Fraction Word Problems: Multiplication and Division

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Question 1 of 6

How many

1 \frac{1}{6}

$1 1 / 6$ sections of rope are there in a total length of

42

$42$ m?

(36)

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Transforming a Fraction from Mixed to Improper

=

$=$

\frac{(c \times A) + b}{c}

$\frac{(\color{#9a00c7}{c}\times\color{#00880A}{A})+\color{#007DDC}{b}}{\color{#9a00c7}{c}}$

First, list down the values stated in the problem

Total length of the rope	$=$ $=$	$42$ $42$ m

Sections it will be cut into	$=$ $=$	$\color{#00880A}{1}\frac{\color{#007DDC}{1}}{\color{#9a00c7}{6}}=\frac{(\color{#9a00c7}{6}\times\color{#00880A}{1})+\color{#007DDC}{1}}{\color{#9a00c7}{6}}=\frac{6+1}{6}=\frac{7}{6}$

Divide the two values

	$42 \div \frac{7}{6}$ $42-:7/6$

$=$ $=$	$\frac{42}{1} \times \frac{6}{7}$ $42/1xx6/7$	Reciprocate the divisor and change the operation to multiplication

$=$ $=$	$\frac{7 \times 6}{1} \times \frac{6}{7 \times 1}$ $(7xx6)/1xx6/(7xx1)$	$42$ $42$ and $7$ $7$ are both multiples of $7$ $7$

$=$ $=$	$\frac{\color{#CC0000}{7}\times6}{1}\times\frac{6}{\color{#CC0000}{7}\times1}$	Cancel diagonally

$=$ $=$	$\frac{6}{1} \times \frac{6}{1}$ $6/1xx6/1$

$=$ $=$	$\frac{36}{1}$ $36/1$

$=$ $=$	$36$ $36$

36

$36$

Question 2 of 6

A race track is

3 \frac{2}{3}

$3 2/3$ km long per lap. How many laps will a racer need to make in order to complete a

77

$77$ km race?

(21) laps

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Transforming a Fraction from Mixed to Improper

=

$=$

\frac{(c \times A) + b}{c}

$\frac{(\color{#9a00c7}{c}\times\color{#00880A}{A})+\color{#007DDC}{b}}{\color{#9a00c7}{c}}$

First, list down the values stated in the problem

Total length of the race	$=$ $=$	$77$ $77$ km

Length of a lap	$=$ $=$	$3 \frac{2}{3}$ $3 2/3$ km

Divide the two values

Start by converting the mixed fraction to an improper fraction

		$77\div\color{#00880A}{3}\frac{\color{#007DDC}{2}}{\color{#9a00c7}{3}}$

	$=$ $=$	$\frac{77}{1}\div\frac{(\color{#9a00c7}{3}\times\color{#00880A}{3})+\color{#007DDC}{2}}{\color{#9a00c7}{3}}$

	$=$ $=$	$\frac{77}{1} \div \frac{9 + 2}{3}$ $77/1-:(9+2)/3$

	$=$ $=$	$\frac{77}{1} \div \frac{11}{3}$ $77/1-:11/3$

Get the reciprocal (flip) of the divisor

\frac{11}{3}

$11/3$ becomes

\frac{3}{11}

$3/11$

Proceed with multiplying this to the dividend.

	$\frac{77}{1} \div$ $77/1-:$ $\frac{11}{3}$ $11/3$

$=$ $=$	$\frac{77}{1} \times$ $77/1xx$ $\frac{3}{11}$ $3/11$

$=$ $=$	$\frac{77\div\color{#CC0000}{11}}{1}\times\frac{3}{11\div\color{#CC0000}{11}}$	Reduce the fractions

$=$ $=$	$\frac{7}{1} \times \frac{3}{1}$ $7/1xx3/1$

$=$ $=$	$\frac{21}{1}$ $21/1$

$=$ $=$	$21$ $21$ laps

21

$21$ laps

Question 3 of 6

3. Question
How many $20$ $20$ -minute periods are there in $5$ $5$ hours?
- (15) periods
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To divide fractions, flip the divisor and then multiply as normal

Method One

Use an illustration to represent the problem

$1$ $1$ circle represents $1$ $1$ hour or $60$ $60$ minutes, so $5$ $5$ circles are drawn.

Since there are three $20$ $20$ minutes in $60$ $60$ minutes, each circle is divided into $3$ $3$ parts.

Count the total number of parts

$5$ $5$ circles $\times 3$ $xx3$ parts $=$ $=$ $15$ $15$

There are $15$ $15$ $20$ $20$ -minute periods in $5$ $5$ hours

$15$ $15$ periods

Method Two

First, list down the values stated in the problem

Total no. of hours $=$ $=$ $5$ $5$ hours

Length of a period
(in hours) $=$ $=$ $20$ $20$ minutes $\div 60$ $-:60$ minutes $= \frac{20}{60} = \frac{1}{3}$ $=20/60=1/3$

Divide the two values

Start by getting the reciprocal (flip) of the divisor

$\frac{1}{3}$ $1/3$ becomes $\frac{3}{1}$ $3/1$

Proceed with multiplying this to the dividend.

$5 \div$ $5-:$ $\frac{1}{3}$ $1/3$

$=$ $=$ $\frac{5}{1} \times$ $5/1xx$ $\frac{3}{1}$ $3/1$

$=$ $=$ $\frac{15}{1}$ $15/1$

$=$ $=$ $15$ $15$

There are $15$ $15$ $20$ $20$ -minute periods in $5$ $5$ hours

$15$ $15$ periods

Question 4 of 6

Bill swims for exactly the same amount of time each day. If he swims for a total of

7 \frac{1}{2}

$7 1/2$ hours in

5

$5$ days, for how long did he swim each day?

1. $3 \frac{1}{2}$ $3 1/2$
2. $1 \frac{1}{2}$ $1 1/2$
3. $1 \frac{1}{3}$ $1 1/3$
4. $2 \frac{1}{3}$ $2 1/3$

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Transforming a Fraction from Mixed to Improper

=

$=$

\frac{(c \times A) + b}{c}

$\frac{(\color{#9a00c7}{c}\times\color{#00880A}{A})+\color{#007DDC}{b}}{\color{#9a00c7}{c}}$

Transforming an Improper to Mixed Fraction

\frac{b}{c} = Q \frac{R}{c}

$\frac{\color{#007DDC}{b}}{\color{#9a00c7}{c}}=\color{#00880A}{Q}\frac{\color{#e65021}{R}}{\color{#9a00c7}{c}}$

(

$($ $b$ $b$

\div

$-:$ $c$ $c$

) =

$)=$ $Q$ $Q$ and $R$ $R$ is the remainder

First, list down the values stated in the problem

Total no. of swimming hours	$=$ $=$	$7 \frac{1}{2}$ $7 1/2$ hours

No. of days	$=$ $=$	$5$ $5$ days

Divide the two values

Start by converting the mixed fraction to an improper fraction

		$\color{#00880A}{7}\frac{\color{#007DDC}{1}}{\color{#9a00c7}{2}}\div5$

	$=$ $=$	$\frac{(\color{#9a00c7}{2}\times\color{#00880A}{7})+\color{#007DDC}{1}}{\color{#9a00c7}{2}}\div\frac{5}{1}$

	$=$ $=$	$\frac{14 + 1}{2} \div \frac{5}{1}$ $(14+1)/2-:5/1$

	$=$ $=$	$\frac{15}{2} \div \frac{5}{1}$ $15/2-:5/1$

Get the reciprocal (flip) of the divisor

\frac{5}{1}

$5/1$ becomes

$1/5$

Proceed with multiplying this to the dividend.

	$15/2-:$ $5/1$

$=$	$15/2xx$ $1/5$

$=$	$\frac{15\div\color{#CC0000}{5}}{2}\times\frac{1}{5\div\color{#CC0000}{5}}$	Reduce the fractions

$=$	$3/2xx1/1$

$=$	$3/2$

Convert the fraction from improper to mixed

Start by dividing the numerator by the denominator

Arrange the numbers for long division

$2$ goes into

$3$ once. So write

$1$ above the line.

Multiply

$1$ to

$2$ and write the answer below

$3$

Subtract

$2$ from

$3$ and write the answer one line below

Since

$2$ cannot go into

$1$ anymore, $1$ is left as the Remainder and $1$ is the Quotient

Substitute values into the given formula

$\frac{\color{#007DDC}{b}}{\color{#9a00c7}{c}}$	$=$	$\color{#00880A}{Q}\frac{\color{#e65021}{R}}{\color{#9a00c7}{c}}$

$\frac{\color{#007DDC}{3}}{\color{#9a00c7}{2}}$	$=$	$\color{#00880A}{1}\frac{\color{#e65021}{1}}{\color{#9a00c7}{2}}$

Bill swam for

$1 1/2$ hours per day.

$1 1/2$ hours

Question 5 of 6

How many

$1 1/3$ m sections of cable can be cut from a

$72$ m roll?

(54) sections

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Transforming a Fraction from Mixed to Improper

$=$

$\frac{(\color{#9a00c7}{c}\times\color{#00880A}{A})+\color{#007DDC}{b}}{\color{#9a00c7}{c}}$

First, list down the values stated in the problem

Total length of the rope	$=$	$72$ m

Sections it will be cut into	$=$	$\color{#00880A}{1}\frac{\color{#007DDC}{1}}{\color{#9a00c7}{3}}=\frac{(\color{#9a00c7}{3}\times\color{#00880A}{1})+\color{#007DDC}{1}}{\color{#9a00c7}{3}}=\frac{3+1}{3}=\frac{4}{3}$

Divide the two values

	$72-:4/3$

$=$	$72/1xx3/4$	Reciprocate the divisor and change the operation to multiplication

$=$	$\frac{72\div\color{#CC0000}{4}}{1}\times\frac{3}{4\div\color{#CC0000}{4}}$	Reduce the fractions

$=$	$18/1xx3/1$	Cancel diagonally

$=$	$54/1$

$=$	$54$

$72$ m roll of cable can have

$54$ sections of

$1 1/3$ m.

$54$

Question 6 of 6

6. Question
A $$1600$ TV has a $1/5$ off discount. Find the following:
- $(i)$ Discount: $$$ (320)
  
  $(ii)$ Sale Price: $$$ (1280)
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To multiply fractions, simply multiply the numerators and denominators separately.

$(i)$ Find the discount price of the TV.

First, list down the values stated in the problem

Original Price $=$ $$1600$

Discount Rate $=$ $1/5$

Multiply the two values

$1600xx1/5$

$=$ $1600/1xx1/5$

$=$ $1600/5$

$=$ $1600-:5$

$=$ $$320$

The discount price is $$320$

$(ii)$ Find the sale price of the TV.

First, list down the known values in the problem

Original Price $=$ $$1600$

Discount Price $=$ $$320$

Subtract the two values

Original price $-$ Discount price $=$ Sale price

$$1600$ $-$ $$320$ $=$ $$1280$

The sale price is $$1280$

$(i) $320$

$(ii) $1280$

Fraction Word Problems: Multiplication and Division

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Quiz summary

Information

1. Question

Hint

Correct!

Transforming a Fraction from Mixed to Improper

2. Question

Hint

Keep Going!

Transforming a Fraction from Mixed to Improper

3. Question

Hint

Fantastic!

4. Question

Hint

Excellent!

Transforming a Fraction from Mixed to Improper

Transforming an Improper to Mixed Fraction

5. Question

Hint

Nice Job!

Transforming a Fraction from Mixed to Improper

6. Question

Hint

Well Done!

Quizzes

Total no. of hours	$=$ $=$	$5$ $5$ hours

Length of a period (in hours)	$=$ $=$	$20$ $20$ minutes $\div 60$ $-:60$ minutes $= \frac{20}{60} = \frac{1}{3}$ $=20/60=1/3$

		$5 \div$ $5-:$ $\frac{1}{3}$ $1/3$

	$=$ $=$	$\frac{5}{1} \times$ $5/1xx$ $\frac{3}{1}$ $3/1$

	$=$ $=$	$\frac{15}{1}$ $15/1$

	$=$ $=$	$15$ $15$

Original price	$-$	Discount price	$=$	Sale price
$$1600$	$-$	$$320$	$=$	$$1280$