# 4.2 Proper fractions, improper fractions, and mixed numbers  (Page 2/2)

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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.

## Mixed number = natural number + proper fraction

Mixed numbers are the sum of a natural number and a proper fraction. Mixed number = (natural number) + (proper fraction)

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 ⋅)

$5\frac{7}{8}$ means $5+\frac{7}{8}$ and not $5\cdot \frac{7}{8}$ , which means 5 times $\frac{7}{8}$ or 5 multiplied by $\frac{7}{8}$ .

Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers.

## Converting improper fractions to 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.

## Converting an improper fraction to a mixed number

To convert an improper fraction to a mixed number, divide the numerator by the denominator.
1. The whole number part of the mixed number is the quotient.
2. The fractional part of the mixed number is the remainder written over the divisor (the denominator of the improper fraction).

## Sample set a

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 $\frac{\text{104}}{4}=\text{26}$ .

## Practice set a

Convert each improper fraction to its corresponding mixed number.

$\frac{9}{2}$

$4\frac{1}{2}$

$\frac{\text{11}}{3}$

$3\frac{2}{3}$

$\frac{\text{14}}{\text{11}}$

$1\frac{3}{\text{11}}$

$\frac{\text{31}}{\text{13}}$

$2\frac{5}{\text{13}}$

$\frac{\text{79}}{4}$

$\text{19}\frac{3}{4}$

$\frac{\text{496}}{8}$

62

## Converting mixed numbers to improper fractions

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.

## Converting a mixed number to an improper fraction

To convert a mixed number to an improper fraction,
1. Multiply the denominator of the fractional part of the mixed number by the whole number part.
2. To this product, add the numerator of the fractional part.
3. Place this result over the denominator of the fractional part.

## Sample set b

Convert each mixed number to an improper fraction.

$5\frac{7}{8}$

1. Multiply: $8\cdot 5=\text{40}$ .
2. Add: $\text{40}+\text{7}=\text{47}$ .
3. Place 47 over 8: $\frac{\text{47}}{8}$ .

Thus, $5\frac{7}{8}=\frac{\text{47}}{8}$ .

$\text{16}\frac{2}{3}$

1. Multiply: $\text{3}\cdot \text{16}=\text{48}$ .
2. Add: $\text{48}+\text{2}=\text{50}$ .
3. Place 50 over 3: $\frac{\text{50}}{3}$

Thus, $\text{16}\frac{2}{3}=\frac{\text{50}}{3}$

## Practice set b

Convert each mixed number to its corresponding improper fraction.

$8\frac{1}{4}$

$\frac{\text{33}}{4}$

$5\frac{3}{5}$

$\frac{\text{28}}{5}$

$1\frac{4}{\text{15}}$

$\frac{\text{19}}{\text{15}}$

$\text{12}\frac{2}{7}$

$\frac{\text{86}}{7}$

## Exercises

For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number.

$\frac{3}{2}$

improper fraction

$\frac{4}{9}$

$\frac{5}{7}$

proper fraction

$\frac{1}{8}$

$6\frac{1}{4}$

mixed number

$\frac{\text{11}}{8}$

$\frac{1,\text{001}}{\text{12}}$

improper fraction

$\text{191}\frac{4}{5}$

$1\frac{9}{\text{13}}$

mixed number

$\text{31}\frac{6}{7}$

$3\frac{1}{\text{40}}$

mixed number

$\frac{\text{55}}{\text{12}}$

$\frac{0}{9}$

proper fraction

$\frac{8}{9}$

$\text{101}\frac{1}{\text{11}}$

mixed number

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.

$\frac{\text{11}}{6}$

$\frac{\text{14}}{3}$

$4\frac{2}{3}$

$\frac{\text{25}}{4}$

$\frac{\text{35}}{4}$

$8\frac{3}{4}$

$\frac{\text{71}}{8}$

$\frac{\text{63}}{7}$

$9$

$\frac{\text{121}}{\text{11}}$

$\frac{\text{165}}{\text{12}}$

$\text{13}\frac{9}{\text{12}}$ or $\text{13}\frac{3}{\text{4}}$

$\frac{\text{346}}{\text{15}}$

$\frac{5,\text{000}}{9}$

$\text{555}\frac{5}{9}$

$\frac{\text{23}}{5}$

$\frac{\text{73}}{2}$

$\text{36}\frac{1}{2}$

$\frac{\text{19}}{2}$

$\frac{\text{316}}{\text{41}}$

$7\frac{\text{29}}{\text{41}}$

$\frac{\text{800}}{3}$

For the following 15 problems, convert each of the mixed num­bers to its corresponding improper fraction.

$4\frac{1}{8}$

$\frac{\text{33}}{8}$

$1\frac{5}{\text{12}}$

$6\frac{7}{9}$

$\frac{\text{61}}{9}$

$\text{15}\frac{1}{4}$

$\text{10}\frac{5}{\text{11}}$

$\frac{\text{115}}{\text{11}}$

$\text{15}\frac{3}{\text{10}}$

$8\frac{2}{3}$

$\frac{\text{26}}{3}$

$4\frac{3}{4}$

$\text{21}\frac{2}{5}$

$\frac{\text{107}}{5}$

$\text{17}\frac{9}{\text{10}}$

$9\frac{\text{20}}{\text{21}}$

$\frac{\text{209}}{\text{21}}$

$5\frac{1}{\text{16}}$

$\text{90}\frac{1}{\text{100}}$

$\frac{\text{9001}}{\text{100}}$

$\text{300}\frac{\text{43}}{1,\text{000}}$

$\text{19}\frac{7}{8}$

$\frac{\text{159}}{8}$

Why does $0\frac{4}{7}$ not qualify as a mixed number?

See the definition of a mixed number.

Why does 5 qualify as a mixed number?

See the definition of a mixed number.

… because it may be written as $5\frac{0}{n}$ , where $n$ is any positive whole number.

## Calculator problems

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction.

$\text{35}\frac{\text{11}}{\text{12}}$

$\text{27}\frac{5}{\text{61}}$

$\frac{1,\text{652}}{\text{61}}$

$\text{83}\frac{\text{40}}{\text{41}}$

$\text{105}\frac{\text{21}}{\text{23}}$

$\frac{2,\text{436}}{\text{23}}$

$\text{72}\frac{\text{605}}{\text{606}}$

$\text{816}\frac{\text{19}}{\text{25}}$

$\frac{\text{20},\text{419}}{\text{25}}$

$\text{708}\frac{\text{42}}{\text{51}}$

$6,\text{012}\frac{4,\text{216}}{8,\text{117}}$

$\frac{\text{48},\text{803},\text{620}}{8,\text{117}}$

## Exercises for review

( [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] ) Find the least common multiple of 28 and 36.

252

( [link] ) Specify the numerator and denominator of the fraction $\frac{\text{12}}{\text{19}}$ .

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