Understanding Fractions: Additional Practice (Set 2)

One fifth of $45$ is $45 \div 5 = 9$, then take $3$ of those parts.

$3 \times 9 = 27$

So $\frac{3}{5}$ of $45$ is $27$.

Winning $\frac{5}{8}$ means losing the other $\frac{3}{8}$. One eighth of $24$ is $24 \div 8 = 3$, so the losses are $3 \times 3 = 9$.

$\frac{3}{8} \text{ of } 24 = 9$

So the team lost $9$ games. (Check: $\frac{5}{8}$ of $24$ is $15$ wins, and $15 + 9 = 24$.)

A fraction equals a whole number when the numerator is a multiple of the denominator. Check each by dividing.

$\frac{15}{3} = 5, \quad \frac{8}{1} = 8, \quad \frac{12}{12} = 1$

But $\frac{9}{4} = 9 \div 4$ is not whole, since $4$ goes into $9$ twice with $1$ left over.

A proper fraction has a denominator larger than its numerator.

$d > 12$

The smallest whole number greater than $12$ is $13$, giving $\frac{12}{13}$, which is just under $1$.

These are unit fractions, all with numerator $1$, so the larger the denominator, the smaller the fraction.

$\frac{1}{4} < \frac{1}{3} < \frac{1}{2}$

From least to greatest the order is $\frac{1}{4}, \frac{1}{3}, \frac{1}{2}$, since fourths are the smallest pieces and halves the largest.

Each whole is $8$ eighths, so $3$ wholes is $3 \times 8 = 24$ eighths, plus the $2$ extra eighths.

$24 + 2 = 26 \;\Rightarrow\; \frac{26}{8}$

So the total is $\frac{26}{8}$, an improper fraction between $3$ and $4$.

Each fraction is one unit fraction short of $1$: $\frac{5}{6} = 1 - \frac{1}{6}$ and $\frac{7}{8} = 1 - \frac{1}{8}$. The one missing the smaller piece is closer to $1$, hence larger.

$\frac{1}{8} < \frac{1}{6} \;\Rightarrow\; \frac{7}{8} > \frac{5}{6}$

Since $\frac{7}{8}$ falls only $\frac{1}{8}$ short while $\frac{5}{6}$ falls $\frac{1}{6}$ short, $\frac{7}{8}$ is greater.

If two fifths is $8$, then one fifth is half of that, $8 \div 2 = 4$. The whole is $5$ fifths.

$5 \times 4 = 20$

So the whole number is $20$, and $\frac{2}{5}$ of $20$ is indeed $8$.

A zero numerator means you took none of the parts, which is a valid amount of $0$. A zero denominator means dividing by $0$, which has no answer.

$\frac{0}{5} = 0 \div 5 = 0, \qquad \frac{5}{0} = 5 \div 0 \;\to\; \text{undefined}$

So only the denominator being $0$ causes a problem.

One ninth of $72$ is $72 \div 9 = 8$, then take $5$ of those parts.

$5 \times 8 = 40$

So $\frac{5}{9}$ of $72$ is $40$.

Both lengths are measured in eighths of a meter, so subtract the numerators over the common denominator $8$.

$\frac{7}{8} - \frac{3}{8} = \frac{4}{8}$

Since $7 - 3 = 4$ eighths remain, the leftover length is $\frac{4}{8}$ of a meter.

Sharing equally is division: each friend gets $7 \div 3$, and the fraction bar means divide.

$7 \div 3 = \frac{7}{3}$

So each friend gets $\frac{7}{3}$ of a sandwich, which is two whole sandwiches and one third more.