NCERT Solutions for Class 7 Maths Chapter 2: Fractions and Decimals — Complete Guide with Step-by-Step Solutions

• Proper fraction: Numerator $<$ denominator. Example: $\frac{3}{7}$. Value is less than $1$.
  - Improper fraction: Numerator $\geq$ denominator. Example: $\frac{9}{4}$. Value is $\geq 1$.
  - Mixed fraction (mixed number): A whole number and a proper fraction combined. Example: $2\frac{1}{4}$.

Converting mixed to improper:

Converting improper to mixed:

Golden Rule: Always convert mixed fractions to improper fractions before multiplying or dividing.

The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$ — you flip the numerator and denominator.

Key property: A number times its reciprocal always equals $1$:

The reciprocal is the foundation of fraction division: to divide by a fraction, multiply by its reciprocal.

Problem: Multiply $\frac{2}{5} \times 7$.

Solution:

Answer: $2\frac{4}{5}$.

Problem: Multiply $\frac{6}{7} \times 14$.

Solution:
Notice that $14$ and $7$ share a common factor of $7$. Simplify first:

Alternatively: $\frac{6}{7} \times \frac{14}{1} = \frac{6}{1} \times \frac{2}{1} = 12$ (after cancelling $7$ with $14$).

Tip: Always look for common factors to cancel BEFORE multiplying. This keeps numbers small and reduces errors.

Problem: Find $\frac{3}{4}$ of $28$.

Solution:
"$\frac{3}{4}$ of $28$" means $\frac{3}{4} \times 28$:

Answer: $21$.

Problem: Multiply $2\frac{3}{5} \times 4$.

Solution:
Step 1: Convert to improper fraction: $2\frac{3}{5} = \frac{13}{5}$.

Step 2: Multiply:

Answer: $10\frac{2}{5}$.

Problem: A car runs $16$ km using $1$ litre of petrol. How many km will it run using $2\frac{3}{4}$ litres?

Solution:
Distance $= 16 \times 2\frac{3}{4} = 16 \times \frac{11}{4}$

Cancel $16$ with $4$: $\frac{16}{4} = 4$.

Answer: $44$ km.

Problem: Evaluate $\frac{5}{8} \times 24$.

Solution:
Cancel $8$ with $24$: $\frac{24}{8} = 3$.

Answer: $15$.

Problem: A piece of cloth is $5$ m long. Raj needs $\frac{2}{5}$ of the cloth. How many metres does he need?

Solution:

Answer: $2$ metres.

Problem: What happens when you multiply a fraction by $1$? By $0$? By a number greater than $1$?

Solution:
- $\frac{3}{4} \times 1 = \frac{3}{4}$ (unchanged — $1$ is the multiplicative identity)
- $\frac{3}{4} \times 0 = 0$ (anything times zero is zero)
- $\frac{3}{4} \times 3 = \frac{9}{4} = 2\frac{1}{4}$ (result is greater than the fraction)
- $\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$ (result is less than the fraction — multiplying by a proper fraction shrinks the value)

Key insight: Multiplying by a proper fraction always gives a result smaller than the original number.

Problem: Multiply $\frac{3}{7} \times \frac{2}{5}$.

Solution:

Answer: $\frac{6}{35}$.

Problem: Multiply $2\frac{1}{3} \times 3\frac{1}{2}$.

Solution:
Convert to improper fractions first:

Answer: $8\frac{1}{6}$.

Problem: Which is greater: $\frac{2}{7}$ of $\frac{3}{4}$ or $\frac{3}{5}$ of $\frac{5}{8}$?

Solution:

Compare $\frac{3}{14}$ and $\frac{3}{8}$. Both have numerator $3$. The fraction with the smaller denominator is larger.

Since $8 < 14$, we have $\frac{3}{8} > \frac{3}{14}$.

Answer: $\frac{3}{5}$ of $\frac{5}{8}$ is greater.

Problem: Find $\frac{1}{2}$ of $4\frac{2}{9}$.

Solution:

Answer: $2\frac{1}{9}$.

Problem: Multiply $\frac{8}{15} \times \frac{5}{12}$.

Solution:
Before multiplying, cancel common factors across numerators and denominators:
- $8$ and $12$ share factor $4$: $\frac{8}{4} = 2$, $\frac{12}{4} = 3$.
- $5$ and $15$ share factor $5$: $\frac{5}{5} = 1$, $\frac{15}{5} = 3$.

Answer: $\frac{2}{9}$.

Tip: Cross-cancellation saves time and reduces the chance of arithmetic errors with large numbers.

Problem: Find the product $\frac{7}{11} \times \frac{11}{7}$.

Solution:

Key insight: A fraction multiplied by its reciprocal always equals $1$.

Problem: A rectangular plot has length $3\frac{1}{2}$ m and breadth $2\frac{2}{3}$ m. Find its area.

Solution:
Area $= \text{length} \times \text{breadth}$

Cancel $2$ with $8$: $\frac{8}{2} = 4$.

Answer: $9\frac{1}{3}$ m$^2$.

Problem: Evaluate $\frac{2}{3} \times \frac{3}{4} \times \frac{4}{5}$.

Solution:

Cancel $3$ in numerator and denominator. Cancel $4$ in numerator and denominator.

Answer: $\frac{2}{5}$.

Problem: Divide $\frac{3}{5} \div 4$.

Solution:
The reciprocal of $4$ is $\frac{1}{4}$.

Answer: $\frac{3}{20}$.

Problem: Divide $\frac{7}{8} \div \frac{2}{3}$.

Solution:

Answer: $1\frac{5}{16}$.

Problem: Divide $2\frac{1}{3} \div \frac{3}{5}$.

Solution:

Answer: $3\frac{8}{9}$.

Problem: Divide $6 \div \frac{2}{3}$.

Solution:

Interpretation: How many groups of $\frac{2}{3}$ fit into $6$? The answer is $9$.

Answer: $9$.

Problem: A wire of length $12\frac{1}{2}$ m is to be cut into pieces of length $\frac{5}{4}$ m each. How many pieces will there be?

Solution:
Number of pieces $= 12\frac{1}{2} \div \frac{5}{4}$

Cancel $25$ with $5$ ($\frac{25}{5} = 5$) and $4$ with $2$ ($\frac{4}{2} = 2$):

Answer: $10$ pieces.

Problem: Divide $\frac{4}{9} \div 1\frac{1}{3}$.

Solution:
Convert: $1\frac{1}{3} = \frac{4}{3}$.

Cancel $4$ with $4$ and $3$ with $9$:

Answer: $\frac{1}{3}$.

Problem: A tank holds $5\frac{1}{4}$ litres. If each bottle holds $\frac{3}{4}$ litres, how many bottles can be filled?

Solution:

Cancel $4$ with $4$:

Answer: $7$ bottles.

Problem: Can you divide $\frac{5}{7} \div 0$?

Solution:
No. Division by zero is undefined.

To divide by $0$, we would need the reciprocal of $0$, which is $\frac{1}{0}$ — but this does not exist because no number multiplied by $0$ gives $1$.

Therefore, $\frac{5}{7} \div 0$ is undefined.

Problem: What is $\frac{5}{8} \div \frac{5}{8}$?

Solution:

General rule: Any non-zero number divided by itself equals $1$.

Answer: $1$.

Problem: A piece of land of area $4\frac{1}{2}$ hectares is to be divided equally among $3$ families. What area does each family get?

Solution:

Answer: Each family gets $1\frac{1}{2}$ hectares.

Problem: Find $0.2 \times 6$.

Solution:

Working: $2 \times 6 = 12$. Decimal places in $0.2 = 1$. So place $1$ decimal place: $1.2$.

Problem: Find $1.3 \times 2.1$.

Solution:
Multiply ignoring decimals: $13 \times 21 = 273$.

Total decimal places: $1 + 1 = 2$.

Answer: $2.73$.

Problem: Find $0.01 \times 0.3$.

Solution:
$1 \times 3 = 3$. Total decimal places: $2 + 1 = 3$.

Answer: $0.003$.

Problem: The side of a square is $2.5$ cm. Find its area.

Solution:

$25 \times 25 = 625$. Decimal places: $1 + 1 = 2$.

Answer: $6.25$ cm$^2$.

Problem: Evaluate (a) $2.35 \times 10$ (b) $2.35 \times 100$ (c) $2.35 \times 1000$.

Solution:
(a) $2.35 \times 10 = 23.5$ (shift decimal $1$ place right)

(b) $2.35 \times 100 = 235$ (shift decimal $2$ places right)

(c) $2.35 \times 1000 = 2350$ (shift decimal $3$ places right)

Rule: Multiplying by $10^n$ shifts the decimal point $n$ places to the right.

Problem: Find $0.5 \times 0.4 \times 0.2$.

Solution:
$5 \times 4 \times 2 = 40$. Total decimal places: $1 + 1 + 1 = 3$.

Answer: $0.04$.

Problem: Cloth costs Rs. $45.50$ per metre. Find the cost of $3.5$ metres.

Solution:

$4550 \times 35 = 159250$. Total decimal places: $2 + 1 = 3$.

Answer: Rs. $159.25$.

Problem: Without calculating exactly, estimate $4.8 \times 3.2$ and then verify.

Solution:
Estimate: $4.8 \approx 5$ and $3.2 \approx 3$. So $4.8 \times 3.2 \approx 15$.

Exact: $48 \times 32 = 1536$. Decimal places: $1 + 1 = 2$.

The estimate ($15$) is close to the exact answer ($15.36$), confirming our calculation is correct.

Tip: Always estimate before computing to catch decimal placement errors.

Problem: Divide $6.5 \div 5$.

Solution:

Working: $65 \div 5 = 13$. Place decimal: $1.3$.

Answer: $1.3$.

Problem: Divide $44.4 \div 0.4$.

Solution:
Multiply both by $10$ to make the divisor a whole number:

Answer: $111$.

Problem: Divide $3.25 \div 0.5$.

Solution:
Multiply both by $10$:

Answer: $6.5$.

Problem: Divide $0.0045 \div 0.09$.

Solution:
Multiply both by $100$:

Answer: $0.05$.

Problem: Evaluate (a) $23.5 \div 10$ (b) $23.5 \div 100$ (c) $23.5 \div 1000$.

Solution:
(a) $23.5 \div 10 = 2.35$ (shift decimal $1$ place left)

(b) $23.5 \div 100 = 0.235$ (shift decimal $2$ places left)

(c) $23.5 \div 1000 = 0.0235$ (shift decimal $3$ places left)

Rule: Dividing by $10^n$ shifts the decimal point $n$ places to the left.

Problem: A car travels $43.2$ km in $1.2$ hours. Find the speed.

Solution:

Multiply both by $10$:

Answer: $36$ km/hr.

Problem: $7.5$ kg of rice is to be packed into bags of $0.25$ kg each. How many bags are needed?

Solution:

Multiply both by $100$:

Answer: $30$ bags.

Problem: Divide $16.8 \div 0.07$.

Solution:
Multiply both by $100$:

Verification: $240 \times 0.07 = 16.8$. Correct.

Answer: $240$.

Problem: $0.144 \div 0.012$.

Solution:
Multiply both by $1000$:

Answer: $12$.

Problem: $2.4$ metres of wire costs Rs. $7.20$. Find the cost per metre.

Solution:

Multiply both by $10$:

Answer: Rs. $3$ per metre.

Answer 1:

Answer 2:

Answer 3:

Answer 4:
$15 \times 4 = 60$. Decimal places: $2 + 1 = 3$. Answer: $0.060 = 0.06$.

Answer 5:
$2.46 \div 0.6 = 24.6 \div 6 = 4.1$.

Answer 6:
$8\frac{1}{4} \div \frac{3}{4} = \frac{33}{4} \times \frac{4}{3} = \frac{33}{3} = 11$ pieces.

Answer 7:
$42.50 \times 3.5 = 4250 \times 35 \div 1000 = 148750 \div 1000 = \text{Rs. } 148.75$.

Answer 8: