Question

1. fill in the chart. the first one is done for you.

division expression	unit forms	improper fractions	mixed numbers	standard algorithm (write your answer in whole numbers and fractional units. then check.)

$$\begin{array}{r}1\\frac{1}{3}\\\\3\\overline{)4}\\\\-3\\\\\\hline1\\end{array}$$

$
check
$3 \times 1\frac{1}{3} = 1\frac{1}{3} + 1\frac{1}{3} + 1\frac{1}{3}$
$= 3 + \frac{3}{3}$
$= 3 + 1$
$= 4$ |

b. $\underline{\quad} \div \underline{\quad}$	$\underline{\quad}$ fifths $\div$ 5 = $\underline{\quad}$ fifths		$1\frac{2}{5}$
c. $\underline{\quad} \div \underline{\quad}$	$\underline{\quad}$ halves $\div$ 2 = $\underline{\quad}$ halves			$\

$$\begin{array}{r}\\\\2\\overline{)7}\\\\\\end{array}$$

$ |

d. $7 \div 4$

$\frac{7}{4}$

Explanation:

Step1: Solve part b: Reverse-engineer from mixed number

The mixed number is $1\frac{2}{5}$. Convert to improper fraction:
$1\frac{2}{5} = \frac{1\times5 + 2}{5} = \frac{7}{5}$
This improper fraction equals $\frac{\text{dividend}}{5}$, so dividend = 7. Division expression: $7\div5$.
Unit forms: 7 fifths $\div5$ = $\frac{7}{5}$ (7 fifths).
Standard algorithm:

$$\begin{array}{r} 1\frac{2}{5} \\ 5\enclose{longdiv}{7} \\ \underline{-5} \\ 2 \\ \end{array}$$

Check: $5\times1\frac{2}{5} = 5\times1 + 5\times\frac{2}{5} = 5 + 2 = 7$

Step2: Solve part c: Use given division

Standard algorithm shows $7\div2$.
Unit forms: 7 halves $\div2$ = $\frac{7}{2}$ (7 halves).
Improper fraction: $\frac{7}{2}$
Mixed number: $3\frac{1}{2}$
Check: $2\times3\frac{1}{2} = 2\times3 + 2\times\frac{1}{2} = 6 + 1 = 7$

Step3: Solve part d: Convert improper to mixed

Improper fraction $\frac{7}{4}$ converts to mixed number: $1\frac{3}{4}$.
Unit forms: 7 fourths $\div4$ = $\frac{7}{4}$ (7 fourths).
Standard algorithm:

$$\begin{array}{r} 1\frac{3}{4} \\ 4\enclose{longdiv}{7} \\ \underline{-4} \\ 3 \\ \end{array}$$

Check: $4\times1\frac{3}{4} = 4\times1 + 4\times\frac{3}{4} = 4 + 3 = 7$

Answer:

Division Expression	Unit Forms	Improper Fractions	Mixed Numbers	Standard Algorithm (and Check)

$$\begin{array}{r}1\frac{1}{3} \\ 3\enclose{longdiv}{4} \\ \underline{-3} \\ 1\end{array}$$

<br> Check: $3\times1\frac{1}{3}=4$ |

b. $7\div5$

7 fifths $\div5$ = 7 fifths

$\frac{7}{5}$

$1\frac{2}{5}$

$$\begin{array}{r}1\frac{2}{5} \\ 5\enclose{longdiv}{7} \\ \underline{-5} \\ 2\end{array}$$

<br> Check: $5\times1\frac{2}{5}=7$ |

c. $7\div2$

7 halves $\div2$ = 7 halves

$\frac{7}{2}$

$3\frac{1}{2}$

$$\begin{array}{r}3\frac{1}{2} \\ 2\enclose{longdiv}{7} \\ \underline{-6} \\ 1\end{array}$$

<br> Check: $2\times3\frac{1}{2}=7$ |

d. $7\div4$

7 fourths $\div4$ = 7 fourths

$\frac{7}{4}$

$1\frac{3}{4}$

$$\begin{array}{r}1\frac{3}{4} \\ 4\enclose{longdiv}{7} \\ \underline{-4} \\ 3\end{array}$$

<br> Check: $4\times1\frac{3}{4}=7$ |