Understanding Fractions: Practice

Each whole bar holds $4$ fourths, and there are $2$ bars.

$2 \times 4 = 8$

So $2$ wholes is $\frac{8}{4}$, which is $8$ fourths.

The whole group is $12$ marbles, so $12$ is the denominator. The number that are red, $5$, is the numerator.

$\text{red} = \frac{5}{12}$

Count in fourths: $\frac{4}{4} = 1$ uses four of them, and $\frac{7}{4}$ is three more fourths past that.

$\frac{7}{4} = 1 + \frac{3}{4}$

Since $\frac{3}{4}$ is less than a whole, $\frac{7}{4}$ lands between $1$ and $2$.

Each whole is $3$ thirds, and there are $2$ wholes, so there are $2 \times 3 = 6$ thirds.

$2 = \frac{6}{3}$

This is an improper fraction equal to the whole number $2$.

One fifth of $30$ is $30 \div 5 = 6$, then take $2$ of those parts.

$2 \times 6 = 12$

So $\frac{2}{5}$ of $30$ is $12$.

The bar is divided into $8$ equal parts, and $5$ are shaded.

$\frac{\text{shaded}}{\text{equal parts}} = \frac{5}{8}$

Both fractions count sixths, so the one with more parts is larger. Compare the numerators: $5$ against $1$.

$\frac{5}{6} > \frac{1}{6}$

Five sixths is more than one sixth.

Sharing equally is division: each child gets $5 \div 4$, and the bar means divide.

$5 \div 4 = \frac{5}{4}$

Each child gets $\frac{5}{4}$ of a bar, which is one whole bar and one fourth more.

All three share the denominator $8$, so order them by their numerators from smallest to largest.

$1 < 3 < 5 \;\Rightarrow\; \frac{1}{8} < \frac{3}{8} < \frac{5}{8}$

One sixth of $18$ is $18 \div 6 = 3$, then take $5$ of those parts.

$5 \times 3 = 15$

So $\frac{5}{6}$ of $18$ is $15$.

Both fractions have numerator $3$, so compare the denominators backwards: the larger denominator gives the smaller fraction. Tenths are smaller parts than fourths.

$\frac{3}{10} < \frac{3}{4}$

A fraction equals $1$ when you take all the parts the whole was cut into, which happens when the top and bottom match.

$\frac{n}{n} = 1$

So the numerator must equal the denominator, as in $\frac{3}{3}$ or $\frac{8}{8}$.