Understanding Fractions: Additional Practice (Set 1)

One seventh of $63$ is $63 \div 7 = 9$, then take $4$ of those parts.

$4 \times 9 = 36$

So $\frac{4}{7}$ of $63$ is $36$.

Since one fourth is $6$, three fourths is three of those parts.

$3 \times 6 = 18$

So $\frac{3}{4}$ of the number is $18$. (The whole number is $24$, and $\frac{3}{4}$ of $24$ is indeed $18$.)

A fraction equals $1$ when its numerator and denominator are equal. Check each one.

$\frac{10}{10} = 1, \quad \frac{3}{3} = 1, \quad \frac{25}{25} = 1$

But $\frac{8}{1} = 8 \div 1 = 8$, not $1$, so it is the exception.

Count in fifths: $\frac{15}{5} = 3$ and $\frac{20}{5} = 4$, so $\frac{17}{5}$ lands between them.

$\frac{15}{5} < \frac{17}{5} < \frac{20}{5}$

Since $15 < 17 < 20$, the fraction $\frac{17}{5}$ sits between $3$ and $4$.

One third of $12$ slices is $12 \div 3 = 4$ slices, and one fourth of $12$ is $12 \div 4 = 3$ slices, so $4 + 3 = 7$ slices are eaten.

$12 - 7 = 5$

That leaves $5$ slices.

Each fraction falls one unit fraction short of $1$: $\frac{4}{5} = 1 - \frac{1}{5}$ and $\frac{9}{10} = 1 - \frac{1}{10}$. The one missing the smaller piece is closer to $1$.

$\frac{1}{10} < \frac{1}{5} \;\Rightarrow\; \frac{9}{10} > \frac{4}{5}$

Since $\frac{9}{10}$ is only $\frac{1}{10}$ short, it is the greater fraction.

Cut the half-pizza into $4$ equal pieces. If the whole pizza were cut the same way, each half would hold $4$ pieces, so the whole holds $8$ equal pieces.

$\text{each piece} = \frac{1}{8} \text{ of the whole pizza}$

Each friend gets one of those pieces, which is $\frac{1}{8}$ of the whole.

First simplify what you can: $\frac{6}{6} = 1$, while $\frac{8}{9}$ is a proper fraction, so it is less than $1$.

$\frac{6}{6} = 1 > \frac{8}{9}$

Since $\frac{8}{9}$ falls one ninth short of a whole, $\frac{6}{6}$ is greater.

A fraction is improper when its denominator is less than or equal to its numerator.

$\square \le 8$

Of the choices, only $6$ is at most $8$, so $\frac{8}{6}$ is improper (its value is more than $1$). The values $9$, $10$, and $12$ all give proper fractions.

Both amounts are measured in fourths of a liter, so subtract the numerators over the common denominator $4$.

$\frac{3}{4} - \frac{1}{4} = \frac{2}{4}$

Since $3 - 1 = 2$ fourths remain, the leftover is $\frac{2}{4}$ of a liter.

Paperbacks are $\frac{2}{7}$, so the rest are $\frac{5}{7}$. One seventh of $35$ is $35 \div 7 = 5$, so the non-paperbacks are $5 \times 5 = 25$.

$\frac{5}{7} \text{ of } 35 = 25$

So $25$ books are not paperbacks. (Check: $\frac{2}{7}$ of $35$ is $10$ paperbacks, and $10 + 25 = 35$.)

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

$\frac{1}{2} > \frac{1}{4} > \frac{1}{6}$

From greatest to least the order is $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}$.