Understanding Fractions: Practice

First find how many pages she read. One eighth of $240$ is $240 \div 8 = 30$, so she read $3 \times 30 = 90$ pages.

$240 - 90 = 150$

Subtracting the pages read from the total leaves $150$ pages.

All four fractions have the same numerator $2$, so the one with the smallest denominator is greatest, because smaller denominators make bigger parts.

$\frac{2}{5} > \frac{2}{7} > \frac{2}{9} > \frac{2}{11}$

Fifths are the largest parts here, so $\frac{2}{5}$ is greatest.

Halfway between $0$ and $1$ is the point $\frac{1}{2}$. In twelfths, that is the count where the numerator is half of $12$.

$\frac{1}{2} = \frac{6}{12}$

Six of the twelve equal steps lands at the midpoint.

Together you eat $3 + 4 = 7$ slices out of $10$, so the slices left are $10 - 7 = 3$.

$\text{left} = \frac{3}{10}$

The whole stays $10$ slices; only the count of remaining slices changes.

With equal numerators, the same number of parts is taken from each whole, so the larger parts win. A smaller denominator means fewer, bigger parts.

$\frac{5}{4} > \frac{5}{9}$

So the fraction with denominator $4$ is larger.

The whole jar is $\frac{9}{9}$, and the beads are either red or blue, so the blue fraction is what is left after the red.

$\frac{9}{9} - \frac{4}{9} = \frac{5}{9}$

Since $9 - 4 = 5$ of the nine equal shares are blue, the blue fraction is $\frac{5}{9}$.

If one third of the number is $7$, then the whole is made of $3$ such thirds.

$3 \times 7 = 21$

So the whole number is $21$, and indeed $\frac{1}{3}$ of $21$ is $7$.

Each whole is $6$ sixths, so $2$ wholes is $2 \times 6 = 12$ sixths, plus the $5$ extra sixths.

$12 + 5 = 17 \;\Rightarrow\; \frac{17}{6}$

So the total is $\frac{17}{6}$, an improper fraction between $2$ and $3$.

Half of $\frac{2}{3}$ means splitting the $2$ thirds into two equal shares, which is $1$ third.

$\frac{2}{3} \div 2 = \frac{1}{3}$

Halving a count of two thirds leaves one third, so you need $\frac{1}{3}$ cup.

A proper fraction has a numerator smaller than its denominator.

$\square < 5$

Of the choices, only $3$ is less than $5$, so $\frac{3}{5}$ is proper. The values $5$, $6$, and $7$ would make the fraction equal to or greater than $1$.

Count in fourths: $\frac{8}{4} = 2$ and $\frac{12}{4} = 3$, so $\frac{11}{4}$ lands between those two.

$\frac{8}{4} < \frac{11}{4} < \frac{12}{4}$

Since $8 < 11 < 12$, the fraction $\frac{11}{4}$ sits between $2$ and $3$.

If $\frac{3}{4}$ are filled, then $\frac{1}{4}$ are empty, and one quarter of $28$ is $28 \div 4 = 7$.

$\frac{1}{4} \text{ of } 28 = 7$

So $7$ seats are empty. (As a check, $\frac{3}{4}$ of $28$ is $21$ filled, and $21 + 7 = 28$.)