Class 5
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 5 (FBISE / NBF)
Mastery Guide: Addition, Subtraction, Multiplication, Division & BODMAS with Fractions
📖 Chapter 3: Fractions
📅 Updated: Sep 09, 2026
Teacher Pedagogical Roadmap
Grade 5 Mathematics • FBISE / SNC Latest Curriculum
Instructional Blueprint: Unit 3 — Fractions (Operations, Comparisons & Applications)
Target Learning Outcomes
- Compare and order proper, improper, and mixed fractions in ascending/descending order.
- Add and subtract 2 or 3 unlike fractions and mixed numbers using LCM.
- Multiply fractions using repeated addition, area models, and cross-cancellation.
- Divide fractions using the Keep-Change-Flip (KCF) reciprocal method.
- Solve multi-step real-world fraction word problems (length, mass, capacity, time).
Pacing & Time Budget (60 Min)
- 00-15m: Visual Models, Like/Unlike Comparisons & Cross Multiplication.
- 15-30m: Dual Addition/Subtraction via LCM & Mixed Number Conversions.
- 30-45m: Multiplication & Cross-Cancellation + The KCF Division Rule.
- 45-60m: Fraction Detective: Multi-step Word Problems & Real-Life Applications.
Differentiation Strategies
- Struggling: Use visual circle/bar fraction models before performing numerical LCM calculations.
- Advanced: 3-term compound fraction operations and multi-unit rate conversions.
1
The Magic of Fractions: Proper, Improper & Mixed Numbers
A fraction represents a part of a whole or a part of a set.
🌍 Textbook Opening Challenge (Page 63):
"The Earth is made up of land and water. If $\frac{2}{3}$ part of the Earth is covered with water, what part of the Earth is land?"
Solution: The whole Earth $= 1 = \frac{3}{3}$.
$\text{Land part} = 1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \mathbf{\frac{1}{3}}$. Exactly one-third of our planet is land!
| Fraction Type | Condition | Kid-Friendly Analogy | Examples |
|---|---|---|---|
| Proper Fraction | $\text{Numerator} < \text{Denominator}$ | A child carrying a small backpack (Normal & balanced, value $< 1$). | $\frac{1}{2}, \frac{3}{4}, \frac{5}{9}, \frac{6}{7}$ |
| Improper Fraction | $\text{Numerator} \ge \text{Denominator}$ | A baby carrying a huge elephant on top! (Top-heavy, value $\ge 1$). | $\frac{7}{3}, \frac{8}{5}, \frac{21}{4}, \frac{9}{9}$ |
| Mixed Number | $\text{Whole Number} + \text{Proper Fraction}$ | 2 full whole pizzas and 1 slice from the next pizza ($2\frac{1}{4}$). | $1\frac{3}{5}, 2\frac{1}{4}, 5\frac{7}{8}, 44\frac{1}{6}$ |
🔄 The C-Method for Converting Mixed $\leftrightarrow$ Improper:
To convert $2\frac{3}{4}$ to an improper fraction: Go in a clockwise circle ($\mathbf{C}$) — Multiply denominator by whole number ($4 \times 2 = 8$), then ADD numerator ($8 + 3 = 11$). Keep denominator $4 \rightarrow \mathbf{\frac{11}{4}}$.
To convert $\frac{11}{4}$ back to mixed number: Divide $11 \div 4 = 2$ quotient with remainder $3 \rightarrow \mathbf{2\frac{3}{4}}$.
2
Comparing & Ordering Fractions: The 4 Pro Methods
Case 1: Like Denominators (Same Bottom)
When denominators are identical, each piece is the exact same size. Simply compare numerators!
Textbook Example (Page 64): Asad ate $\frac{1}{7}$ of bread, Sana ate $\frac{3}{7}$.
Since $1 < 3 \implies \mathbf{\frac{1}{7} < \frac{3}{7}}$. Sana ate more bread!
Since $1 < 3 \implies \mathbf{\frac{1}{7} < \frac{3}{7}}$. Sana ate more bread!
Case 2: Like Numerators (Same Top)
When numerators are equal, the fraction with the LARGER denominator is SMALLER (because the cake was split into more pieces)!
Textbook Example (Page 65): Hassan won $\frac{3}{5}$ games, Nabeel won $\frac{3}{10}$.
Since $5 < 10 \implies \mathbf{\frac{3}{5} > \frac{3}{10}}$. Hassan played better!
Since $5 < 10 \implies \mathbf{\frac{3}{5} > \frac{3}{10}}$. Hassan played better!
Case 3: Unlike Denominators (The LCM Method)
1. Find LCM of denominators.
2. Convert fractions to equivalent fractions with LCM denominator.
3. Compare numerators.
Example (Page 65): Urwa reads $\frac{1}{4}$ book, Amna reads $\frac{2}{5}$.
$\text{LCM}(4, 5) = 20 \implies \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$ and $\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$.
As $5 < 8 \implies \mathbf{\frac{1}{4} < \frac{2}{5}}$. Urwa read less!
$\text{LCM}(4, 5) = 20 \implies \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$ and $\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$.
As $5 < 8 \implies \mathbf{\frac{1}{4} < \frac{2}{5}}$. Urwa read less!
Case 4: Cross Multiplication ("Cross-Bow Trick")
For two fractions $\frac{a}{b}$ and $\frac{c}{d}$, multiply diagonally:
$\text{Left Product} = a \times d$, $\text{Right Product} = b \times c$.
Example (Page 67): Compare $\frac{3}{4}$ and $\frac{7}{8}$.
$\text{Left} = 3 \times 8 = 24$, $\text{Right} = 4 \times 7 = 28$.
Since $24 < 28 \implies \mathbf{\frac{3}{4} < \frac{7}{8}}$.
$\text{Left} = 3 \times 8 = 24$, $\text{Right} = 4 \times 7 = 28$.
Since $24 < 28 \implies \mathbf{\frac{3}{4} < \frac{7}{8}}$.
📊 Ascending vs Descending Order:
- Ascending Order: From Least to Greatest (Climbing up the stairs: $\frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}$).
- Descending Order: From Greatest to Least (Sliding down the slide: $\frac{3}{4} > \frac{2}{3} > \frac{1}{2} > \frac{1}{3}$).
3
Addition & Subtraction of Fractions & Mixed Numbers
To add or subtract fractions, they MUST speak the same language (have the same denominator).
4-Step Addition / Subtraction Protocol:
- Convert Mixed Numbers: Turn any mixed numbers into improper fractions ($w\frac{n}{d} \rightarrow \frac{w \times d + n}{d}$).
- Find the Common Denominator: Calculate the $\text{LCM}$ of all denominators.
- Create Equivalent Fractions: Multiply top and bottom of each fraction by the factor needed to match the LCM.
- Add or Subtract Numerators: Combine numerators over the common denominator, then simplify to lowest terms / mixed numbers.
Worked Walkthrough (Textbook Page 70): Solve $\frac{2}{5} + \frac{3}{10} + \frac{1}{20}$
$\bullet$ Step 1: $\text{LCM}(5, 10, 20) = 20$.
$\bullet$ Step 2: $\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$, $\frac{3 \times 2}{10 \times 2} = \frac{6}{20}$, $\frac{1 \times 1}{20 \times 1} = \frac{1}{20}$.
$\bullet$ Step 3: $\frac{8 + 6 + 1}{20} = \frac{15}{20}$.
$\bullet$ Step 4: Simplify by dividing top and bottom by $5 \implies \mathbf{\frac{3}{4}}$.
$\bullet$ Step 1: $\text{LCM}(5, 10, 20) = 20$.
$\bullet$ Step 2: $\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$, $\frac{3 \times 2}{10 \times 2} = \frac{6}{20}$, $\frac{1 \times 1}{20 \times 1} = \frac{1}{20}$.
$\bullet$ Step 3: $\frac{8 + 6 + 1}{20} = \frac{15}{20}$.
$\bullet$ Step 4: Simplify by dividing top and bottom by $5 \implies \mathbf{\frac{3}{4}}$.
4
Multiplication of Fractions & The Cross-Cancellation Secret
Unlike addition, multiplication does NOT need a common denominator!
1. Whole Number $\times$ Fraction
Multiplication is repeated addition! (Textbook Page 74):
$7 \times \frac{3}{4} = \frac{7 \times 3}{4} = \frac{21}{4} = \mathbf{5\frac{1}{4}}$.
2. Fraction $\times$ Fraction
Multiply across:
$\frac{\text{Top } \times \text{ Top}}{\text{Bottom } \times \text{ Bottom}} \implies \frac{1}{6} \times \frac{2}{3} = \frac{1 \times 2}{6 \times 3} = \frac{2}{18} = \mathbf{\frac{1}{9}}$.
⚡ The Cross-Cancellation Ninja Secret:
Before multiplying huge numbers, always simplify numerator and denominator diagonally using common factors!
Example: $\frac{17}{5} \times 1\frac{1}{4} = \frac{17}{5} \times \frac{5}{4}$. Cross out the $5$ on top and $5$ on bottom $\rightarrow \frac{17 \times 1}{1 \times 4} = \mathbf{\frac{17}{4} = 4\frac{1}{4}\text{ cm}}$ (Textbook Page 76).
5
Division of Fractions: The K-C-F (Keep-Change-Flip) Rule
Dividing by a fraction is the exact same as multiplying by its reciprocal (inverted fraction)!
1. KEEP
Keep the 1st fraction exactly as it is.
2. CHANGE
Change the division sign $\div$ to $\times$.
3. FLIP
Flip the 2nd fraction upside down ($\frac{a}{b} \rightarrow \frac{b}{a}$).
Textbook Page 77 Lab Experiment: How many $\frac{1}{3}\text{ L}$ glasses can be filled from $7\frac{1}{3}\text{ L}$ of water?
$\bullet$ Convert mixed number: $7\frac{1}{3} = \frac{22}{3}\text{ L}$.
$\bullet$ Set up division: $\frac{22}{3} \div \frac{1}{3}$.
$\bullet$ Apply K-C-F: $\frac{22}{3} \times \frac{3}{1} = \frac{22 \times 3}{3 \times 1} = \mathbf{22\text{ glasses}}$.
$\bullet$ Convert mixed number: $7\frac{1}{3} = \frac{22}{3}\text{ L}$.
$\bullet$ Set up division: $\frac{22}{3} \div \frac{1}{3}$.
$\bullet$ Apply K-C-F: $\frac{22}{3} \times \frac{3}{1} = \frac{22 \times 3}{3 \times 1} = \mathbf{22\text{ glasses}}$.
6
The Fraction Detective: How to Choose the Right Operation
| Operation | Detective Clue Words | Real-World Scenario |
|---|---|---|
| Addition ($+$) | Total, altogether, combined sum, both together | Total time studying Math and Urdu ($2\frac{1}{2} + 1\frac{1}{4} = 3\frac{3}{4}\text{ hrs}$). |
| Subtraction ($-$) | Left over, remaining, how much more/less, difference | Apple juice remaining after serving guests ($5\frac{6}{7} - 3\frac{2}{5} = 2\frac{16}{35}\text{ L}$). |
| Multiplication ($\times$) | "Fraction of", repeated sets, total for many items | Cloth for 7 identical dresses ($7 \times 5\frac{1}{2}\text{ m} = 38\frac{1}{2}\text{ m}$). |
| Division ($\div$) | Split equally, cut into pieces, shared among, find number of packets | Packing $44\frac{1}{6}\text{ kg}$ sugar into $4\frac{5}{12}\text{ kg}$ packets ($44\frac{1}{6} \div 4\frac{5}{12} = 10\text{ packets}$). |
📖 100% Solved Textbook Exercises (FBISE Grade 5)
Exercises 1 to 3 + Review Exercise 3Exercise 1 (Textbook Page 68) — Comparing & Ordering Fractions
Q1. Compare the following pairs of fractions using symbol $<$ or $>$:
(i) $\frac{4}{5}$ and $\frac{3}{5}$:
Like denominators ($5$). Since $4 > 3 \implies \mathbf{\frac{4}{5} > \frac{3}{5}}$.
Like denominators ($5$). Since $4 > 3 \implies \mathbf{\frac{4}{5} > \frac{3}{5}}$.
(ii) $\frac{1}{4}$ and $\frac{1}{5}$:
Like numerators ($1$). Since $4 < 5 \implies \mathbf{\frac{1}{4} > \frac{1}{5}}$.
Like numerators ($1$). Since $4 < 5 \implies \mathbf{\frac{1}{4} > \frac{1}{5}}$.
(iii) $\frac{5}{6}$ and $\frac{6}{7}$:
$\text{Cross: } 5 \times 7 = 35 < 6 \times 6 = 36 \implies \mathbf{\frac{5}{6} < \frac{6}{7}}$.
$\text{Cross: } 5 \times 7 = 35 < 6 \times 6 = 36 \implies \mathbf{\frac{5}{6} < \frac{6}{7}}$.
(iv) $2\frac{1}{3}$ and $\frac{8}{5}$:
$2\frac{1}{3} = \frac{7}{3} = \frac{35}{15}$, $\frac{8}{5} = \frac{24}{15} \implies \mathbf{2\frac{1}{3} > \frac{8}{5}}$.
$2\frac{1}{3} = \frac{7}{3} = \frac{35}{15}$, $\frac{8}{5} = \frac{24}{15} \implies \mathbf{2\frac{1}{3} > \frac{8}{5}}$.
(v) $\frac{7}{3}$ and $1\frac{3}{5}$:
$\frac{7}{3} = \frac{35}{15}$, $1\frac{3}{5} = \frac{8}{5} = \frac{24}{15} \implies \mathbf{\frac{7}{3} > 1\frac{3}{5}}$.
$\frac{7}{3} = \frac{35}{15}$, $1\frac{3}{5} = \frac{8}{5} = \frac{24}{15} \implies \mathbf{\frac{7}{3} > 1\frac{3}{5}}$.
(vi) $\frac{3}{4}$ and $\frac{2}{5}$:
$\text{Cross: } 3 \times 5 = 15 > 4 \times 2 = 8 \implies \mathbf{\frac{3}{4} > \frac{2}{5}}$.
$\text{Cross: } 3 \times 5 = 15 > 4 \times 2 = 8 \implies \mathbf{\frac{3}{4} > \frac{2}{5}}$.
(vii) $1\frac{5}{9}$ and $\frac{5}{2}$:
$\frac{14}{9} = \frac{28}{18}$, $\frac{5}{2} = \frac{45}{18} \implies \mathbf{1\frac{5}{9} < \frac{5}{2}}$.
$\frac{14}{9} = \frac{28}{18}$, $\frac{5}{2} = \frac{45}{18} \implies \mathbf{1\frac{5}{9} < \frac{5}{2}}$.
(viii) $3\frac{1}{2}$ and $1\frac{2}{3}$:
$\frac{7}{2} = \frac{21}{6}$, $\frac{5}{3} = \frac{10}{6} \implies \mathbf{3\frac{1}{2} > 1\frac{2}{3}}$.
$\frac{7}{2} = \frac{21}{6}$, $\frac{5}{3} = \frac{10}{6} \implies \mathbf{3\frac{1}{2} > 1\frac{2}{3}}$.
Q2. Vegetable Quantities:
Carrots $= \frac{3}{4}$, Radishes $= \frac{7}{8}$.
Convert to denominator 8: $\frac{3}{4} = \frac{6}{8}$.
As $\frac{6}{8} < \frac{7}{8} \implies$ Carrots ($\frac{3}{4}$) are less in quantity.
Carrots $= \frac{3}{4}$, Radishes $= \frac{7}{8}$.
Convert to denominator 8: $\frac{3}{4} = \frac{6}{8}$.
As $\frac{6}{8} < \frac{7}{8} \implies$ Carrots ($\frac{3}{4}$) are less in quantity.
Q3. Cake Consumption:
Hafsa ate $\frac{4}{5}$, Aliza ate $\frac{3}{7}$.
$\text{LCM}(5, 7) = 35$: $\frac{4 \times 7}{35} = \frac{28}{35}$, $\frac{3 \times 5}{35} = \frac{15}{35}$.
As $28 > 15 \implies$ Hafsa ate more cake.
Hafsa ate $\frac{4}{5}$, Aliza ate $\frac{3}{7}$.
$\text{LCM}(5, 7) = 35$: $\frac{4 \times 7}{35} = \frac{28}{35}$, $\frac{3 \times 5}{35} = \frac{15}{35}$.
As $28 > 15 \implies$ Hafsa ate more cake.
Q4. Decreasing Order:
Fractions: $\frac{7}{3}$, $1\frac{5}{3} = \frac{8}{3}$, $2\frac{3}{4} = \frac{11}{4}$.
$\text{LCM}(3, 4) = 12$: $\frac{28}{12}, \frac{32}{12}, \frac{33}{12}$.
Decreasing order: $\mathbf{2\frac{3}{4}, 1\frac{5}{3}, \frac{7}{3}}$ (i.e. $\frac{11}{4} > \frac{8}{3} > \frac{7}{3}$).
Fractions: $\frac{7}{3}$, $1\frac{5}{3} = \frac{8}{3}$, $2\frac{3}{4} = \frac{11}{4}$.
$\text{LCM}(3, 4) = 12$: $\frac{28}{12}, \frac{32}{12}, \frac{33}{12}$.
Decreasing order: $\mathbf{2\frac{3}{4}, 1\frac{5}{3}, \frac{7}{3}}$ (i.e. $\frac{11}{4} > \frac{8}{3} > \frac{7}{3}$).
Q5. Closest to School:
Zain $= \frac{3}{5}\text{ km}$, Sami $= \frac{5}{4}\text{ km}$, Hamid $= \frac{1}{2}\text{ km}$.
$\text{LCM}(5, 4, 2) = 20$: Zain $\frac{12}{20}$, Sami $\frac{25}{20}$, Hamid $\frac{10}{20}$.
Since $\frac{10}{20} < \frac{12}{20} < \frac{25}{20} \implies$ Hamid lives closest to school ($\frac{1}{2}\text{ km}$).
Zain $= \frac{3}{5}\text{ km}$, Sami $= \frac{5}{4}\text{ km}$, Hamid $= \frac{1}{2}\text{ km}$.
$\text{LCM}(5, 4, 2) = 20$: Zain $\frac{12}{20}$, Sami $\frac{25}{20}$, Hamid $\frac{10}{20}$.
Since $\frac{10}{20} < \frac{12}{20} < \frac{25}{20} \implies$ Hamid lives closest to school ($\frac{1}{2}\text{ km}$).
Q6. Arrange the following fractions in ascending order (Least to Greatest):
(i) $\frac{3}{2}, \frac{1}{3}, \frac{3}{4}, \frac{5}{3}$:
$\text{LCM}(2, 3, 4) = 12 \rightarrow \frac{18}{12}, \frac{4}{12}, \frac{9}{12}, \frac{20}{12}$.
Order: $\mathbf{\frac{1}{3}, \frac{3}{4}, \frac{3}{2}, \frac{5}{3}}$.
$\text{LCM}(2, 3, 4) = 12 \rightarrow \frac{18}{12}, \frac{4}{12}, \frac{9}{12}, \frac{20}{12}$.
Order: $\mathbf{\frac{1}{3}, \frac{3}{4}, \frac{3}{2}, \frac{5}{3}}$.
(ii) $\frac{3}{8}, \frac{1}{4}, \frac{3}{3}, \frac{4}{3}$:
$\text{LCM}(8, 4, 3) = 24 \rightarrow \frac{9}{24}, \frac{6}{24}, \frac{24}{24}, \frac{32}{24}$.
Order: $\mathbf{\frac{1}{4}, \frac{3}{8}, \frac{3}{3}, \frac{4}{3}}$.
$\text{LCM}(8, 4, 3) = 24 \rightarrow \frac{9}{24}, \frac{6}{24}, \frac{24}{24}, \frac{32}{24}$.
Order: $\mathbf{\frac{1}{4}, \frac{3}{8}, \frac{3}{3}, \frac{4}{3}}$.
(iii) $1\frac{1}{6}, 2\frac{1}{3}, \frac{5}{2}, \frac{4}{3}$:
Improper: $\frac{7}{6}, \frac{7}{3}, \frac{5}{2}, \frac{4}{3} \rightarrow \text{LCM}=6$: $\frac{7}{6}, \frac{14}{6}, \frac{15}{6}, \frac{8}{6}$.
Order: $\mathbf{1\frac{1}{6}, \frac{4}{3}, 2\frac{1}{3}, \frac{5}{2}}$.
Improper: $\frac{7}{6}, \frac{7}{3}, \frac{5}{2}, \frac{4}{3} \rightarrow \text{LCM}=6$: $\frac{7}{6}, \frac{14}{6}, \frac{15}{6}, \frac{8}{6}$.
Order: $\mathbf{1\frac{1}{6}, \frac{4}{3}, 2\frac{1}{3}, \frac{5}{2}}$.
(iv) $\frac{7}{10}, \frac{4}{5}, 1\frac{1}{10}, 2\frac{1}{5}$:
Improper: $\frac{7}{10}, \frac{8}{10}, \frac{11}{10}, \frac{22}{10}$.
Order: $\mathbf{\frac{7}{10}, \frac{4}{5}, 1\frac{1}{10}, 2\frac{1}{5}}$.
Improper: $\frac{7}{10}, \frac{8}{10}, \frac{11}{10}, \frac{22}{10}$.
Order: $\mathbf{\frac{7}{10}, \frac{4}{5}, 1\frac{1}{10}, 2\frac{1}{5}}$.
Q7. Arrange the following fractions in descending order (Greatest to Least):
(i) $\frac{7}{6}, \frac{4}{6}, \frac{5}{4}, \frac{5}{3}$:
$\text{LCM}=12 \rightarrow \frac{14}{12}, \frac{8}{12}, \frac{15}{12}, \frac{20}{12}$.
Order: $\mathbf{\frac{5}{3}, \frac{5}{4}, \frac{7}{6}, \frac{4}{6}}$.
$\text{LCM}=12 \rightarrow \frac{14}{12}, \frac{8}{12}, \frac{15}{12}, \frac{20}{12}$.
Order: $\mathbf{\frac{5}{3}, \frac{5}{4}, \frac{7}{6}, \frac{4}{6}}$.
(ii) $\frac{6}{7}, 1\frac{1}{7}, 2\frac{1}{2}, \frac{3}{2}$:
Improper: $\frac{6}{7}, \frac{8}{7}, \frac{5}{2}, \frac{3}{2} \rightarrow \text{LCM}=14$: $\frac{12}{14}, \frac{16}{14}, \frac{35}{14}, \frac{21}{14}$.
Order: $\mathbf{2\frac{1}{2}, \frac{3}{2}, 1\frac{1}{7}, \frac{6}{7}}$.
Improper: $\frac{6}{7}, \frac{8}{7}, \frac{5}{2}, \frac{3}{2} \rightarrow \text{LCM}=14$: $\frac{12}{14}, \frac{16}{14}, \frac{35}{14}, \frac{21}{14}$.
Order: $\mathbf{2\frac{1}{2}, \frac{3}{2}, 1\frac{1}{7}, \frac{6}{7}}$.
(iii) $\frac{7}{6}, \frac{4}{3}, \frac{5}{2}, \frac{3}{2}$:
$\text{LCM}=6 \rightarrow \frac{7}{6}, \frac{8}{6}, \frac{15}{6}, \frac{9}{6}$.
Order: $\mathbf{\frac{5}{2}, \frac{3}{2}, \frac{4}{3}, \frac{7}{6}}$.
$\text{LCM}=6 \rightarrow \frac{7}{6}, \frac{8}{6}, \frac{15}{6}, \frac{9}{6}$.
Order: $\mathbf{\frac{5}{2}, \frac{3}{2}, \frac{4}{3}, \frac{7}{6}}$.
(iv) $\frac{7}{20}, \frac{9}{10}, \frac{7}{5}, 1\frac{1}{10}$:
$\text{LCM}=20 \rightarrow \frac{7}{20}, \frac{18}{20}, \frac{28}{20}, \frac{22}{20}$.
Order: $\mathbf{\frac{7}{5}, 1\frac{1}{10}, \frac{9}{10}, \frac{7}{20}}$.
$\text{LCM}=20 \rightarrow \frac{7}{20}, \frac{18}{20}, \frac{28}{20}, \frac{22}{20}$.
Order: $\mathbf{\frac{7}{5}, 1\frac{1}{10}, \frac{9}{10}, \frac{7}{20}}$.
Q8. Longest and Shortest Ribbon:
Lengths: $\frac{3}{4}\text{ m}, \frac{4}{5}\text{ m}, \frac{5}{8}\text{ m}$. $\text{LCM}(4, 5, 8) = 40$.
$\frac{3 \times 10}{40} = \frac{30}{40}$, $\frac{4 \times 8}{40} = \frac{32}{40}$, $\frac{5 \times 5}{40} = \frac{25}{40}$.
$\bullet$ Longest: $\mathbf{\frac{4}{5}\text{ m}}$ ($32/40$).
$\bullet$ Shortest: $\mathbf{\frac{5}{8}\text{ m}}$ ($25/40$).
Lengths: $\frac{3}{4}\text{ m}, \frac{4}{5}\text{ m}, \frac{5}{8}\text{ m}$. $\text{LCM}(4, 5, 8) = 40$.
$\frac{3 \times 10}{40} = \frac{30}{40}$, $\frac{4 \times 8}{40} = \frac{32}{40}$, $\frac{5 \times 5}{40} = \frac{25}{40}$.
$\bullet$ Longest: $\mathbf{\frac{4}{5}\text{ m}}$ ($32/40$).
$\bullet$ Shortest: $\mathbf{\frac{5}{8}\text{ m}}$ ($25/40$).
Q9. Garden Plants Comparison:
Flower plants $= 3\frac{2}{3} = \frac{11}{3} = \frac{22}{6}$.
Fruit plants $= 4\frac{1}{2} = \frac{9}{2} = \frac{27}{6}$.
Since $27 > 22 \implies$ Fruit plants ($4\frac{1}{2}$) are more in number.
Flower plants $= 3\frac{2}{3} = \frac{11}{3} = \frac{22}{6}$.
Fruit plants $= 4\frac{1}{2} = \frac{9}{2} = \frac{27}{6}$.
Since $27 > 22 \implies$ Fruit plants ($4\frac{1}{2}$) are more in number.
Exercise 2 (Textbook Pages 73–74) — Addition & Subtraction of Fractions
Q1. Add the following fractions:
a) $\frac{1}{2} + \frac{2}{4}$:
$\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = \mathbf{1}$.
$\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = \mathbf{1}$.
b) $5\frac{2}{3} + 2\frac{5}{7}$:
$\frac{17}{3} + \frac{19}{7} = \frac{119 + 57}{21} = \frac{176}{21} = \mathbf{8\frac{8}{21}}$.
$\frac{17}{3} + \frac{19}{7} = \frac{119 + 57}{21} = \frac{176}{21} = \mathbf{8\frac{8}{21}}$.
c) $3\frac{4}{5} + \frac{5}{7}$:
$\frac{19}{5} + \frac{5}{7} = \frac{133 + 25}{35} = \frac{158}{35} = \mathbf{4\frac{18}{35}}$.
$\frac{19}{5} + \frac{5}{7} = \frac{133 + 25}{35} = \frac{158}{35} = \mathbf{4\frac{18}{35}}$.
d) $4\frac{7}{10} + \frac{6}{8}$:
$\frac{47}{10} + \frac{3}{4} = \frac{94 + 15}{20} = \frac{109}{20} = \mathbf{5\frac{9}{20}}$.
$\frac{47}{10} + \frac{3}{4} = \frac{94 + 15}{20} = \frac{109}{20} = \mathbf{5\frac{9}{20}}$.
e) $\frac{7}{9} + \frac{6}{8} + \frac{6}{3}$:
$\frac{7}{9} + \frac{3}{4} + 2 = \frac{28 + 27 + 72}{36} = \frac{127}{36} = \mathbf{3\frac{19}{36}}$.
$\frac{7}{9} + \frac{3}{4} + 2 = \frac{28 + 27 + 72}{36} = \frac{127}{36} = \mathbf{3\frac{19}{36}}$.
f) $1\frac{3}{10} + 6\frac{14}{20} + 2\frac{15}{40}$:
$\frac{13}{10} + \frac{67}{10} + \frac{19}{8} = 8 + \frac{19}{8} = \frac{83}{8} = \mathbf{10\frac{3}{8}}$.
$\frac{13}{10} + \frac{67}{10} + \frac{19}{8} = 8 + \frac{19}{8} = \frac{83}{8} = \mathbf{10\frac{3}{8}}$.
g) $\frac{24}{6} + \frac{31}{12} + \frac{43}{24}$:
$4 + \frac{62 + 43}{24} = 4 + \frac{105}{24} = 4 + \frac{35}{8} = \frac{67}{8} = \mathbf{8\frac{3}{8}}$.
$4 + \frac{62 + 43}{24} = 4 + \frac{105}{24} = 4 + \frac{35}{8} = \frac{67}{8} = \mathbf{8\frac{3}{8}}$.
h) $\frac{7}{8} + 4\frac{1}{4} + \frac{15}{16}$:
$\frac{7}{8} + \frac{17}{4} + \frac{15}{16} = \frac{14 + 68 + 15}{16} = \frac{97}{16} = \mathbf{6\frac{1}{16}}$.
$\frac{7}{8} + \frac{17}{4} + \frac{15}{16} = \frac{14 + 68 + 15}{16} = \frac{97}{16} = \mathbf{6\frac{1}{16}}$.
Q2. Subtract the following fractions:
a) $\frac{3}{2} - \frac{2}{24}$:
$\frac{3}{2} - \frac{1}{12} = \frac{18 - 1}{12} = \frac{17}{12} = \mathbf{1\frac{5}{12}}$.
$\frac{3}{2} - \frac{1}{12} = \frac{18 - 1}{12} = \frac{17}{12} = \mathbf{1\frac{5}{12}}$.
b) $2\frac{16}{18} - 1\frac{4}{6}$:
$2\frac{8}{9} - 1\frac{2}{3} = \frac{26}{9} - \frac{15}{9} = \frac{11}{9} = \mathbf{1\frac{2}{9}}$.
$2\frac{8}{9} - 1\frac{2}{3} = \frac{26}{9} - \frac{15}{9} = \frac{11}{9} = \mathbf{1\frac{2}{9}}$.
c) $3\frac{5}{14} - 1\frac{2}{21}$:
$\frac{47}{14} - \frac{23}{21} = \frac{141 - 46}{42} = \frac{95}{42} = \mathbf{2\frac{11}{42}}$.
$\frac{47}{14} - \frac{23}{21} = \frac{141 - 46}{42} = \frac{95}{42} = \mathbf{2\frac{11}{42}}$.
d) $\frac{5}{6} - \frac{6}{11}$:
$\text{LCM}=66 \implies \frac{55 - 36}{66} = \mathbf{\frac{19}{66}}$.
$\text{LCM}=66 \implies \frac{55 - 36}{66} = \mathbf{\frac{19}{66}}$.
e) $4\frac{1}{6} - \frac{17}{18}$:
$\frac{25}{6} - \frac{17}{18} = \frac{75 - 17}{18} = \frac{58}{18} = \frac{29}{9} = \mathbf{3\frac{2}{9}}$.
$\frac{25}{6} - \frac{17}{18} = \frac{75 - 17}{18} = \frac{58}{18} = \frac{29}{9} = \mathbf{3\frac{2}{9}}$.
f) $\frac{21}{12} - \frac{8}{10}$:
$\frac{7}{4} - \frac{4}{5} = \frac{35 - 16}{20} = \mathbf{\frac{19}{20}}$.
$\frac{7}{4} - \frac{4}{5} = \frac{35 - 16}{20} = \mathbf{\frac{19}{20}}$.
g) $2\frac{13}{24} - \frac{4}{18}$:
$\frac{61}{24} - \frac{2}{9} = \frac{183 - 16}{72} = \frac{167}{72} = \mathbf{2\frac{23}{72}}$.
$\frac{61}{24} - \frac{2}{9} = \frac{183 - 16}{72} = \frac{167}{72} = \mathbf{2\frac{23}{72}}$.
h) $5\frac{1}{8} - \frac{5}{15}$:
$\frac{41}{8} - \frac{1}{3} = \frac{123 - 8}{24} = \frac{115}{24} = \mathbf{4\frac{19}{24}}$.
$\frac{41}{8} - \frac{1}{3} = \frac{123 - 8}{24} = \frac{115}{24} = \mathbf{4\frac{19}{24}}$.
Q3. Raheel Marathon Distance:
Monday $= \frac{1}{4}\text{ km}$, Tuesday $= \frac{7}{8}\text{ km}$, Wednesday $= \frac{15}{6} = \frac{5}{2}\text{ km}$.
$\text{Total} = \frac{1}{4} + \frac{7}{8} + \frac{5}{2} = \frac{2 + 7 + 20}{8} = \frac{29}{8} = \mathbf{3\frac{5}{8}\text{ km}}$.
Monday $= \frac{1}{4}\text{ km}$, Tuesday $= \frac{7}{8}\text{ km}$, Wednesday $= \frac{15}{6} = \frac{5}{2}\text{ km}$.
$\text{Total} = \frac{1}{4} + \frac{7}{8} + \frac{5}{2} = \frac{2 + 7 + 20}{8} = \frac{29}{8} = \mathbf{3\frac{5}{8}\text{ km}}$.
Q4. Saad Study Hours:
Math $= 2\frac{1}{2} = \frac{5}{2}\text{ h}$, Urdu $= 1\frac{1}{4} = \frac{5}{4}\text{ h}$.
a) More time: Math by $\frac{5}{2} - \frac{5}{4} = \frac{5}{4} = \mathbf{1\frac{1}{4}\text{ hours}}$.
b) Total time: $\frac{5}{2} + \frac{5}{4} = \frac{15}{4} = \mathbf{3\frac{3}{4}\text{ hours}}$.
Math $= 2\frac{1}{2} = \frac{5}{2}\text{ h}$, Urdu $= 1\frac{1}{4} = \frac{5}{4}\text{ h}$.
a) More time: Math by $\frac{5}{2} - \frac{5}{4} = \frac{5}{4} = \mathbf{1\frac{1}{4}\text{ hours}}$.
b) Total time: $\frac{5}{2} + \frac{5}{4} = \frac{15}{4} = \mathbf{3\frac{3}{4}\text{ hours}}$.
Q5. Electrician Wire:
Total $= 18\frac{8}{9} = \frac{170}{9}\text{ m}$. Uses $\frac{2}{9}\text{ m}$ and $2\frac{1}{3} = \frac{7}{3} = \frac{21}{9}\text{ m}$.
a) Wire used: $\frac{2}{9} + \frac{21}{9} = \frac{23}{9} = \mathbf{2\frac{5}{9}\text{ m}}$.
b) Wire left: $\frac{170}{9} - \frac{23}{9} = \frac{147}{9} = \frac{49}{3} = \mathbf{16\frac{1}{3}\text{ m}}$.
Total $= 18\frac{8}{9} = \frac{170}{9}\text{ m}$. Uses $\frac{2}{9}\text{ m}$ and $2\frac{1}{3} = \frac{7}{3} = \frac{21}{9}\text{ m}$.
a) Wire used: $\frac{2}{9} + \frac{21}{9} = \frac{23}{9} = \mathbf{2\frac{5}{9}\text{ m}}$.
b) Wire left: $\frac{170}{9} - \frac{23}{9} = \frac{147}{9} = \frac{49}{3} = \mathbf{16\frac{1}{3}\text{ m}}$.
Q6 & Q7. Open Decomposition:
Q6: Write $\frac{10}{12}$ as sum of 3 fractions:
$\mathbf{\frac{2}{12} + \frac{3}{12} + \frac{5}{12} = \frac{10}{12} = \frac{5}{6}}$ (or $\frac{1}{6} + \frac{1}{4} + \frac{5}{12}$).
Q7: Write $\frac{4}{7}$ as difference of 2 fractions:
$\mathbf{\frac{6}{7} - \frac{2}{7} = \frac{4}{7}}$ (or $1 - \frac{3}{7} = \frac{4}{7}$).
Q6: Write $\frac{10}{12}$ as sum of 3 fractions:
$\mathbf{\frac{2}{12} + \frac{3}{12} + \frac{5}{12} = \frac{10}{12} = \frac{5}{6}}$ (or $\frac{1}{6} + \frac{1}{4} + \frac{5}{12}$).
Q7: Write $\frac{4}{7}$ as difference of 2 fractions:
$\mathbf{\frac{6}{7} - \frac{2}{7} = \frac{4}{7}}$ (or $1 - \frac{3}{7} = \frac{4}{7}$).
Exercise 3 (Textbook Pages 79–80) — Multiplication & Division of Fractions
Q1. Solve the following with the help of figures (Repeated Addition):
a) $\frac{1}{2} \times 12$:
$\frac{12}{2} = \mathbf{6}$ (12 half-circles form 6 full circles).
$\frac{12}{2} = \mathbf{6}$ (12 half-circles form 6 full circles).
b) $\frac{5}{9} \times 7$:
$\frac{35}{9} = \mathbf{3\frac{8}{9}}$ (7 sets of $5/9$ slices).
$\frac{35}{9} = \mathbf{3\frac{8}{9}}$ (7 sets of $5/9$ slices).
c) $\frac{6}{7} \times 3$:
$\frac{18}{7} = \mathbf{2\frac{4}{7}}$ (3 sets of $6/7$ bars).
$\frac{18}{7} = \mathbf{2\frac{4}{7}}$ (3 sets of $6/7$ bars).
d) $\frac{3}{4} \times 4$:
$\frac{12}{4} = \mathbf{3}$ (4 three-quarter circles form 3 full circles).
$\frac{12}{4} = \mathbf{3}$ (4 three-quarter circles form 3 full circles).
e) $\frac{2}{3} \times 5$:
$\frac{10}{3} = \mathbf{3\frac{1}{3}}$ (5 sets of two-third blocks).
$\frac{10}{3} = \mathbf{3\frac{1}{3}}$ (5 sets of two-third blocks).
f) $\frac{5}{6} \times 9$:
$\frac{45}{6} = \frac{15}{2} = \mathbf{7\frac{1}{2}}$ (9 five-sixth sectors).
$\frac{45}{6} = \frac{15}{2} = \mathbf{7\frac{1}{2}}$ (9 five-sixth sectors).
Q2. Multiply the following fractions:
a) $\frac{4}{5} \times \frac{1}{4}$:
Cancel 4: $\mathbf{\frac{1}{5}}$.
Cancel 4: $\mathbf{\frac{1}{5}}$.
b) $\frac{3}{7} \times \frac{14}{18}$:
$\frac{3}{7} \times \frac{7}{9} = \frac{3}{9} = \mathbf{\frac{1}{3}}$.
$\frac{3}{7} \times \frac{7}{9} = \frac{3}{9} = \mathbf{\frac{1}{3}}$.
c) $\frac{5}{3} \times \frac{75}{20}$:
$\frac{5}{3} \times \frac{15}{4} = \frac{5 \times 5}{4} = \frac{25}{4} = \mathbf{6\frac{1}{4}}$.
$\frac{5}{3} \times \frac{15}{4} = \frac{5 \times 5}{4} = \frac{25}{4} = \mathbf{6\frac{1}{4}}$.
d) $\frac{58}{60} \times \frac{4}{20} \times \frac{10}{20}$:
$\frac{29}{30} \times \frac{1}{5} \times \frac{1}{2} = \mathbf{\frac{29}{300}}$.
$\frac{29}{30} \times \frac{1}{5} \times \frac{1}{2} = \mathbf{\frac{29}{300}}$.
e) $\frac{2}{7} \times \frac{4}{5} \times \frac{3}{7}$:
$\frac{2 \times 4 \times 3}{7 \times 5 \times 7} = \mathbf{\frac{24}{245}}$.
$\frac{2 \times 4 \times 3}{7 \times 5 \times 7} = \mathbf{\frac{24}{245}}$.
f) $\frac{30}{28} \times \frac{2}{8} \times \frac{6}{9}$:
$\frac{15}{14} \times \frac{1}{4} \times \frac{2}{3} = \frac{5}{14 \times 2} = \mathbf{\frac{5}{28}}$.
$\frac{15}{14} \times \frac{1}{4} \times \frac{2}{3} = \frac{5}{14 \times 2} = \mathbf{\frac{5}{28}}$.
g) $1\frac{3}{5} \times 2\frac{3}{7} \times 3\frac{3}{4}$:
$\frac{8}{5} \times \frac{17}{7} \times \frac{15}{4} = 2 \times \frac{17}{7} \times 3 = \frac{102}{7} = \mathbf{14\frac{4}{7}}$.
$\frac{8}{5} \times \frac{17}{7} \times \frac{15}{4} = 2 \times \frac{17}{7} \times 3 = \frac{102}{7} = \mathbf{14\frac{4}{7}}$.
h) $5\frac{10}{35} \times 8\frac{2}{5} \times 9\frac{3}{11}$:
$\frac{37}{7} \times \frac{42}{5} \times \frac{102}{11} = \frac{37 \times 6 \times 102}{55} = \frac{22644}{55} = \mathbf{411\frac{39}{55}}$.
$\frac{37}{7} \times \frac{42}{5} \times \frac{102}{11} = \frac{37 \times 6 \times 102}{55} = \frac{22644}{55} = \mathbf{411\frac{39}{55}}$.
i) $9\frac{1}{9} \times 10\frac{1}{3} \times 5\frac{1}{2}$:
$\frac{82}{9} \times \frac{31}{3} \times \frac{11}{2} = \frac{41 \times 31 \times 11}{27} = \frac{13981}{27} = \mathbf{517\frac{22}{27}}$.
$\frac{82}{9} \times \frac{31}{3} \times \frac{11}{2} = \frac{41 \times 31 \times 11}{27} = \frac{13981}{27} = \mathbf{517\frac{22}{27}}$.
Q3. Divide the following fractions (Apply Keep-Change-Flip):
a) $\frac{4}{12} \div \frac{4}{18}$:
$\frac{1}{3} \times \frac{18}{4} = \frac{6}{4} = \mathbf{\frac{3}{2} = 1\frac{1}{2}}$.
$\frac{1}{3} \times \frac{18}{4} = \frac{6}{4} = \mathbf{\frac{3}{2} = 1\frac{1}{2}}$.
b) $\frac{3}{25} \div \frac{9}{45}$:
$\frac{3}{25} \times \frac{45}{9} = \frac{3}{25} \times 5 = \mathbf{\frac{3}{5}}$.
$\frac{3}{25} \times \frac{45}{9} = \frac{3}{25} \times 5 = \mathbf{\frac{3}{5}}$.
c) $\frac{5}{60} \div \frac{7}{20}$:
$\frac{1}{12} \times \frac{20}{7} = \frac{5}{3 \times 7} = \mathbf{\frac{5}{21}}$.
$\frac{1}{12} \times \frac{20}{7} = \frac{5}{3 \times 7} = \mathbf{\frac{5}{21}}$.
d) $2\frac{58}{60} \div 4\frac{4}{20}$:
$\frac{89}{30} \div \frac{21}{5} = \frac{89}{30} \times \frac{5}{21} = \mathbf{\frac{89}{126}}$.
$\frac{89}{30} \div \frac{21}{5} = \frac{89}{30} \times \frac{5}{21} = \mathbf{\frac{89}{126}}$.
e) $5\frac{5}{9} \div 9\frac{6}{7}$:
$\frac{50}{9} \div \frac{69}{7} = \frac{50}{9} \times \frac{7}{69} = \mathbf{\frac{350}{621}}$.
$\frac{50}{9} \div \frac{69}{7} = \frac{50}{9} \times \frac{7}{69} = \mathbf{\frac{350}{621}}$.
f) $6\frac{30}{32} \div 7\frac{2}{4}$:
$\frac{111}{16} \div \frac{15}{2} = \frac{111}{16} \times \frac{2}{15} = \frac{37}{8 \times 5} = \mathbf{\frac{37}{40}}$.
$\frac{111}{16} \div \frac{15}{2} = \frac{111}{16} \times \frac{2}{15} = \frac{37}{8 \times 5} = \mathbf{\frac{37}{40}}$.
g) $2\frac{5}{6} \div 4\frac{2}{9}$:
$\frac{17}{6} \div \frac{38}{9} = \frac{17}{6} \times \frac{9}{38} = \frac{17 \times 3}{2 \times 38} = \mathbf{\frac{51}{76}}$.
$\frac{17}{6} \div \frac{38}{9} = \frac{17}{6} \times \frac{9}{38} = \frac{17 \times 3}{2 \times 38} = \mathbf{\frac{51}{76}}$.
h) $3\frac{1}{2} \div \frac{4}{9}$:
$\frac{7}{2} \times \frac{9}{4} = \frac{63}{8} = \mathbf{7\frac{7}{8}}$.
$\frac{7}{2} \times \frac{9}{4} = \frac{63}{8} = \mathbf{7\frac{7}{8}}$.
i) $\frac{1}{7} \div 2\frac{6}{7}$:
$\frac{1}{7} \div \frac{20}{7} = \frac{1}{7} \times \frac{7}{20} = \mathbf{\frac{1}{20}}$.
$\frac{1}{7} \div \frac{20}{7} = \frac{1}{7} \times \frac{7}{20} = \mathbf{\frac{1}{20}}$.
Q4. Cloth for 7 Dresses:
1 dress $= 5\frac{1}{2}\text{ m} = \frac{11}{2}\text{ m}$.
7 dresses $= 7 \times \frac{11}{2} = \frac{77}{2} = \mathbf{38\frac{1}{2}\text{ metres}}$.
1 dress $= 5\frac{1}{2}\text{ m} = \frac{11}{2}\text{ m}$.
7 dresses $= 7 \times \frac{11}{2} = \frac{77}{2} = \mathbf{38\frac{1}{2}\text{ metres}}$.
Q5. Hina's Cake Slices:
Total pieces $= 8$. Hina ate $\frac{1}{2}$.
$\text{Pieces eaten} = \frac{1}{2} \times 8 = \mathbf{4\text{ pieces}}$.
Total pieces $= 8$. Hina ate $\frac{1}{2}$.
$\text{Pieces eaten} = \frac{1}{2} \times 8 = \mathbf{4\text{ pieces}}$.
Q6. Madeeha's Pizza Portion:
Nida had $\frac{3}{12} = \frac{1}{4}$ pizza. Gave $\frac{1}{8}$ of it.
$\text{Madeeha got} = \frac{1}{8} \times \frac{1}{4} = \mathbf{\frac{1}{32}\text{ of whole pizza}}$.
Nida had $\frac{3}{12} = \frac{1}{4}$ pizza. Gave $\frac{1}{8}$ of it.
$\text{Madeeha got} = \frac{1}{8} \times \frac{1}{4} = \mathbf{\frac{1}{32}\text{ of whole pizza}}$.
Q7. Classroom Ribbons:
Blue $= 5\frac{7}{8} = \frac{47}{8}\text{ m}$. Red $= \frac{2}{3} \times \frac{47}{8} = \frac{47}{12} = \mathbf{3\frac{11}{12}\text{ m}}$.
Total length: $\frac{47}{8} + \frac{47}{12} = \frac{141 + 94}{24} = \frac{235}{24} = \mathbf{9\frac{19}{24}\text{ metres}}$.
Blue $= 5\frac{7}{8} = \frac{47}{8}\text{ m}$. Red $= \frac{2}{3} \times \frac{47}{8} = \frac{47}{12} = \mathbf{3\frac{11}{12}\text{ m}}$.
Total length: $\frac{47}{8} + \frac{47}{12} = \frac{141 + 94}{24} = \frac{235}{24} = \mathbf{9\frac{19}{24}\text{ metres}}$.
Q8. Missing Factor:
$x \times 6\frac{1}{6} = 4\frac{5}{9} \implies x = 4\frac{5}{9} \div 6\frac{1}{6}$.
$x = \frac{41}{9} \div \frac{37}{6} = \frac{41}{9} \times \frac{6}{37} = \frac{41 \times 2}{3 \times 37} = \mathbf{\frac{82}{111}}$.
$x \times 6\frac{1}{6} = 4\frac{5}{9} \implies x = 4\frac{5}{9} \div 6\frac{1}{6}$.
$x = \frac{41}{9} \div \frac{37}{6} = \frac{41}{9} \times \frac{6}{37} = \frac{41 \times 2}{3 \times 37} = \mathbf{\frac{82}{111}}$.
Q9. Sugar Packaging:
Total $= 44\frac{1}{6} = \frac{265}{6}\text{ kg}$. Packet $= 4\frac{5}{12} = \frac{53}{12}\text{ kg}$.
a) Packets: $\frac{265}{6} \times \frac{12}{53} = 5 \times 2 = \mathbf{10\text{ packets}}$.
b) 9 packets mass: $9 \times \frac{53}{12} = \frac{159}{4} = \mathbf{39\frac{3}{4}\text{ kg}}$.
Total $= 44\frac{1}{6} = \frac{265}{6}\text{ kg}$. Packet $= 4\frac{5}{12} = \frac{53}{12}\text{ kg}$.
a) Packets: $\frac{265}{6} \times \frac{12}{53} = 5 \times 2 = \mathbf{10\text{ packets}}$.
b) 9 packets mass: $9 \times \frac{53}{12} = \frac{159}{4} = \mathbf{39\frac{3}{4}\text{ kg}}$.
Review Exercise 3 (Textbook Pages 81–82) — Comprehensive Mastery & Word Problems
Q1. Encircle the correct option:
a) $\frac{1}{3} + \frac{2}{5} = \underline{\hspace{20pt}}$
$\frac{5+6}{15} = \frac{11}{15} \implies$ (ii) $\frac{11}{15}$.
$\frac{5+6}{15} = \frac{11}{15} \implies$ (ii) $\frac{11}{15}$.
b) $\frac{7}{9} - \frac{2}{3} = \underline{\hspace{20pt}}$
$\frac{7-6}{9} = \frac{1}{9} \implies$ (iii) $\frac{1}{9}$.
$\frac{7-6}{9} = \frac{1}{9} \implies$ (iii) $\frac{1}{9}$.
c) The product of $\frac{3}{4}$ and $\frac{4}{3}$ is:
$\frac{3 \times 4}{4 \times 3} = 1 \implies$ (iii) $1$.
$\frac{3 \times 4}{4 \times 3} = 1 \implies$ (iii) $1$.
d) Which figure shows $5 \times \frac{1}{3}$?
Five individual $\frac{1}{3}$ sectors $\implies$ (iv).
Five individual $\frac{1}{3}$ sectors $\implies$ (iv).
e) Smallest fraction among $\frac{2}{5}, \frac{1}{3}, \frac{2}{3}, \frac{3}{5}$:
$\frac{1}{3} = \frac{5}{15} < \frac{6}{15} < \frac{9}{15} < \frac{10}{15} \implies$ (ii) $\frac{1}{3}$.
$\frac{1}{3} = \frac{5}{15} < \frac{6}{15} < \frac{9}{15} < \frac{10}{15} \implies$ (ii) $\frac{1}{3}$.
Q2. Solve the following:
a) $\frac{7}{20} + 4\frac{3}{10}$:
$\frac{7}{20} + \frac{43}{10} = \frac{7 + 86}{20} = \frac{93}{20} = \mathbf{4\frac{13}{20}}$.
$\frac{7}{20} + \frac{43}{10} = \frac{7 + 86}{20} = \frac{93}{20} = \mathbf{4\frac{13}{20}}$.
b) $\frac{8}{5} + \frac{19}{15} + 5\frac{4}{30}$:
$\frac{8}{5} + \frac{19}{15} + \frac{77}{15} = \frac{8}{5} + \frac{96}{15} = \frac{8}{5} + \frac{32}{5} = \frac{40}{5} = \mathbf{8}$.
$\frac{8}{5} + \frac{19}{15} + \frac{77}{15} = \frac{8}{5} + \frac{96}{15} = \frac{8}{5} + \frac{32}{5} = \frac{40}{5} = \mathbf{8}$.
c) $3\frac{14}{50} - 2\frac{9}{25}$:
$3\frac{7}{25} - 2\frac{9}{25} = \frac{82}{25} - \frac{59}{25} = \mathbf{\frac{23}{25}}$.
$3\frac{7}{25} - 2\frac{9}{25} = \frac{82}{25} - \frac{59}{25} = \mathbf{\frac{23}{25}}$.
d) $1\frac{16}{44} \div \frac{4}{11}$:
$\frac{15}{11} \times \frac{11}{4} = \mathbf{\frac{15}{4} = 3\frac{3}{4}}$.
$\frac{15}{11} \times \frac{11}{4} = \mathbf{\frac{15}{4} = 3\frac{3}{4}}$.
e) $2\frac{6}{31} \times \frac{62}{24}$:
$\frac{68}{31} \times \frac{31}{12} = \frac{68}{12} = \mathbf{\frac{17}{3} = 5\frac{2}{3}}$.
$\frac{68}{31} \times \frac{31}{12} = \frac{68}{12} = \mathbf{\frac{17}{3} = 5\frac{2}{3}}$.
f) $1\frac{4}{7} + 2\frac{13}{28} + \frac{3}{4}$:
$\frac{11}{7} + \frac{69}{28} + \frac{3}{4} = \frac{44 + 69 + 21}{28} = \frac{134}{28} = \mathbf{\frac{67}{14} = 4\frac{11}{14}}$.
$\frac{11}{7} + \frac{69}{28} + \frac{3}{4} = \frac{44 + 69 + 21}{28} = \frac{134}{28} = \mathbf{\frac{67}{14} = 4\frac{11}{14}}$.
g) $3\frac{6}{8} + 3\frac{3}{4}$:
$3\frac{3}{4} + 3\frac{3}{4} = \mathbf{7\frac{1}{2} = \frac{15}{2}}$.
$3\frac{3}{4} + 3\frac{3}{4} = \mathbf{7\frac{1}{2} = \frac{15}{2}}$.
h) $1\frac{3}{9} \times 25$:
$\frac{4}{3} \times 25 = \mathbf{\frac{100}{3} = 33\frac{1}{3}}$.
$\frac{4}{3} \times 25 = \mathbf{\frac{100}{3} = 33\frac{1}{3}}$.
i) $6\frac{2}{3} \div 3\frac{1}{12}$:
$\frac{20}{3} \times \frac{12}{37} = \mathbf{\frac{80}{37} = 2\frac{6}{37}}$.
$\frac{20}{3} \times \frac{12}{37} = \mathbf{\frac{80}{37} = 2\frac{6}{37}}$.
Q3. Fraction of a Number:
Find $2\frac{3}{4}$ of $36$:
$\frac{11}{4} \times 36 = 11 \times 9 = \mathbf{99}$.
Find $2\frac{3}{4}$ of $36$:
$\frac{11}{4} \times 36 = 11 \times 9 = \mathbf{99}$.
Q4. Months in Fraction of Year:
Months in $\frac{3}{4}\text{ year}$ (1 year = 12 months):
$\frac{3}{4} \times 12 = 3 \times 3 = \mathbf{9\text{ months}}$.
Months in $\frac{3}{4}\text{ year}$ (1 year = 12 months):
$\frac{3}{4} \times 12 = 3 \times 3 = \mathbf{9\text{ months}}$.
Q5. Product Added to Quotient:
$\bullet\text{ Product} = \frac{4}{5} \times \frac{12}{2} = \frac{24}{5}$.
$\bullet\text{ Quotient} = \frac{4}{45} \div \frac{4}{5} = \frac{4}{45} \times \frac{5}{4} = \frac{1}{9}$.
$\bullet\text{ Sum} = \frac{24}{5} + \frac{1}{9} = \frac{216 + 5}{45} = \mathbf{\frac{221}{45} = 4\frac{41}{45}}$.
$\bullet\text{ Product} = \frac{4}{5} \times \frac{12}{2} = \frac{24}{5}$.
$\bullet\text{ Quotient} = \frac{4}{45} \div \frac{4}{5} = \frac{4}{45} \times \frac{5}{4} = \frac{1}{9}$.
$\bullet\text{ Sum} = \frac{24}{5} + \frac{1}{9} = \frac{216 + 5}{45} = \mathbf{\frac{221}{45} = 4\frac{41}{45}}$.
Q6. Marwa's Apple Juice Left:
Total $= 5\frac{6}{7} = \frac{41}{7}\text{ L}$. Served $= 3\frac{2}{5} = \frac{17}{5}\text{ L}$.
$\text{Left} = \frac{41}{7} - \frac{17}{5} = \frac{205 - 119}{35} = \mathbf{\frac{86}{35} = 2\frac{16}{35}\text{ litres}}$.
Total $= 5\frac{6}{7} = \frac{41}{7}\text{ L}$. Served $= 3\frac{2}{5} = \frac{17}{5}\text{ L}$.
$\text{Left} = \frac{41}{7} - \frac{17}{5} = \frac{205 - 119}{35} = \mathbf{\frac{86}{35} = 2\frac{16}{35}\text{ litres}}$.
Q7. Total Meat Bought:
Chicken $3\frac{1}{2}\text{ kg} + \text{Fish } 1\frac{1}{2}\text{ kg} + \text{Mutton } 2\frac{1}{4}\text{ kg}$.
$= 5 + 2\frac{1}{4} = \mathbf{7\frac{1}{4}\text{ kg} = \frac{29}{4}\text{ kg}}$.
Chicken $3\frac{1}{2}\text{ kg} + \text{Fish } 1\frac{1}{2}\text{ kg} + \text{Mutton } 2\frac{1}{4}\text{ kg}$.
$= 5 + 2\frac{1}{4} = \mathbf{7\frac{1}{4}\text{ kg} = \frac{29}{4}\text{ kg}}$.
Q8. Omair's Exercise in 30 Days:
Daily $= 2\frac{1}{5} = \frac{11}{5}\text{ hours}$.
$30\text{ days} = 30 \times \frac{11}{5} = 6 \times 11 = \mathbf{66\text{ hours}}$.
Daily $= 2\frac{1}{5} = \frac{11}{5}\text{ hours}$.
$30\text{ days} = 30 \times \frac{11}{5} = 6 \times 11 = \mathbf{66\text{ hours}}$.
Q9. Square Perimeter & Verification:
Side $= 1\frac{7}{8} = \frac{15}{8}\text{ m}$.
$\bullet\text{ Formula: } 4 \times \frac{15}{8} = \mathbf{\frac{15}{2} = 7\frac{1}{2}\text{ metres}}$.
$\bullet\text{ Verification: } \frac{15}{8} + \frac{15}{8} + \frac{15}{8} + \frac{15}{8} = \frac{60}{8} = \mathbf{7\frac{1}{2}\text{ m}}$.
Side $= 1\frac{7}{8} = \frac{15}{8}\text{ m}$.
$\bullet\text{ Formula: } 4 \times \frac{15}{8} = \mathbf{\frac{15}{2} = 7\frac{1}{2}\text{ metres}}$.
$\bullet\text{ Verification: } \frac{15}{8} + \frac{15}{8} + \frac{15}{8} + \frac{15}{8} = \frac{60}{8} = \mathbf{7\frac{1}{2}\text{ m}}$.
Q10. Room Width from Area:
$\text{Area} = 209\frac{1}{18} = \frac{3763}{18}\text{ m}^2$, $\text{Length} = 17\frac{2}{3} = \frac{53}{3}\text{ m}$.
$\text{Width} = \frac{3763}{18} \div \frac{53}{3} = \frac{3763}{18} \times \frac{3}{53} = \mathbf{\frac{71}{6} = 11\frac{5}{6}\text{ metres}}$.
$\text{Area} = 209\frac{1}{18} = \frac{3763}{18}\text{ m}^2$, $\text{Length} = 17\frac{2}{3} = \frac{53}{3}\text{ m}$.
$\text{Width} = \frac{3763}{18} \div \frac{53}{3} = \frac{3763}{18} \times \frac{3}{53} = \mathbf{\frac{71}{6} = 11\frac{5}{6}\text{ metres}}$.
Q11. Ordering 4 Fractions:
Fractions: $\frac{29}{30}, \frac{3}{5}, \frac{14}{15}, \frac{13}{6}$. $\text{LCM} = 30$.
Equivalents: $\frac{29}{30}, \frac{18}{30}, \frac{28}{30}, \frac{65}{30}$.
$\bullet$ Ascending: $\mathbf{\frac{3}{5}, \frac{14}{15}, \frac{29}{30}, \frac{13}{6}}$.
$\bullet$ Descending: $\mathbf{\frac{13}{6}, \frac{29}{30}, \frac{14}{15}, \frac{3}{5}}$.
Fractions: $\frac{29}{30}, \frac{3}{5}, \frac{14}{15}, \frac{13}{6}$. $\text{LCM} = 30$.
Equivalents: $\frac{29}{30}, \frac{18}{30}, \frac{28}{30}, \frac{65}{30}$.
$\bullet$ Ascending: $\mathbf{\frac{3}{5}, \frac{14}{15}, \frac{29}{30}, \frac{13}{6}}$.
$\bullet$ Descending: $\mathbf{\frac{13}{6}, \frac{29}{30}, \frac{14}{15}, \frac{3}{5}}$.
More Chapter Notes for Class 5 (FBISE)
Mathematics
Mathematics • Chapter 1
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Mastery Guide: Large Numbers, Place Value Systems, Arithmetic Operations & BODMAS Rule
Mathematics • Chapter 2
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Mastery Guide: Prime Factorization, HCF, LCM & Real-World Word Problems
Mathematics • Chapter 4
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Mastery Guide: Place Value, Decimal Arithmetic, Fraction Conversions & Percentage Applications
Mathematics • Chapter 5
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Mastery Guide: Metric Conversions, Multi-Unit Arithmetic & 24-Hour Time Calculations
Mathematics • Chapter 6
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Mastery Guide: The Unitary Method, Direct Proportion, Rates & Financial Applications
Mathematics • Chapter 7
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Mastery Guide: Angles, Triangles, Quadrilaterals, Symmetry & 3D Nets
Mathematics • Chapter 8
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Mastery Guide: Perimeter, Area, Parallelograms, Triangles & Commercial Applications
Mathematics • Chapter 9
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Mastery Guide: Averages, Bar & Line Graphs, Pie Charts & Theoretical Probability
Self-Assessment Practice
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