Mixed Numbers: Practice

Divide $23$ by $5$. Five goes into $23$ four times, using $20$, leaving $3$.

$23 \div 5 = 4 \text{ remainder } 3$

The quotient $4$ is the whole part and the remainder $3$ over $5$ is the fraction, so $\tfrac{23}{5} = 4\tfrac{3}{5}$.

Multiply the whole number by the denominator and add the numerator, keeping the denominator $4$.

$5 \times 4 + 3 = 23$

So $5\tfrac{3}{4} = \tfrac{23}{4}$.

Divide $38$ by $7$. Seven goes into $38$ five times, using $35$, leaving $3$.

$38 \div 7 = 5 \text{ remainder } 3$

The quotient $5$ is the whole part and the remainder $3$ over $7$ is the fraction, so $\tfrac{38}{7} = 5\tfrac{3}{7}$.

First divide $20$ by $8$: it goes twice, using $16$, leaving $4$, so $\tfrac{20}{8} = 2\tfrac{4}{8}$. Now simplify the fraction part by its greatest common factor $4$.

$\tfrac{4}{8} = \tfrac{1}{2}$

So in lowest terms $\tfrac{20}{8} = 2\tfrac{1}{2}$.

The denominators already match, so add the whole parts and the fraction parts separately. The wholes give $2 + 1 = 3$, and the fractions give $\tfrac15 + \tfrac35 = \tfrac45$.

$2\tfrac{1}{5} + 1\tfrac{3}{5} = 3 + \tfrac{4}{5} = 3\tfrac{4}{5}$

The fraction part stays in fifths; do not add the denominators.

The denominators match, so subtract the whole parts and the fraction parts separately. The wholes give $4 - 1 = 3$, and the fractions give $\tfrac56 - \tfrac16 = \tfrac46$, which simplifies.

$\tfrac{4}{6} = \tfrac{2}{3}$

So $4\tfrac{5}{6} - 1\tfrac{1}{6} = 3\tfrac{2}{3}$.

Put both in the same form. Convert the mixed number to an improper fraction: $2 \times 3 + 1 = 7$, so $2\tfrac13 = \tfrac73$.

$\tfrac{7}{3} > \tfrac{5}{3}$

With the same denominator, the larger numerator wins, so $2\tfrac13 > \tfrac53$.

Divide $45$ by $6$: it goes seven times, using $42$, leaving $3$, so $\tfrac{45}{6} = 7\tfrac{3}{6}$. Simplify the fraction part by its greatest common factor $3$.

$\tfrac{3}{6} = \tfrac{1}{2}$

So $\tfrac{45}{6} = 7\tfrac{1}{2}$.

Add the whole parts $3 + 1 = 4$ and the fraction parts $\tfrac25 + \tfrac45 = \tfrac65$. The fraction total is more than one whole.

$\tfrac{6}{5} = 1\tfrac{1}{5}$

Carry the $1$: $4 + 1\tfrac15 = 5\tfrac15$.

Add the whole parts $1 + 1 = 2$. For the fractions, the least common denominator of $3$ and $6$ is $6$: $\tfrac23 = \tfrac46$, so $\tfrac46 + \tfrac16 = \tfrac56$.

$1\tfrac{2}{3} + 1\tfrac{1}{6} = 2 + \tfrac{5}{6} = 2\tfrac{5}{6}$

The fraction part is in lowest terms.

Divide $50$ by $9$. Nine goes into $50$ five times, using $45$, leaving $5$.

$50 \div 9 = 5 \text{ remainder } 5$

The quotient $5$ is the whole part and the remainder $5$ over $9$ is the fraction, so $\tfrac{50}{9} = 5\tfrac{5}{9}$.

Convert both to improper fractions: $3\tfrac12 = \tfrac72$ and $1\tfrac34 = \tfrac74$. Dividing means multiplying by the reciprocal of the divisor.

$\tfrac{7}{2} \times \tfrac{4}{7} = \tfrac{7 \times 4}{2 \times 7} = \tfrac{28}{14} = 2$

Only the divisor is flipped, never the first fraction.