Fractions: Understanding Parts of a Whole

B · How many equal parts the whole is divided into

The denominator (bottom number) names the piece size — it says the whole was cut into 8 equal parts. The numerator 5 counts how many are taken.

C · $\frac{4}{6}$

The whole was cut into 6 equal slices (denominator 6) and 4 were taken (numerator 4), giving $\frac{4}{6}$, which simplifies to $\frac{2}{3}$.

A · $\frac{1}{3}$

For unit fractions (numerator 1), a smaller denominator means larger pieces. Since 3 is the smallest denominator, $\frac{1}{3}$ is the largest.

D · $\frac{3}{5}$

A proper fraction has numerator less than denominator, so its value is below 1. Only $\frac{3}{5}$ satisfies this; $\frac{7}{5}$ and $\frac{5}{5}$ are improper and $1\frac{2}{5}$ is a mixed number.

A · $0.25$

To convert a fraction to a decimal, divide numerator by denominator: $1 \div 4 = 0.25$.

A · True

A fraction $\frac{a}{b}$ requires the whole to be divided into $b$ **equal** parts; unequal parts make the fraction meaningless.

A · $\frac{8}{12}$

Multiplying numerator and denominator of $\frac{2}{3}$ by 4 gives $\frac{8}{12}$. The other options cannot be obtained by multiplying or dividing both parts by the same number.

A · The denominators differ, so the pieces are not the same size

Halves and thirds are different-sized pieces, so they must first be rewritten with a common denominator: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. Also, $\frac{2}{5}$ is smaller than $\frac{1}{2}$ alone, which is impossible for a sum.

A · $\frac{8}{15}$

Multiply numerators together and denominators together: $2 \times 4 = 8$ and $3 \times 5 = 15$, giving $\frac{8}{15}$. No common denominator is needed for multiplication.

C · $1\frac{2}{5}$

Divide 7 by 5: quotient 1 becomes the whole number, remainder 2 goes over the same denominator 5, giving $1\frac{2}{5}$.

C · $\frac{3}{4}$

The HCF of 18 and 24 is 6. Dividing both by 6 gives $\frac{3}{4}$, which is in simplest form because 3 and 4 share no common factor other than 1.

C · $2$

Dividing by $\frac{1}{4}$ is the same as multiplying by its reciprocal $\frac{4}{1}$: $\frac{1}{2} \times 4 = 2$. This makes sense because there are two quarters in a half.

A · $\frac{3}{8}$ is smaller because eighths are thinner than fourths

Both take 3 pieces, but eighths are half the size of fourths, so $\frac{3}{8}$ is smaller. A bigger denominator means smaller pieces.

D · $20\%$

Convert the fraction to a decimal first: $1 \div 5 = 0.2$. Then multiply by 100 to get the percentage: $0.2 \times 100 = 20\%$.

A · Both A and R are true, and R correctly explains A.

$\frac{1}{2}$ becomes $\frac{50}{100}$ by multiplying both parts by 50, so A is true. The equivalence rule stated in R is exactly the reason A holds, so R explains A.

C · I and II only

Numerator 9 exceeds denominator 4, so I is true. Dividing gives quotient 2 and remainder 1, so $\frac{9}{4} = 2\frac{1}{4}$ and II is true. Since $2\frac{1}{4}$ is greater than 2, III is false — it lies between 2 and 3.

B · $\frac{3}{8}$

To find half of $\frac{3}{4}$, multiply by $\frac{1}{2}$: $\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$.

B · $\frac{1}{3}$

First, convert $\frac{2}{3}$ to have a common denominator with $\frac{5}{6}$: $\frac{2}{3} = \frac{4}{6}$. Then, $\frac{5}{6} - \frac{4}{6} = \frac{1}{6}$.