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}

Concept: On a number line, divide each unit (whole) into equal parts equal to the denominator. Count the required number of parts from 0.

(a) $\frac{2}{3}$ and $\frac{5}{3}$:
- Divide each whole into 3 equal parts.
- $\frac{2}{3}$: mark 2 parts from 0 → lies between 0 and 1.
- $\frac{5}{3}$: mark 5 parts from 0 → $\frac{5}{3} = 1\frac{2}{3}$, lies between 1 and 2.

(b) $\frac{3}{4}$ and $\frac{5}{4}$:
- Divide each whole into 4 equal parts.
- $\frac{3}{4}$: mark 3 parts from 0 → lies between 0 and 1.
- $\frac{5}{4}$: mark 5 parts from 0 → $\frac{5}{4} = 1\frac{1}{4}$, lies between 1 and 2.

(c) $\frac{4}{8}$ and $\frac{9}{8}$:
- Divide each whole into 8 equal parts.
- $\frac{4}{8}$: mark 4 parts from 0 → $\frac{4}{8} = \frac{1}{2}$, lies exactly at the midpoint between 0 and 1.
- $\frac{9}{8}$: mark 9 parts from 0 → $\frac{9}{8} = 1\frac{1}{8}$, lies just after 1.

Concept: A fraction $\frac{a}{b}$ is greater than 1 (a whole) when the numerator is greater than the denominator (a > b).

Rule:
- If numerator > denominator → fraction > 1 (improper fraction, greater than a whole)
- If numerator $=$ denominator → fraction $= 1$
- If numerator < denominator → fraction < 1

Examples:
- $\frac{5}{3}$: 5 > 3 → greater than 1 ✓ (circle it)
- $\frac{4}{4}$: $4 = 4$ → equal to 1 (not greater)
- $\frac{3}{5}$: 3 < 5 → less than 1 (do not circle)
- $\frac{7}{4}$: 7 > 4 → greater than 1 ✓ (circle it)

Reasoning: When the numerator is larger than the denominator, we have more parts than needed to make one whole, so the fraction is greater than 1.

Concept: Compare each fraction to 1.
- Fraction > 1 if numerator > denominator.
- Fraction $= 1$ if numerator $=$ denominator.
- Fraction < 1 if numerator < denominator.
Then compare the two fractions based on their relation to 1.

(a) $\frac{8}{7}$ ___ $\frac{9}{15}$
$\frac{8}{7}$: 8 > 7 → greater than 1.
$\frac{9}{15}$: 9 < 15 → less than 1.
Since one is > 1 and the other is < 1:
\frac{8}{7} > \frac{9}{15}

(b) $\frac{13}{20}$ ___ $\frac{17}{15}$
$\frac{13}{20}$: 13 < 20 → less than 1.
$\frac{17}{15}$: 17 > 15 → greater than 1.
\frac{13}{20} < \frac{17}{15}

(c) $\frac{7}{6}$ ___ $\frac{8}{8}$
$\frac{7}{6}$: 7 > 6 → greater than 1.
$\frac{8}{8}$: $8 = 8$ → equal to 1.
\frac{7}{6} > \frac{8}{8}

(d) $\frac{6}{6}$ ___ $\frac{19}{12}$
$\frac{6}{6} = 1$.
$\frac{19}{12}$: 19 > 12 → greater than 1.
\frac{6}{6} < \frac{19}{12}

(e) $\frac{12}{9}$ ___ $\frac{4}{5}$
$\frac{12}{9}$: 12 > 9 → greater than 1.
$\frac{4}{5}$: 4 < 5 → less than 1.
\frac{12}{9} > \frac{4}{5}

(f) $\frac{15}{5}$ ___ $\frac{16}{4}$
$\frac{15}{5} = 3$ (both greater than 1).
$\frac{16}{4} = 4$ (both greater than 1).
Since 3 < 4:
\frac{15}{5} < \frac{16}{4}

Concept: A fraction $\frac{a}{b} = \frac{1}{2}$ when $2a = b$, i.e., the denominator is exactly twice the numerator.

Rule: $\frac{a}{b} = \frac{1}{2}$ if $b = 2 \times a$.

Examples to check:
- $\frac{2}{4}$: $4 = 2 \times 2$ ✓ → equal to $\frac{1}{2}$ (circle)
- $\frac{3}{6}$: $6 = 2 \times 3$ ✓ → equal to $\frac{1}{2}$ (circle)
- $\frac{4}{8}$: $8 = 2 \times 4$ ✓ → equal to $\frac{1}{2}$ (circle)
- $\frac{5}{10}$: $10 = 2 \times 5$ ✓ → equal to $\frac{1}{2}$ (circle)
- $\frac{6}{12}$: $12 = 2 \times 6$ ✓ → equal to $\frac{1}{2}$ (circle)

(Circle whichever fractions from the image satisfy this rule.)

Concept: A fraction \frac{a}{b} < \frac{1}{2} when 2a < b, i.e., twice the numerator is less than the denominator.

Rule: \frac{a}{b} < \frac{1}{2} if 2 \times a < b.

Examples:
- $\frac{3}{8}$: 2 \times 3 = 6 < 8 ✓ → less than $\frac{1}{2}$ (circle)
- $\frac{2}{9}$: 2 \times 2 = 4 < 9 ✓ → less than $\frac{1}{2}$ (circle)
- $\frac{4}{7}$: 2 \times 4 = 8 > 7 → greater than $\frac{1}{2}$ (do not circle)
- $\frac{1}{3}$: 2 \times 1 = 2 < 3 ✓ → less than $\frac{1}{2}$ (circle)

Reasoning: If you double the numerator and it is still less than the denominator, the fraction is less than half.

Concept: Compare each fraction to $\frac{1}{2}$: if 2 \times \text{numerator} < \text{denominator}, fraction < \frac{1}{2}; if 2 \times \text{numerator} > \text{denominator}, fraction > \frac{1}{2}.

(a) $\frac{2}{9}$ and $\frac{4}{7}$
$\frac{2}{9}$: 2 \times 2 = 4 < 9 → less than $\frac{1}{2}$.
$\frac{4}{7}$: 2 \times 4 = 8 > 7 → greater than $\frac{1}{2}$.
\frac{2}{9} < \frac{4}{7}

(b) $\frac{11}{14}$ and $\frac{7}{20}$
$\frac{11}{14}$: 2 \times 11 = 22 > 14 → greater than $\frac{1}{2}$.
$\frac{7}{20}$: 2 \times 7 = 14 < 20 → less than $\frac{1}{2}$.
\frac{11}{14} > \frac{7}{20}

(c) $\frac{5}{7}$ and $\frac{3}{9}$
$\frac{5}{7}$: 2 \times 5 = 10 > 7 → greater than $\frac{1}{2}$.
$\frac{3}{9}$: 2 \times 3 = 6 < 9 → less than $\frac{1}{2}$.
\frac{5}{7} > \frac{3}{9}

(d) $\frac{6}{7}$ and $\frac{4}{10}$
$\frac{6}{7}$: 2 \times 6 = 12 > 7 → greater than $\frac{1}{2}$.
$\frac{4}{10}$: 2 \times 4 = 8 < 10 → less than $\frac{1}{2}$.
\frac{6}{7} > \frac{4}{10}

(e) $\frac{9}{17}$ and $\frac{3}{15}$
$\frac{9}{17}$: 2 \times 9 = 18 > 17 → slightly greater than $\frac{1}{2}$.
$\frac{3}{15}$: 2 \times 3 = 6 < 15 → less than $\frac{1}{2}$.
\frac{9}{17} > \frac{3}{15}

(f) $\frac{7}{12}$ and $\frac{3}{11}$
$\frac{7}{12}$: 2 \times 7 = 14 > 12 → greater than $\frac{1}{2}$.
$\frac{3}{11}$: 2 \times 3 = 6 < 11 → less than $\frac{1}{2}$.
\frac{7}{12} > \frac{3}{11}

(g) $\frac{1}{3}$ and $\frac{5}{9}$
$\frac{1}{3}$: 2 \times 1 = 2 < 3 → less than $\frac{1}{2}$.
$\frac{5}{9}$: 2 \times 5 = 10 > 9 → greater than $\frac{1}{2}$.
\frac{1}{3} < \frac{5}{9}

(h) $\frac{3}{9}$ and $\frac{4}{7}$
$\frac{3}{9}$: 2 \times 3 = 6 < 9 → less than $\frac{1}{2}$.
$\frac{4}{7}$: 2 \times 4 = 8 > 7 → greater than $\frac{1}{2}$.
\frac{3}{9} < \frac{4}{7}

Given: Length of one ant $= \frac{1}{4}$ cm. Number of ants $= 16$.

Step 1: Total length $= 16 \times \frac{1}{4}$ cm

Step 2:
$$16 \times \frac{1}{4} = \frac{16}{4} = 4 \text{ cm}$$

Using the number line: Mark $\frac{1}{4}$ intervals. Starting from 0, count 16 jumps of $\frac{1}{4}$ each:
$$\underbrace{\frac{1}{4} + \frac{1}{4} + \cdots + \frac{1}{4}}_{16 \text{ times}} = \frac{16}{4} = 4$$
You land on 4 on the number line.

Answer: The total length of 16 ants in a line is $4$ cm.