Fractions

Concept: Equivalent fractions are obtained by multiplying (or dividing) both numerator and denominator by the same non-zero number.

(a) $\frac{1}{3}$
$$\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12}$$

(b) $\frac{1}{4}$
$$\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}$$

(c) $\frac{1}{5}$
$$\frac{1}{5} = \frac{2}{10} = \frac{3}{15} = \frac{4}{20}$$

(d) $\frac{1}{6}$
$$\frac{1}{6} = \frac{2}{12} = \frac{3}{18} = \frac{4}{24}$$

How to generate equivalent fractions: Multiply both the numerator and the denominator of a fraction by the same number (2, 3, 4, …). The value of the fraction does not change.

**A. How many $\frac{1}{6}$s make $\frac{1}{3}$?**

We need to find $n$ such that $n \times \frac{1}{6} = \frac{1}{3}$.
$$\frac{1}{3} = \frac{2}{6}$$
So $n = 2$.
**Answer: 2 pieces of $\frac{1}{6}$ make $\frac{1}{3}$.

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B(a). How many $\frac{1}{8}$s make $\frac{1}{4}$?**

$$\frac{1}{4} = \frac{2}{8}$$
**Answer: 2 pieces of $\frac{1}{8}$ make $\frac{1}{4}$.

B(b). How many $\frac{1}{8}$s make $\frac{1}{2}$?**

$$\frac{1}{2} = \frac{4}{8}$$
**Answer: 4 pieces of $\frac{1}{8}$ make $\frac{1}{2}$.

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C(a). How many $\frac{1}{12}$s make $\frac{1}{2}$?**

$$\frac{1}{2} = \frac{6}{12}$$
**Answer: 6 pieces of $\frac{1}{12}$ make $\frac{1}{2}$.

C(b). How many $\frac{1}{12}$s make $\frac{1}{3}$?**

$$\frac{1}{3} = \frac{4}{12}$$
**Answer: 4 pieces of $\frac{1}{12}$ make $\frac{1}{3}$.**

Concept: A whole = 1. We need combinations of fractions that add up to 1.

**• Using only $\frac{1}{6}$ and $\frac{1}{12}$ pieces:**
One possible way: $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{12} + \frac{1}{12} = \frac{4}{6} + \frac{2}{12} = \frac{8}{12} + \frac{2}{12} + \frac{2}{12}$
Simpler: $4 \times \frac{1}{6} + 4 \times \frac{1}{12} = \frac{4}{6} + \frac{4}{12} = \frac{8}{12} + \frac{4}{12} = \frac{12}{12} = 1$ ✓

Another way: $2 \times \frac{1}{6} + 8 \times \frac{1}{12} = \frac{2}{6} + \frac{8}{12} = \frac{4}{12} + \frac{8}{12} = \frac{12}{12} = 1$ ✓

**• Using $\frac{1}{12}$, $\frac{1}{4}$, and $\frac{1}{2}$ pieces:**
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1 \text{ (but uses only 1/2 and 1/4)}$$
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} = \frac{6}{12} + \frac{3}{12} + \frac{3}{12} = \frac{12}{12} = 1 \checkmark$$

• Using any five pieces of the same size:
Five pieces of $\frac{1}{5}$: $5 \times \frac{1}{5} = 1$ ✓

• Using any seven pieces:
Seven pieces of $\frac{1}{7}$: $7 \times \frac{1}{7} = 1$ ✓
Or: $3 \times \frac{1}{6} + 4 \times \frac{1}{8} = \frac{3}{6} + \frac{4}{8} = \frac{1}{2} + \frac{1}{2} = 1$ ✓ (7 pieces total)

Concept: To find equivalent fractions, multiply both numerator and denominator by the same number.

(a) $\frac{1}{7}$
$$\frac{1}{7} = \frac{1 \times 2}{7 \times 2} = \frac{2}{14}$$
Other answers: $\frac{3}{21},\ \frac{4}{28}$, etc.

(b) $\frac{2}{3}$
$$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$
Other answers: $\frac{6}{9},\ \frac{8}{12}$, etc.

(c) $\frac{3}{4}$
$$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$$
Other answers: $\frac{9}{12},\ \frac{12}{16}$, etc.

(d) $\frac{3}{5}$
$$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$$
Other answers: $\frac{9}{15},\ \frac{12}{20}$, etc.

Concept: Two fractions are equivalent if cross-multiplication gives equal products, i.e., $\frac{a}{b} = \frac{c}{d}$ if $a \times d = b \times c$.

(a) $\frac{2}{3}$ and $\frac{3}{4}$:
$2 \times 4 = 8$ and $3 \times 3 = 9$. Since $8 \neq 9$, they are NOT equivalent. ✗

(b) $\frac{3}{5}$ and $\frac{6}{10}$:
$3 \times 10 = 30$ and $5 \times 6 = 30$. Since $30 = 30$, they are equivalent. ✓

(c) $\frac{4}{12}$ and $\frac{2}{6}$:
$4 \times 6 = 24$ and $12 \times 2 = 24$. Since $24 = 24$, they are equivalent. ✓

(d) $\frac{6}{9}$ and $\frac{1}{3}$:
$6 \times 3 = 18$ and $9 \times 1 = 9$. Since $18 \neq 9$, they are NOT equivalent. ✗

Concept: Multiply or divide both numerator and denominator by the same number to get equivalent fractions.

(a) $\frac{2}{5} = \square$
$$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$$
**Answer: $\frac{4}{10}$** (other answers like $\frac{6}{15}$ are also correct)

(b) $\frac{3}{4} = \square$
$$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$$
**Answer: $\frac{6}{8}$** (other answers like $\frac{9}{12}$ are also correct)

(c) $\frac{4}{7} = \frac{8}{\square}$
Numerator: $4 \times 2 = 8$, so denominator: $7 \times 2 = 14$.
$$\frac{4}{7} = \frac{8}{14}$$
**Answer: $\square = 14$

(d)** $\frac{5}{9} = \frac{25}{\square}$
Numerator: $5 \times 5 = 25$, so denominator: $9 \times 5 = 45$.
$$\frac{5}{9} = \frac{25}{45}$$
**Answer: $\square = 45$**

Concept: When denominators are the same, the fraction with the larger numerator is greater.

(a) $\frac{1}{4}$ ___ $\frac{3}{4}$
Numerators: $1 < 3$, denominators same.
$$\frac{1}{4} < \frac{3}{4}$$

(b) $\frac{3}{5}$ ___ $\frac{4}{5}$
Numerators: $3 < 4$, denominators same.
$$\frac{3}{5} < \frac{4}{5}$$

(c) $\frac{5}{7}$ ___ $\frac{2}{7}$
Numerators: $5 > 2$, denominators same.
$$\frac{5}{7} > \frac{2}{7}$$

(d) $\frac{7}{8}$ ___ $\frac{3}{8}$
Numerators: $7 > 3$, denominators same.
$$\frac{7}{8} > \frac{3}{8}$$

(e) $\frac{5}{10}$ ___ $\frac{6}{10}$
Numerators: $5 < 6$, denominators same.
$$\frac{5}{10} < \frac{6}{10}$$

(f) $\frac{2}{6}$ ___ $\frac{1}{6}$
Numerators: $2 > 1$, denominators same.
$$\frac{2}{6} > \frac{1}{6}$$

Concept: When numerators are the same, the fraction with the smaller denominator is greater (because the whole is divided into fewer, larger parts).

(a) $\frac{3}{8}$ ___ $\frac{3}{7}$
Numerators equal. Denominators: $8 > 7$, so $\frac{1}{8} < \frac{1}{7}$, therefore:
$$\frac{3}{8} < \frac{3}{7}$$

(b) $\frac{4}{9}$ ___ $\frac{4}{10}$
Numerators equal. Denominators: $9 < 10$, so $\frac{1}{9} > \frac{1}{10}$, therefore:
$$\frac{4}{9} > \frac{4}{10}$$

(c) $\frac{2}{7}$ ___ $\frac{2}{5}$
Numerators equal. Denominators: $7 > 5$, so $\frac{1}{7} < \frac{1}{5}$, therefore:
$$\frac{2}{7} < \frac{2}{5}$$

(d) $\frac{5}{7}$ ___ $\frac{5}{6}$
Numerators equal. Denominators: $7 > 6$, so $\frac{1}{7} < \frac{1}{6}$, therefore:
$$\frac{5}{7} < \frac{5}{6}$$

(e) $\frac{6}{9}$ ___ $\frac{6}{10}$
Numerators equal. Denominators: $9 < 10$, so $\frac{1}{9} > \frac{1}{10}$, therefore:
$$\frac{6}{9} > \frac{6}{10}$$

(f) $\frac{7}{9}$ ___ $\frac{7}{11}$
Numerators equal. Denominators: $9 < 11$, so $\frac{1}{9} > \frac{1}{11}$, therefore:
$$\frac{7}{9} > \frac{7}{11}$$