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The terms 1 and $\frac{2}{3}$ can be represented as $1+\frac{2}{3}$ or $1\frac{2}{3}$ .
Thus,
$\frac{5}{3}=1\frac{2}{3}$ .
Improper fraction = mixed number.
There are 6 one thirds, or $\frac{6}{3}$ , or 2.
$6\left(\frac{1}{3}\right)=\frac{6}{3}=2$
Thus,
$\frac{6}{3}=2$
Improper fraction = whole number.
The following important fact is illustrated in the preceding examples.
For example $1\frac{1}{3}$ can be expressed as $1+\frac{1}{3}$ The fraction $5\frac{7}{8}$ can be expressed as $5+\frac{7}{8}$ .
It is important to note that a number such as $5+\frac{7}{8}$ does not indicate multiplication. To indicate multiplication, we would need to use a multiplication symbol (such as ⋅)
Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers.
To understand how we might convert an improper fraction to a mixed number, let's consider the fraction, $\frac{4}{3}$ .
$\begin{array}{ccc}\frac{4}{3}& =& \underset{1}{\underbrace{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}}+\frac{1}{3}\hfill \\ & =& 1+\frac{1}{3}\hfill \\ & =& 1\frac{1}{3}\hfill \end{array}$
Thus, $\frac{4}{3}=1\frac{1}{3}$ .
We can illustrate a procedure for converting an improper fraction to a mixed number using this example. However, the conversion is more easily accomplished by dividing the numerator by the denominator and using the result to write the mixed number.
Convert each improper fraction to its corresponding mixed number.
$\frac{5}{3}$ Divide 5 by 3.
The improper fraction $\frac{5}{3}=1\frac{2}{3}$ .
$\frac{\text{46}}{9}$ . Divide 46 by 9.
The improper fraction $\frac{\text{46}}{9}=5\frac{1}{9}$ .
$\frac{\text{83}}{\text{11}}$ . Divide 83 by 11.
The improper fraction $\frac{\text{83}}{\text{11}}=7\frac{6}{\text{11}}$ .
$\frac{\text{104}}{4}$ Divide 104 by 4.
$\frac{\text{104}}{4}=\text{26}\frac{0}{4}=\text{26}$
The improper fraction $\frac{\text{104}}{4}=\text{26}$ .
Convert each improper fraction to its corresponding mixed number.
To understand how to convert a mixed number to an improper fraction, we'll recall
mixed number = (natural number) + (proper fraction)
and consider the following diagram.
Recall that multiplication describes repeated addition.
Notice that $\frac{5}{3}$ can be obtained from $1\frac{2}{3}$ using multiplication in the following way.
Multiply: $3\cdot 1=3$
Add: $3+2=5$ . Place the 5 over the 3: $\frac{5}{3}$
The procedure for converting a mixed number to an improper fraction is illustrated in this example.
Convert each mixed number to an improper fraction.
$5\frac{7}{8}$
Thus, $5\frac{7}{8}=\frac{\text{47}}{8}$ .
$\text{16}\frac{2}{3}$
Thus, $\text{16}\frac{2}{3}=\frac{\text{50}}{3}$
Convert each mixed number to its corresponding improper fraction.
For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number.
$\frac{4}{9}$
$\frac{1}{8}$
$\frac{\text{11}}{8}$
$\text{191}\frac{4}{5}$
$\text{31}\frac{6}{7}$
$\frac{\text{55}}{\text{12}}$
$\frac{8}{9}$
For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.
$\frac{\text{11}}{6}$
$\frac{\text{25}}{4}$
$\frac{\text{71}}{8}$
$\frac{\text{121}}{\text{11}}$
$\frac{\text{165}}{\text{12}}$
$\text{13}\frac{9}{\text{12}}$ or $\text{13}\frac{3}{\text{4}}$
$\frac{\text{346}}{\text{15}}$
$\frac{\text{23}}{5}$
$\frac{\text{19}}{2}$
$\frac{\text{316}}{\text{41}}$
$7\frac{\text{29}}{\text{41}}$
$\frac{\text{800}}{3}$
For the following 15 problems, convert each of the mixed numbers to its corresponding improper fraction.
$1\frac{5}{\text{12}}$
$\text{15}\frac{1}{4}$
$\text{10}\frac{5}{\text{11}}$
$\frac{\text{115}}{\text{11}}$
$\text{15}\frac{3}{\text{10}}$
$4\frac{3}{4}$
$\text{17}\frac{9}{\text{10}}$
$9\frac{\text{20}}{\text{21}}$
$\frac{\text{209}}{\text{21}}$
$5\frac{1}{\text{16}}$
$\text{90}\frac{1}{\text{100}}$
$\frac{\text{9001}}{\text{100}}$
$\text{300}\frac{\text{43}}{\mathrm{1,}\text{000}}$
Why does $0\frac{4}{7}$ not qualify as a mixed number?
Why does 5 qualify as a mixed number?
… because it may be written as $5\frac{0}{\mathrm{n}}$ , where $n$ is any positive whole number.
$\text{35}\frac{\text{11}}{\text{12}}$
$\text{27}\frac{5}{\text{61}}$
$\frac{\mathrm{1,}\text{652}}{\text{61}}$
$\text{83}\frac{\text{40}}{\text{41}}$
$\text{105}\frac{\text{21}}{\text{23}}$
$\frac{\mathrm{2,}\text{436}}{\text{23}}$
$\text{72}\frac{\text{605}}{\text{606}}$
$\text{816}\frac{\text{19}}{\text{25}}$
$\frac{\text{20},\text{419}}{\text{25}}$
$\text{708}\frac{\text{42}}{\text{51}}$
$\mathrm{6,}\text{012}\frac{\mathrm{4,}\text{216}}{\mathrm{8,}\text{117}}$
$\frac{\text{48},\text{803},\text{620}}{\mathrm{8,}\text{117}}$
( [link] ) Round 2,614,000 to the nearest thousand.
( [link] ) Find the product. $\text{1,004}\cdot \text{1,005}$ .
1,009,020
( [link] ) Determine if 41,826 is divisible by 2 and 3.
( [link] ) Specify the numerator and denominator of the fraction $\frac{\text{12}}{\text{19}}$ .
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