Mixed Numbers and Improper Fractions: Converting Both Ways

How many $2$s fit into $9$, and what is left?

$4\tfrac{1}{2}$

$9 \div 2 = 4$ remainder $1$, so the answer is $4\tfrac{1}{2}$.

$3\tfrac{2}{5}$

$17 \div 5 = 3$ remainder $2$, so the answer is $3\tfrac{2}{5}$.

$\tfrac{11}{4}$

$2 \times 4 = 8$, then $8 + 3 = 11$. The answer is $\tfrac{11}{4}$.

$\tfrac{31}{6}$

$5 \times 6 = 30$, then $30 + 1 = 31$. The answer is $\tfrac{31}{6}$.

$4$

$20 \div 5 = 4$ remainder $0$, so there is no fraction part. The answer is the whole number $4$.

Convert first, then simplify the leftover fraction.

$3\tfrac{1}{4}$

$26 \div 8 = 3$ remainder $2$, giving $3\tfrac{2}{8}$.

Divide both parts of $\tfrac{2}{8}$ by $2$ to get $\tfrac{1}{4}$, so the answer is $3\tfrac{1}{4}$.

Put both in the same form first.

$3\tfrac{1}{2}$

$\tfrac{13}{4} = 3\tfrac{1}{4}$, and $\tfrac{1}{4}$ is less than $\tfrac{1}{2}$.

So $3\tfrac{1}{2}$ is bigger.

Two and a quarter cups.

$9 \div 4 = 2$ remainder $1$, so $\tfrac{9}{4} = 2\tfrac{1}{4}$.

Fill the cup twice, then once more to the quarter mark.

Is $\tfrac{5}{7}$ bigger or smaller than $1$?

She added the whole number to the top instead of multiplying first. The answer is $\tfrac{23}{7}$.

Each whole is $\tfrac{7}{7}$, so three wholes are $3 \times 7 = 21$ sevenths. Adding the extra $2$ gives $\tfrac{23}{7}$.

Her answer $\tfrac{5}{7}$ is less than one whole, which cannot be right for a number that starts with a $3$.

$4\tfrac{3}{8}$, which is $\tfrac{35}{8}$.

The mixed number is $4\tfrac{3}{8}$.

Converting: $4 \times 8 = 32$, then $32 + 3 = 35$, so the improper fraction is $\tfrac{35}{8}$.