Fractions

Given: 3 guavas together weigh 1 kg and all are of the same size.

Concept: When 1 whole unit is divided equally among 3 parts, each part = $\frac{1}{3}$ of the whole.

$$\text{Weight of each guava} = 1 \div 3 = \frac{1}{3} \text{ kg}$$

Answer: Each guava will roughly weigh $\dfrac{1}{3}$ kg.

Given: 1 kg of rice packed equally into 4 packets.

Concept: When 1 whole unit is divided equally into 4 parts, each part = $\frac{1}{4}$.

$$\text{Weight of each packet} = 1 \div 4 = \frac{1}{4} \text{ kg}$$

Answer: The weight of each packet is $\dfrac{1}{4}$ kg.

Given: 3 glasses shared equally among 4 friends.

Concept: Equal sharing gives a fraction = $\frac{\text{number of glasses}}{\text{number of friends}}$.

$$\text{Each friend's share} = 3 \div 4 = \frac{3}{4} \text{ glass}$$

Answer: Each one drank $\dfrac{3}{4}$ glass of sugarcane juice.

Given: Big fish = $\dfrac{1}{2}$ kg, Small fish = $\dfrac{1}{4}$ kg.

Concept: To add fractions, convert to the same denominator.

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

$$\text{Total weight} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \text{ kg}$$

Answer: Together they weigh $\dfrac{3}{4}$ kg.

First, convert each fraction word to a number:
- Quarter $= \dfrac{1}{4} = 0.25$
- Half $= \dfrac{1}{2} = 0.5$
- Three quarters $= \dfrac{3}{4} = 0.75$
- One and a quarter $= 1\dfrac{1}{4} = 1.25$
- One and a half $= 1\dfrac{1}{2} = 1.5$
- Two and a half $= 2\dfrac{1}{2} = 2.5$

Arranging from smallest to biggest:
\frac{1}{4} < \frac{1}{2} < \frac{3}{4} < 1\frac{1}{4} < 1\frac{1}{2} < 2\frac{1}{2}

Answer: Quarter, Half, Three quarters, One and a quarter, One and a half, Two and a half.

Given: The whole chikki is broken into 2 equal pieces.

Concept: Each equal part of a whole divided into $n$ parts $= \dfrac{1}{n}$.

$$\text{Each piece} = \frac{1}{2} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{2}$ of the whole chikki.

Given: The whole chikki is broken into 4 equal pieces.

$$\text{Each piece} = \frac{1}{4} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{4}$ of the whole chikki.

Given: The whole chikki is broken into 3 equal pieces.

$$\text{Each piece} = \frac{1}{3} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{3}$ of the whole chikki.

Given: The whole chikki is broken into 5 equal pieces.

$$\text{Each piece} = \frac{1}{5} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{5}$ of the whole chikki.

Given: The whole chikki is broken into 6 equal pieces.

$$\text{Each piece} = \frac{1}{6} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{6}$ of the whole chikki.

Given: The whole chikki is broken into 8 equal pieces.

$$\text{Each piece} = \frac{1}{8} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{8}$ of the whole chikki.

Given: The whole chikki is broken into 9 equal pieces.

$$\text{Each piece} = \frac{1}{9} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{9}$ of the whole chikki.

Given: The whole chikki is broken into 10 equal pieces.

$$\text{Each piece} = \frac{1}{10} \text{ of the whole chikki}$$

Answer: Each piece is $\dfrac{1}{10}$ of the whole chikki.

The table of $\dfrac{1}{2}$ is built by repeatedly adding $\dfrac{1}{2}$:

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

Continuing 2 more steps:
$$5 \times \frac{1}{2} = \frac{5}{2} = 2\frac{1}{2}$$
$$6 \times \frac{1}{2} = \frac{6}{2} = 3$$

Answer: The next two steps are $\dfrac{5}{2}$ (or $2\dfrac{1}{2}$) and $3$.

Yes. The table of $\dfrac{1}{4}$ is built by repeatedly adding $\dfrac{1}{4}$:

$$1 \times \frac{1}{4} = \frac{1}{4}$$
$$2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$$
$$3 \times \frac{1}{4} = \frac{3}{4}$$
$$4 \times \frac{1}{4} = \frac{4}{4} = 1$$
$$5 \times \frac{1}{4} = \frac{5}{4} = 1\frac{1}{4}$$
$$6 \times \frac{1}{4} = \frac{6}{4} = 1\frac{1}{2}$$
$$7 \times \frac{1}{4} = \frac{7}{4} = 1\frac{3}{4}$$
$$8 \times \frac{1}{4} = \frac{8}{4} = 2$$

Answer: Yes, the table for $\dfrac{1}{4}$ can be made as shown above.

Making $\dfrac{1}{3}$: Take a paper strip and fold it into 3 equal parts. Each part represents $\dfrac{1}{3}$ of the strip.

Making $\dfrac{1}{6}$: Now take the $\dfrac{1}{3}$ piece and fold it into 2 equal parts. Each of these smaller parts is $\dfrac{1}{2}$ of $\dfrac{1}{3}$.

$$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$$

Answer: Yes, by folding the $\dfrac{1}{3}$ strip in half, we get $\dfrac{1}{6}$.

Picture: Draw 5 pieces, each being $\dfrac{1}{4}$ of a roti (i.e., a circle divided into 4 equal parts, with one part shaded, repeated 5 times).

Addition statement:
$$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{5}{4} = 1\frac{1}{4}$$

Answer: 5 times $\dfrac{1}{4}$ of a roti $= \dfrac{5}{4} = 1\dfrac{1}{4}$ rotis.

Picture: Draw 9 pieces, each being $\dfrac{1}{4}$ of a roti.

Addition statement:
$$\underbrace{\frac{1}{4} + \frac{1}{4} + \cdots + \frac{1}{4}}_{9 \text{ times}} = \frac{9}{4} = 2\frac{1}{4}$$

Answer: 9 times $\dfrac{1}{4}$ of a roti $= \dfrac{9}{4} = 2\dfrac{1}{4}$ rotis.

Each fractional unit $\dfrac{1}{n}$ corresponds to a shape divided into $n$ equal parts with 1 part shaded.

- $\dfrac{1}{3}$ → the picture showing a shape divided into 3 equal parts (1 part shaded).
- $\dfrac{1}{5}$ → the picture showing a shape divided into 5 equal parts (1 part shaded).
- $\dfrac{1}{6}$ → the picture showing a shape divided into 6 equal parts (1 part shaded).
- $\dfrac{1}{8}$ → the picture showing a shape divided into 8 equal parts (1 part shaded).

Answer: Match by counting the number of equal parts in each picture — $\dfrac{1}{3}$ with 3 parts, $\dfrac{1}{5}$ with 5 parts, $\dfrac{1}{6}$ with 6 parts, $\dfrac{1}{8}$ with 8 parts.

Concept: On a number line from 0 to 1, divide the segment into 10 equal parts. Each part has length $\dfrac{1}{10}$.

- $\dfrac{1}{10}$: Mark the point 1 division from 0.
- $\dfrac{3}{10}$: Mark the point 3 divisions from 0.
- $\dfrac{4}{5} = \dfrac{8}{10}$: Mark the point 8 divisions from 0.

$$0 \quad \frac{1}{10} \quad \cdots \quad \frac{3}{10} \quad \cdots \quad \frac{8}{10} \quad \cdots \quad 1$$

Answer: Draw a number line, divide 0–1 into 10 equal parts, and mark $\dfrac{1}{10}$, $\dfrac{3}{10}$, and $\dfrac{8}{10}$ (which equals $\dfrac{4}{5}$).

Sample fractions chosen:
$$\frac{1}{2},\quad \frac{1}{4},\quad \frac{3}{4},\quad \frac{2}{5},\quad \frac{7}{10}$$

Converting to tenths for easy marking:
- $\dfrac{1}{2} = \dfrac{5}{10}$ → 5th mark
- $\dfrac{1}{4} = \dfrac{2.5}{10}$ → halfway between 2nd and 3rd mark
- $\dfrac{3}{4} = \dfrac{7.5}{10}$ → halfway between 7th and 8th mark
- $\dfrac{2}{5} = \dfrac{4}{10}$ → 4th mark
- $\dfrac{7}{10}$ → 7th mark

Answer: (Students may choose any five fractions and mark them appropriately on the number line.)

Concept: Between any two numbers on the number line, there are infinitely many fractions.

For example, between 0 and 1 we have: $\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}, \ldots$ and also $\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{5}{7}, \ldots$ — these go on without end.

Answer: There are infinitely many fractions between 0 and 1.

Given: The segment from 0 to 1 is divided into 2 equal parts, so each part $= \dfrac{1}{2}$.

- Blue line spans 1 part $\Rightarrow$ length $= \dfrac{1}{2}$.
- The black line spans both parts (from 0 to 1) $\Rightarrow$ length $= \dfrac{2}{2} = 1$.

*(Note: The exact length of the black line depends on the figure. Based on the description, if the black line covers the full unit, its length is 1. If it covers one and a half units, it would be $\dfrac{3}{2}$. Students should read the figure carefully.)*

Answer: The blue line is $\dfrac{1}{2}$ units long. The black line's length is written in the box based on how many half-units it spans (e.g., if it spans 3 half-units, the answer is $\dfrac{3}{2}$).

Concept: Count how many equal parts each black line spans, then write the fraction $\dfrac{\text{number of parts spanned}}{\text{total equal parts in one unit}}$.

For each black line shown in the figure:
- Identify the total number of equal divisions between 0 and 1.
- Count how many divisions the black line covers.
- The fraction = $\dfrac{\text{divisions covered}}{\text{total divisions}}$.

Answer: (Students should look at each black line in the figure, count the number of equal parts it spans, and write the corresponding fraction in the box. For example, if the unit is divided into 4 parts and the black line covers 3 parts, the answer is $\dfrac{3}{4}$.)

Concept: To find whole units, divide the numerator by the denominator.

$$\frac{7}{2} = \frac{2}{2} + \frac{2}{2} + \frac{2}{2} + \frac{1}{2} = 1 + 1 + 1 + \frac{1}{2} = 3 + \frac{1}{2}$$

So $7 \div 2 = 3$ remainder $1$.

Answer: There are 3 whole units in $\dfrac{7}{2}$ (with $\dfrac{1}{2}$ remaining).

For $\dfrac{4}{3}$:
$$\frac{4}{3} = \frac{3}{3} + \frac{1}{3} = 1 + \frac{1}{3}$$
$4 \div 3 = 1$ remainder $1$.
$\Rightarrow$ 1 whole unit in $\dfrac{4}{3}$.

For $\dfrac{7}{3}$:
$$\frac{7}{3} = \frac{3}{3} + \frac{3}{3} + \frac{1}{3} = 1 + 1 + \frac{1}{3} = 2 + \frac{1}{3}$$
$7 \div 3 = 2$ remainder $1$.
$\Rightarrow$ 2 whole units in $\dfrac{7}{3}$.

Answer: $\dfrac{4}{3}$ has 1 whole unit; $\dfrac{7}{3}$ has 2 whole units.

Divide numerator by denominator:
$$8 \div 3 = 2 \text{ remainder } 2$$
$$\frac{8}{3} = \frac{6}{3} + \frac{2}{3} = 2 + \frac{2}{3} = 2\frac{2}{3}$$

Answer: There are 2 whole units in $\dfrac{8}{3}$, and it equals $2\dfrac{2}{3}$.

$$11 \div 5 = 2 \text{ remainder } 1$$
$$\frac{11}{5} = \frac{10}{5} + \frac{1}{5} = 2 + \frac{1}{5} = 2\frac{1}{5}$$

Answer: There are 2 whole units in $\dfrac{11}{5}$, and it equals $2\dfrac{1}{5}$.

$$9 \div 4 = 2 \text{ remainder } 1$$
$$\frac{9}{4} = \frac{8}{4} + \frac{1}{4} = 2 + \frac{1}{4} = 2\frac{1}{4}$$

Answer: There are 2 whole units in $\dfrac{9}{4}$, and it equals $2\dfrac{1}{4}$.

Yes, all fractions greater than 1 can be written as mixed numbers.

Reason: Any fraction $\dfrac{p}{q}$ where p > q (i.e., greater than 1) can be written as:
$$\frac{p}{q} = \text{(whole number)} + \frac{\text{remainder}}{q}$$
by dividing $p$ by $q$ to get a quotient (whole part) and a remainder (fractional part).

For example: $\dfrac{13}{4} = 3\dfrac{1}{4}$, since $13 = 3 \times 4 + 1$.

Answer: Yes, every fraction greater than 1 can be written as a mixed number.

$$9 \div 2 = 4 \text{ remainder } 1$$
$$\frac{9}{2} = 4 + \frac{1}{2} = 4\frac{1}{2}$$

Answer: $\dfrac{9}{2} = 4\dfrac{1}{2}$

$$9 \div 5 = 1 \text{ remainder } 4$$
$$\frac{9}{5} = 1 + \frac{4}{5} = 1\frac{4}{5}$$

Answer: $\dfrac{9}{5} = 1\dfrac{4}{5}$

$$21 \div 19 = 1 \text{ remainder } 2$$
$$\frac{21}{19} = 1 + \frac{2}{19} = 1\frac{2}{19}$$

Answer: $\dfrac{21}{19} = 1\dfrac{2}{19}$

$$47 \div 9 = 5 \text{ remainder } 2$$
$$\frac{47}{9} = 5 + \frac{2}{9} = 5\frac{2}{9}$$

Answer: $\dfrac{47}{9} = 5\dfrac{2}{9}$

$$12 \div 11 = 1 \text{ remainder } 1$$
$$\frac{12}{11} = 1 + \frac{1}{11} = 1\frac{1}{11}$$

Answer: $\dfrac{12}{11} = 1\dfrac{1}{11}$

$$19 \div 6 = 3 \text{ remainder } 1$$
$$\frac{19}{6} = 3 + \frac{1}{6} = 3\frac{1}{6}$$

Answer: $\dfrac{19}{6} = 3\dfrac{1}{6}$

Concept: Fractions are equivalent if they simplify to the same value.

$$\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$$

$$\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}$$

$$\frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2}$$

All three fractions simplify to $\dfrac{1}{2}$.

Answer: Yes, $\dfrac{3}{6}$, $\dfrac{4}{8}$, and $\dfrac{5}{10}$ are equivalent fractions because they all equal $\dfrac{1}{2}$.

Concept: Multiply (or divide) both numerator and denominator by the same non-zero number.

$$\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}$$

$$\frac{2}{6} = \frac{2 \times 3}{6 \times 3} = \frac{6}{18}$$

Also, simplifying: $\dfrac{2}{6} = \dfrac{2 \div 2}{6 \div 2} = \dfrac{1}{3}$.

Answer: Two equivalent fractions for $\dfrac{2}{6}$ are $\dfrac{1}{3}$ and $\dfrac{4}{12}$ (or $\dfrac{6}{18}$, etc.).

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

$$\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$$

$$\frac{4}{6} = \frac{4 \times 2}{6 \times 2} = \frac{8}{12}$$

$$\frac{4}{6} = \frac{4 \times 3}{6 \times 3} = \frac{12}{18}$$

$$\frac{4}{6} = \frac{4 \times 4}{6 \times 4} = \frac{16}{24}$$

$$\frac{4}{6} = \frac{4 \times 5}{6 \times 5} = \frac{20}{30}$$

Answer: $\dfrac{4}{6} = \dfrac{2}{3} = \dfrac{8}{12} = \dfrac{12}{18} = \dfrac{16}{24} = \dfrac{20}{30} = \ldots$

Given: 3 rotis shared equally among 4 children.

Fraction each child gets:
$$\text{Each child's share} = 3 \div 4 = \frac{3}{4} \text{ roti}$$

Picture: Divide each of the 3 rotis into 4 equal parts. Each child gets 1 part from each roti, i.e., $\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{3}{4}$.

Division fact: $3 \div 4 = \dfrac{3}{4}$

Addition fact: $3 = \dfrac{3}{4} + \dfrac{3}{4} + \dfrac{3}{4} + \dfrac{3}{4}$

Multiplication fact: $3 = 4 \times \dfrac{3}{4}$

Answer: Each child gets $\dfrac{3}{4}$ roti.

Given: 2 rotis shared equally among 4 children.

Fraction each child gets:
$$2 \div 4 = \frac{2}{4} = \frac{1}{2} \text{ roti}$$

Picture: Divide each of the 2 rotis into 4 equal parts. Each child gets 1 part from each roti $= \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{2}{4} = \dfrac{1}{2}$.

Division fact: $2 \div 4 = \dfrac{1}{2}$

Addition fact: $2 = \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2}$

Multiplication fact: $2 = 4 \times \dfrac{1}{2}$

Answer: Each child gets $\dfrac{1}{2}$ roti.

Part 1 — Anil's share:
$$\text{Anil's share} = 2 \div 5 = \frac{2}{5} \text{ cake}$$

Part 2 — 10 children getting the same share $\dfrac{2}{5}$:

Each child gets $\dfrac{2}{5}$ cake. For 10 children:
$$\text{Total cakes needed} = 10 \times \frac{2}{5} = \frac{20}{5} = 4 \text{ cakes}$$

Verification: $\dfrac{4}{10} = \dfrac{2}{5}$ ✓ (equivalent fraction)

Part 3 — Two groups combined:
- Group 1: 2 cakes, 5 children
- Group 2: 4 cakes, 10 children
- Combined: $2 + 4 = 6$ cakes, $5 + 10 = 15$ children
$$\text{Each child's share} = \frac{6}{15} = \frac{2}{5} \text{ cake}$$

The share remains the same: $\dfrac{2}{5}$ cake.

Answer: Anil gets $\dfrac{2}{5}$ cake. For 10 children, 4 cakes are needed. When the two groups are combined (6 cakes, 15 children), each child still gets $\dfrac{2}{5}$ cake.

Concept: To find an equivalent fraction with denominator 8, multiply numerator and denominator of $\dfrac{5}{4}$ by 2.

$$\frac{5}{4} = \frac{5 \times 2}{4 \times 2} = \frac{10}{8}$$

Answer: 5 glasses shared among 4 friends is the same as 10 glasses shared among 8 friends. $\dfrac{5}{4} = \dfrac{10}{8}$.

Concept: The numerator is multiplied by 3 (from 4 to 12), so the denominator must also be multiplied by 3.

$$\frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9}$$

Answer: 4 kg in 3 bags is the same as 12 kg in 9 bags. $\dfrac{4}{3} = \dfrac{12}{9}$.

Concept: Multiply both numerator and denominator by the same number to get an equivalent fraction. (Multiple answers are possible.)

Multiplying by 2:
$$\frac{7}{5} = \frac{7 \times 2}{5 \times 2} = \frac{14}{10}$$

Multiplying by 3:
$$\frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15}$$

Answer: One possible answer: 7 rotis among 5 children is the same as 14 rotis among 10 children. $\dfrac{7}{5} = \dfrac{14}{10}$.