Question 1

Leena made 5 cups of tea. She used $\frac{1}{4}$ litre of milk for this. How much milk is there in each cup of tea?

Accepted Answer

Leena used $\frac{1}{4}$ litres of milk in 5 cups of tea. So, in 1 cup of tea the volume of milk should be:

$$
\frac{1}{4} \div 5.
$$

Writing this as multiplication, we have:

$$
5 \times (\text{milk per cup}) = \frac{1}{4}.
$$

The reciprocal of 5 (the divisor) is $\frac{1}{5}$.

Multiplying this reciprocal by the dividend ($\frac{1}{4}$), we get

$$
\frac{1}{5} \times \frac{1}{4} = \frac{1}{20}.
$$

So, each cup of tea has $\frac{1}{20}$ litre of milk. — To find milk per cup, divide total milk by number of cups. Division by 5 is multiplication by $\frac{1}{5}$. Thus, milk per cup = $\frac{1}{4} \times \frac{1}{5} = \frac{1}{20}$ litre.

Question 2

Cover an area of $7\frac{1}{2}$ square units with square bricks each of whose sides is $\frac{1}{5}$ units. How many such square bricks are needed?

Accepted Answer

Each square brick has an area of $\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}$ square units.

The total area to be covered is $7\frac{1}{2} = \frac{15}{2}$ square units.

Number of bricks $= \frac{15}{2} \div \frac{1}{25} = \frac{15}{2} \times 25 = \frac{375}{2} = 187.5$.

Since number of bricks must be whole, 188 bricks are needed (assuming partial bricks are not possible). — Area of one brick = $\frac{1}{25}$ sq. units.

Total area = $\frac{15}{2}$ sq. units.

Number of bricks = Total area ÷ Area of one brick = $\frac{15}{2} \times 25 = \frac{375}{2} = 187.5$.

Since bricks cannot be fractional, 188 bricks are needed.

Question 3

Four fountains fill a cistern. The first fountain can fill the cistern in a day. The second can fill it in half a day. The third can fill it in a quarter of a day. The fourth can fill the cistern in one fifth of a day. If they all flow together, in how much time will they fill the cistern?

Accepted Answer

Step 1: Calculate how many times each fountain fills the cistern in one day:
- First fountain: $1 \div 1 = 1$
- Second fountain: $1 \div \frac{1}{2} = 2$
- Third fountain: $1 \div \frac{1}{4} = 4$
- Fourth fountain: $1 \div \frac{1}{5} = 5$

Step 2: Total filling rate per day = $1 + 2 + 4 + 5 = 12$

Step 3: Time taken when all flow together = $\frac{1}{12}$ days. — Each fountain's rate is the reciprocal of the time it takes to fill the cistern.

Add all rates to get combined rate.

Time taken = reciprocal of combined rate = $\frac{1}{12}$ days.

Question 4

What fraction of area of the whole square does the shaded region occupy? (Refer to Fig. 8.4 and subsequent figures.)

Accepted Answer

Let the area of the whole square be 1 square unit.

The top right square occupies $\frac{1}{4}$ of the whole square.

The yellow triangle inside this red square has area half of the red square, so $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$.

The shaded region occupies $\frac{3}{4}$ of the yellow triangle.

Therefore, shaded area = $\frac{3}{4} \times \frac{1}{8} = \frac{3}{32}$ of the whole square. — Stepwise breakdown:
- Whole square area = 1
- Red square area = $\frac{1}{4}$
- Yellow triangle area = half of red square = $\frac{1}{8}$
- Shaded part = $\frac{3}{4}$ of yellow triangle = $\frac{3}{32}$

Question 5

In each of the figures given below, find the fraction of the big square that the shaded region occupies.

Accepted Answer

The question refers to two figures (images img-29.jpeg and img-30.jpeg). The solution involves analyzing the shaded regions in each figure relative to the whole square and calculating the fraction accordingly. Since the figures are not described in text, the exact fractions depend on visual analysis.

(As the textbook states, these will be solved in later chapters.) — The problem requires identifying the fraction of the big square occupied by the shaded region in each figure. Without textual description, the solution depends on visual inspection and fraction calculation of areas.

This is an exercise to practice fractional area calculations.

Question 6

A miser gave to a beggar $\frac{1}{5}$ of $\frac{1}{16}$ of $\frac{1}{4}$ of $\frac{1}{2}$ of $\frac{2}{3}$ of $\frac{3}{4}$ of a dramma. If you know the mathematics of fractions well, tell me O child, how many cowrie shells were given by the miser to the beggar? (1 dramma = 1280 cowrie shells)

Accepted Answer

Calculate the fraction of a dramma given:

$$
\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{1}{5} \times \frac{1}{16} \times \frac{1}{4} = \frac{6}{7680}.
$$

Simplify:

$$
\frac{6}{7680} = \frac{1}{1280}.
$$

Since 1 dramma = 1280 cowrie shells, the miser gave $\frac{1}{1280}$ of a dramma = 1 cowrie shell. — Multiply all fractions stepwise:
- $\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$
- $\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}$
- $\frac{1}{4} \times \frac{1}{5} = \frac{1}{20}$
- $\frac{1}{20} \times \frac{1}{16} = \frac{1}{320}$
- $\frac{1}{320} \times \frac{1}{4} = \frac{1}{1280}$

Thus, fraction of dramma given = $\frac{1}{1280}$.

Since 1 dramma = 1280 cowrie shells, the beggar received 1 cowrie shell.

Question 7

What is the rule for multiplying two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

Accepted Answer

Multiply numerators together and denominators together to get $\frac{a 	imes c}{b 	imes d}$ — The multiplication of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is done by multiplying the numerators to get the new numerator and the denominators to get the new denominator, resulting in $\frac{a 	imes c}{b 	imes d}$. This method reflects finding a part of a part and is the standard rule for fraction multiplication.

Question 8

When multiplying two fractions both less than 1, how does the product compare to the original fractions?

Accepted Answer

The product is smaller than either fraction — When two fractions less than 1 are multiplied, the product represents a part of a part, which is always smaller than either of the original fractions. For example, $\frac{1}{2} 	imes \frac{1}{3} = \frac{1}{6}$, which is smaller than both $\frac{1}{2}$ and $\frac{1}{3}$.

Question 9

Multiply $3$ by $\frac{2}{5}$ using repeated addition.

Accepted Answer

$\frac{6}{5}$ — Given: 3 (whole number), $\frac{2}{5}$ (fraction)
Find: Product of 3 and $\frac{2}{5}$ using repeated addition
Formula: Multiplying a fraction by a whole number is repeated addition of the fraction that many times.
Solution: Step 1: Add $\frac{2}{5}$ three times: $\frac{2}{5} + \frac{2}{5} + \frac{2}{5}$
Step 2: Add numerators: $\frac{2+2+2}{5} = \frac{6}{5}$
Answer: $\frac{6}{5}$
Note: Students often forget to add numerators correctly or keep denominator same.

Question 10

Convert the whole number 4 into a fraction and multiply it by $\frac{3}{7}$ using the multiplication rule.

Accepted Answer

$\frac{12}{7}$ — Given: Whole number 4, fraction $\frac{3}{7}$
Find: Product using multiplication rule
Formula: Convert whole number to fraction by placing over 1, then multiply numerators and denominators.
Solution: Step 1: Convert 4 to $\frac{4}{1}$
Step 2: Multiply: $\frac{4}{1} 	imes \frac{3}{7} = \frac{4 	imes 3}{1 	imes 7} = \frac{12}{7}$
Answer: $\frac{12}{7}$
Note: Students sometimes forget to convert whole number to fraction before multiplying.

Question 11

Explain how multiplying a fraction by a whole number can be represented visually using a rectangle divided into equal parts.

Accepted Answer

Multiplying a fraction by a whole number can be shown by shading parts of a rectangle divided into equal parts. For example, if a rectangle is divided into 5 equal parts and 2 parts are shaded to represent $\frac{2}{5}$, then multiplying by 3 means shading this $\frac{2}{5}$ three times side by side. This visualizes repeated addition and shows the total shaded parts as the product. — This visual method helps students understand multiplication as repeated addition of fractions by showing repeated shaded parts in the rectangle. It makes the abstract concept concrete and easier to grasp.

Question 12

Multiply $\frac{2}{3}$ by $\frac{3}{4}$ and simplify the result.

Accepted Answer

$\frac{1}{2}$ — Given: $\frac{2}{3}$ and $\frac{3}{4}$
Find: Product and simplify
Formula: Multiply numerators and denominators, then simplify.
Solution: Step 1: Multiply numerators: $2 	imes 3 = 6$
Step 2: Multiply denominators: $3 	imes 4 = 12$
Step 3: Product is $\frac{6}{12}$
Step 4: Simplify $\frac{6}{12} = \frac{1}{2}$
Answer: $\frac{1}{2}$
Note: Students often forget to simplify the fraction.

Question 13

Describe the area model used to explain multiplication of fractions $\frac{2}{3}$ and $\frac{3}{4}$.

Accepted Answer

The area model uses a rectangle divided into 3 equal parts horizontally and 4 equal parts vertically. Two parts are shaded horizontally to represent $\frac{2}{3}$, and three parts are shaded vertically to represent $\frac{3}{4}$. The overlapping shaded area, which is 6 parts out of 12 total, represents the product $\frac{6}{12}$, which simplifies to $\frac{1}{2}$. This model shows multiplication as finding a part of a part. — This visual representation helps students understand why multiplying numerators and denominators gives the product fraction. It concretely shows the overlapping area as the product fraction.

Question 14

Multiply the mixed numbers $2 \frac{1}{3}$ and $1 \frac{1}{2}$ and express the answer as a mixed number.

Accepted Answer

3 \frac{1}{2} — Given: $2 \frac{1}{3}$ and $1 \frac{1}{2}$
Find: Product as a mixed number
Formula: Convert mixed numbers to improper fractions, multiply, then simplify and convert back.
Solution: Step 1: Convert $2 \frac{1}{3} = \frac{7}{3}$
Step 2: Convert $1 \frac{1}{2} = \frac{3}{2}$
Step 3: Multiply: $\frac{7}{3} 	imes \frac{3}{2} = \frac{21}{6}$
Step 4: Simplify $\frac{21}{6} = \frac{7}{2}$
Step 5: Convert $\frac{7}{2}$ to mixed number: $3 \frac{1}{2}$
Answer: $3 \frac{1}{2}$
Note: Students often forget to convert back to mixed numbers or simplify.

Question 15

Explain the process of converting a mixed number to an improper fraction with an example.

Accepted Answer

To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, add the numerator, and write the result over the original denominator. For example, $2 \frac{1}{3}$ is converted as (2 × 3 + 1)/3 = $\frac{7}{3}$. This conversion is necessary for performing multiplication or division of mixed numbers. — This method helps in simplifying operations with mixed numbers by converting them into improper fractions which are easier to multiply or divide.

Working With

Working With — Study Notes

Introduction to Multiplication of Fractions

Multiplying a Fraction by a Whole Number

Multiplying a Fraction by a Fraction

Practice Questions — Working With

Working With — Study Guides & Articles

All 8 Chapters in Ganita Prakash