{7}=\frac{5}{7}$
$\frac{1}{8}+\frac{3}{8}$ . The denominators are the same. Add the numerators and place the sum over 8. Reduce.
$\frac{1}{8}+\frac{3}{8}=\frac{1+3}{8}=\frac{4}{8}=\frac{1}{2}$
$\frac{4}{9}+\frac{5}{9}$ . The denominators are the same. Add the numerators and place the sum over 9.
$\frac{4}{9}+\frac{5}{9}=\frac{4+5}{9}=\frac{9}{9}=1$
$\frac{7}{8}+\frac{5}{8}$ . The denominators are the same. Add the numerators and place the sum over 8.
$\frac{7}{8}+\frac{5}{8}=\frac{7+5}{8}=\frac{\text{12}}{8}=\frac{3}{2}$
To see what happens if we mistakenly add the denominators as well as the numerators, let's add
$\frac{1}{2}+\frac{1}{2}$
Adding the numerators and mistakenly adding the denominators produces
$\frac{1}{2}+\frac{1}{2}=\frac{1+1}{2+2}=\frac{2}{4}=\frac{1}{2}$
This means that two $\frac{1}{2}$ 's is the same as one $\frac{1}{2}$ . Preposterous! We do not add denominators .
Find the following sums.
Show why adding both the numerators and denominators is preposterous by adding $\frac{3}{4}$ and $\frac{3}{4}$ and examining the result.
$\frac{3}{4}+\frac{3}{4}=\frac{3+3}{4+4}=\frac{6}{8}=\frac{3}{4}$ , so two $\frac{3}{4}$ ’s= one $\frac{3}{4}$ which is preposterous.
We can picture the concept of subtraction of fractions in much the same way we pictured addition.
From this observation, we can suggest the following rule for subtracting fractions having like denominators:
Find the following differences.
$\frac{3}{5}-\frac{1}{5}$ . The denominators are the same. Subtract the numerators. Place the difference over 5.
$\frac{3}{5}-\frac{1}{5}=\frac{3-1}{5}=\frac{2}{5}$
$\frac{8}{6}-\frac{2}{6}$ . The denominators are the same. Subtract the numerators. Place the difference over 6.
$\frac{8}{6}-\frac{2}{6}=\frac{8-2}{6}=\frac{6}{6}=1$
$\frac{\text{16}}{9}-\frac{2}{9}$ . The denominators are the same. Subtract numerators and place the difference over 9.
$\frac{\text{16}}{9}-\frac{2}{9}=\frac{\text{16}-2}{9}=\frac{\text{14}}{9}$
To see what happens if we mistakenly subtract the denominators, let's consider
$\frac{7}{\text{15}}-\frac{4}{\text{15}}=\frac{7-4}{\text{15}-\text{15}}=\frac{3}{0}$
We get division by zero, which is undefined. We do not subtract denominators.
Find the following differences.
$\frac{\text{10}}{\text{13}}-\frac{8}{\text{13}}$
$\frac{2}{\text{13}}$
$\frac{\text{26}}{\text{10}}-\frac{\text{14}}{\text{10}}$
$\frac{6}{5}$
Show why subtracting both the numerators and the denominators is in error by performing the subtraction $\frac{5}{9}-\frac{2}{9}$ .
$\frac{5}{9}-\frac{2}{9}=\frac{5-2}{9-9}=\frac{3}{0}$ , which is undefined
For the following problems, find the sums and differences. Be sure to reduce.
$\frac{1}{6}+\frac{2}{6}$
$\frac{3}{\text{11}}+\frac{4}{\text{11}}$
$\frac{9}{\text{15}}+\frac{4}{\text{15}}$
$\frac{\text{13}}{\text{15}}$
$\frac{3}{\text{10}}+\frac{2}{\text{10}}$
$\frac{\text{11}}{\text{16}}-\frac{2}{\text{16}}$
$\frac{\text{15}}{\text{23}}-\frac{2}{\text{23}}$
$\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$
$\frac{3}{\text{11}}+\frac{1}{\text{11}}+\frac{5}{\text{11}}$
$\frac{9}{\text{11}}$
$\frac{\text{16}}{\text{20}}+\frac{1}{\text{20}}+\frac{2}{\text{20}}$
$\frac{\text{12}}{8}+\frac{2}{8}+\frac{1}{8}$
$\frac{\text{15}}{8}$
$\frac{1}{\text{15}}+\frac{8}{\text{15}}+\frac{6}{\text{15}}$
$\frac{3}{8}+\frac{2}{\text{8}}-\frac{1}{\text{8}}$
$\frac{1}{2}$
$\frac{\text{11}}{\text{16}}+\frac{9}{\text{16}}-\frac{5}{\text{16}}$
$\frac{4}{\text{20}}-\frac{1}{\text{20}}+\frac{9}{\text{20}}$
$\frac{3}{5}$
$\frac{7}{\text{10}}-\frac{3}{\text{10}}+\frac{\text{11}}{\text{10}}$
$\frac{\text{16}}{5}-\frac{1}{5}-\frac{2}{5}$
$\frac{\text{13}}{5}$
$\frac{\text{21}}{\text{35}}-\frac{\text{17}}{\text{35}}+\frac{\text{31}}{\text{35}}$
$\frac{1}{\text{18}}+\frac{3}{\text{18}}+\frac{1}{\text{18}}+\frac{4}{\text{18}}-\frac{5}{\text{18}}$
$\frac{6}{\text{22}}-\frac{2}{\text{22}}+\frac{4}{\text{22}}-\frac{1}{\text{22}}+\frac{\text{11}}{\text{22}}$
$\frac{9}{\text{11}}$
The following rule for addition and subtraction of two fractions is preposterous. Show why by performing the operations using the rule for the following two problems.
$\frac{3}{\text{10}}-\frac{3}{\text{10}}$
$\frac{8}{\text{15}}+\frac{8}{\text{15}}$
$\frac{\text{16}}{\text{30}}=\frac{8}{5}$ (using the preposterous rule)
Find the total length of the screw.
Two months ago, a woman paid off $\frac{3}{\text{24}}$ of a loan. One month ago, she paid off $\frac{5}{\text{24}}$ of the total loan. This month she will again pay off $\frac{5}{\text{24}}$ of the total loan. At the end of the month, how much of her total loan will she have paid off?
$\frac{\text{13}}{\text{24}}$
Find the inside diameter of the pipe.
( [link] ) Use the numbers 2, 4, and 8 to illustrate the associative property of addition.
( [link] ) Find the prime factors of 495.
${3}^{2}\cdot 5\cdot \text{11}$
( [link] ) Find the value of $\frac{3}{4}\cdot \frac{\text{16}}{\text{25}}\cdot \frac{5}{9}$ .
( [link] ) $\frac{8}{3}$ of what number is $1\frac{7}{9}$ ?
$\frac{2}{3}$
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