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Tearrik was gifted an heirloom by his grandfather. It is a handmade kimono.

Tearrik wants to make a box to hold this beautiful kimono. He plans to cut a piece of wood that is $5$ feet long. He wants the cuts to create equal parts.

External credits: textures.com

Solution

a Tearrik plans to cut the piece of wood into parts that are each $\frac{4}{5}$ feet long.

External credits: textures.com

He wants to know the number of smaller pieces of wood he can produce from the larger piece of wood. Divide the length of the larger piece of wood by the length desired for the smaller pieces to determine that number.

$$\begin{array}{c}5÷\frac{4}{5}\end{array}$$

Dividing a whole number by a fraction is the same as multiplying that whole number by the reciprocal of the fraction. Recall the fact that all whole numbers are fractions whose denominator is $1.$

$5÷\frac{4}{5}$

Rewrite

Rewrite $5$ as $\frac{5}{1}$

$\frac{5}{1}÷\frac{4}{5}$

DivFracByFracDiv

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

$\frac{5}{1}\cdot \frac{5}{4}$

MultFrac

Multiply fractions

$\frac{5\cdot 5}{1\cdot 4}$

Multiply

Multiply

$\frac{25}{4}$

The fraction solved for is an improper fraction. Write it as a mixed number for making sense of how many pieces of wood there are.

$\frac{25}{4}$

WriteSum

Write as a sum

$\frac{24+1}{4}$

WriteSumFrac

Write as a sum of fractions

$\frac{24}{4}+\frac{1}{4}$

CalcQuot

Calculate quotient

$6+\frac{1}{4}$

Rewrite

Rewrite $6+\frac{1}{4}$ as $6\frac{1}{4}$

$6\frac{1}{4}$

The quotient is $6\frac{1}{4}.$ This means that Tearrik can produce $6$ smaller pieces of $\frac{4}{5}$ feet each from the original $5\text{-}$foot piece.

b In Part A, dividing $5$ into $\frac{4}{5}$ was found to be $6\frac{1}{4}.$

$$\begin{array}{c}5÷\frac{4}{5}=6\frac{1}{4}\end{array}$$

This finding is interpreted as Tearrik getting six $\frac{4}{5}\text{-}$foot pieces. The remaining piece is $\frac{1}{4}$ of a $\frac{4}{5}\text{-}$foot piece. The length of the remaining piece of wood can be found by multiplying these fractions.

$\frac{1}{4}\times \frac{4}{5}$

MultFrac

Multiply fractions

$\frac{1\cdot 4}{4\cdot 5}$

CancelCommonFac

Cancel out common factors

$\frac{1\cdot \cancel{4}}{\cancel{4}\cdot 5}$

SimpQuot

Simplify quotient

$\frac{1}{5}$

The length of the remaining piece of wood is a $\frac{1}{5}$ of one foot.

Alternative Solution

Use a Diagram

A diagram can be used to model the division of $5$ by $\frac{4}{5}.$ Divide each foot of the $5\text{-}$foot-long wood into $5$ equal pieces.

External credits: textures.com

Notice that each of the smaller parts represents a $\frac{1}{5}$ of a foot. Then determine how many of the $\frac{4}{5}\text{-}$foot-long pieces are contained within the wood.

External credits: textures.com

There are $6$ of them. The length of the remaining part is a $\frac{1}{5}$ of a foot. Note that the remaining part is also $\frac{1}{4}$ of $\frac{4}{5}.$ This confirms that the result found algebraically is correct.

Tearrik is excited about making the box. A problem arises, however. He realizes a bit of paint would look cool but he does not have any in his home. Tearrik is full of energy and starts to run to the nearest paint shop.

External credits: macrovector

Tearrik runs $\frac{2}{3}$ of the way from the garage to the nearest paint shop.

Solution

a The distance Tearrik ran is given. He ran $\frac{2}{3}$ of the way to the paint shop. This distance is equal to $\frac{3}{5}$ of a mile.

The distance from the house to the paint shop is missing. That distance can be determined by finding $\text{two-thirds}$ of $\text{what}$ $\text{number}$ is $\text{three}$ $\text{fifths}$.

$$\begin{array}{c}\frac{2}{3}\text{ of }\text{what number}\text{ is }\frac{3}{5}?\end{array}$$

This question can be mathematically expressed as follows.

$$\begin{array}{c}\frac{2}{3}\times \boxed{}=\frac{3}{5}\end{array}$$

Now, this multiplication problem can be written as a division problem.

$$\begin{array}{c}\frac{2}{3}\times \boxed{}=\frac{3}{5}\iff \frac{3}{5}÷\frac{2}{3}=\boxed{}\end{array}$$

The quotient of this division represents the distance to the paint shop. Consider that dividing a fraction by a fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.

$\frac{3}{5}÷\frac{2}{3}$

DivFracByFracDiv

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

$\frac{3}{5}\cdot \frac{3}{2}$

MultFrac

Multiply fractions

$\frac{3\cdot 3}{5\cdot 2}$

Multiply

Multiply

$\frac{9}{10}$

The distance to the paint shop is $\frac{9}{10}$ miles.

b The amount of blue paint Tearrik bought is a given, $\frac{3}{4}$ gallons. He poured that amount evenly into $6$ cups. The diagram illustrates the total amount of paint and the unknown amount per cup.

Dividing $\frac{3}{4}$ by $6$ gives how many gallons of paint each cup contains.

$$\begin{array}{c}\frac{3}{4}÷6\end{array}$$

This is a division of a fraction by a whole number. That means the whole number should be written as a fraction to calculate the quotient.

$$\begin{array}{c}\frac{3}{4}÷\frac{6}{1}\end{array}$$

Now, the steps performed when dividing two fractions can be followed.

$\frac{3}{4}÷\frac{6}{1}$

DivFracByFracDiv

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

$\frac{3}{4}\cdot \frac{1}{6}$

MultFrac

Multiply fractions

$\frac{3\cdot 1}{4\cdot 6}$

Multiply

Multiply

$\frac{3}{24}$

The number three is a common factor between the denominator and numerator of the obtained fraction. This fact can be used to simplify the fraction.

$$\begin{array}{rl}\frac{3}{24}& =\frac{3\cdot 1}{3\cdot 8}\\ & \Updownarrow \\ \frac{3}{24}& =\frac{1}{8}\end{array}$$

This means that Tearrik pours $\frac{1}{8}$ gallon into each cup.

Tearrik now has $6$ pieces of wood. Each piece has a length of $\frac{4}{5}$ feet. The total area of the pieces is $1\frac{3}{5}$ square feet.

External credits: textures.com

Solution

Start by finding the length of the greater rectangle. The result of multiplying the length of a small wood piece by $6,$ because there are $6$ pieces, will give the length of the rectangle.

$\frac{4}{5}\times 6$

MoveRightFacToNum

$\frac{a}{c}\cdot b=\frac{a\cdot b}{c}$

$\frac{4\cdot 6}{5}$

Multiply

Multiply

$\frac{24}{5}$

The length of the rectangle is $\frac{24}{5}$ feet.

External credits: textures.com

Recall the formula for the area of a rectangle. It is the rectangle's width times its length.

$$\begin{array}{c}\text{Area}=\text{Width}\times \text{Length}\end{array}$$

Here, the area and length of the rectangle are already known. Its width is what needs to be found. At this point of the process, it is helpful to rearrange the formula to isolate the width to one side.

$$\begin{array}{c}\text{Width}=\text{Area}÷\text{Length}\end{array}$$

The width can then be calculated using the known values.

$$\begin{array}{c}\text{Width}=1\frac{3}{5}÷\frac{24}{5}\end{array}$$

The expression on the right-hand side is a division of a mixed number by a fraction. The mixed number should be converted into an improper fraction.

$1\frac{3}{5}÷\frac{24}{5}$

▼

Write mixed number as a fraction

MixedToFrac

$a\frac{b}{c}=\frac{a\cdot c+b}{c}$

$\frac{1\cdot 5+3}{5}÷\frac{24}{5}$

MultByOne

$a\cdot 1=a$

$\frac{5+3}{5}÷\frac{24}{5}$

AddFrac

Add fractions

$\frac{8}{5}÷\frac{24}{5}$

Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.

$\frac{8}{5}÷\frac{24}{5}$

DivFracByFracDiv

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

$\frac{8}{5}\cdot \frac{5}{24}$

MultFrac

Multiply fractions

$\frac{8\cdot 5}{5\cdot 24}$

SplitIntoFactors

Split into factors

$\frac{8\cdot 5}{5\cdot 8\cdot 3}$

CancelCommonFac

Cancel out common factors

$\frac{\cancel{8}\cdot \cancel{5}}{\cancel{5}\cdot \cancel{8}\cdot 3}$

SimpQuot

Simplify quotient

$\frac{1}{3}$

The width of the rectangle is $\frac{1}{3}$ of a foot. This also represents the width of each small rectangle.

External credits: textures.com

Tearrik realizes that he cannot create a box as he imagined. He asks his mom for help. Together, they cut two of the $\frac{4}{5}\text{-}$foot-long pieces of wood into squares. They manage to form a box by putting the pieces together. After that, they painted the box to match the kimono.