Jaka EduTech
Jaka EduTech
Parent Sign In

© 2026 Jaka EduTech

Jaka EduTech

📖 Mathematics

Std 7
Curriculum Book Guide
1 2 3 4 5 6 7 8
Chapters
1 Large Numbers Around Us 2 Arithmetic Expressions 3 A Peek Beyond the Point 4 Expressions Using Letter-Numbers 5 Parallel and Intersecting Lines 6 Number Play 7 A Tale of Three Intersecting Lines 8 Working with Fractions
8
Chapter 8
Skill: 60%

Working with Fractions

Working with Fractions

Chapter at a Glance

This chapter covers the mathematical foundations and applications of fraction multiplication and division. It establishes calculations using real-world rate-of-movement scenarios (e.g., Aaron and his tortoise) and visualizes fraction products using rectangular area models on a unit square. The chapter details Brahmagupta's historical formulas for multiplying and dividing fractions and introduces mixed fraction conversions. It also discusses rules for how multiplying and dividing by numbers greater than 1 or between 0 and 1 scales the result relative to the original values.

Key Definitions & Terminology

  • Multiplier: The factor in multiplication that scales the other quantity.
  • Multiplicand: The quantity that is scaled or multiplied.
  • Reciprocal: The inverted form of a fraction, where the numerator and denominator are interchanged (e.g., reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$). The product of a fraction and its reciprocal is always 1.
  • Apavartana (Simplification): The historical Indian process of reducing a fraction to its lowest terms by canceling out common factors before performing final multiplication.
  • Dramma: An ancient silver coin equivalent to 1280 cowrie shells.
  • Mixed Fraction: A number representing a whole number combined with a proper fraction (e.g., $1\frac{1}{4}$).

Formulas, Rules & Properties

  • Brahmagupta's Multiplication of Fractions (628 CE):
    $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
  • Brahmagupta's Division of Fractions:
    $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$
  • Multiplying by Reciprocals:
    $$\frac{a}{b} \times \frac{b}{a} = 1$$
  • Product Scaling Properties:
  • Situation 1 (Both factors $> 1$): The product is greater than both numbers (e.g., $\frac{4}{3} \times 4 = \frac{16}{3}$).
  • Situation 2 (Both factors between $0$ and $1$): The product is smaller than both numbers (e.g., $\frac{3}{4} \times \frac{2}{5} = \frac{3}{10}$).
  • Situation 3 (One factor between $0$ and $1$, one factor $> 1$): The product is less than the larger factor but greater than the smaller factor (e.g., $\frac{3}{4} \times 5 = \frac{15}{4}$).
  • General Rule: Multiplying by a value between 0 and 1 decreases a number, while multiplying by a value greater than 1 increases it.
  • Division Scaling Properties:
  • Dividing by a divisor between 0 and 1 yields a quotient greater than the dividend.
  • Dividing by a divisor greater than 1 yields a quotient smaller than the dividend.
  • General Cancellation Product Identity:
    $$\left(1-\frac{1}{2}\right) \times \left(1-\frac{1}{3}\right) \times \left(1-\frac{1}{4}\right) \times \dots \times \left(1-\frac{1}{n}\right) = \frac{1}{n}$$
    This is shown by simplifying terms: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots \times \frac{n-1}{n} = \frac{1}{n}$.

Core Concepts & Topics

  • Fractional Rectangle Area Model:
  • Visualizing the product of two fractions, say $\frac{a}{b} \times \frac{c}{d}$, as the area of a shaded rectangle with side lengths $\frac{a}{b}$ and $\frac{c}{d}$ within a unit square. Intersecting the $b$ rows and $d$ columns divides the unit square into $b \times d$ small rectangles, of which $a \times c$ are shaded.
  • Combined Fountain Rates (Cistern Puzzles):
  • Calculating the rate at which multiple fountains fill a cistern by summing their individual daily rates. The time taken to fill the cistern together is the reciprocal of the combined rate:
    $$\text{Time} = \frac{1}{\text{Rate}_{\text{total}}} = \frac{1}{\sum \frac{1}{t_i}}$$
  • Cowrie Currency Conversions (Miser Riddle):
  • Applying compound fraction multiplications to coins and shells, where $1\text{ dramma} = 1280\text{ cowrie shells}$.
  • Splitting Paths in Probability/Fractions:
  • Finding the fraction of an ant colony reaching food sources by multiplying the splitting ratios (halves) along each path branch.

Worked Examples

  • Tortoise Walking:
  • Problem: A tortoise walks $\frac{1}{4}$ km in 1 hour. How far does it walk in 3 hours?
  • Solution: $3 \times \frac{1}{4} = \frac{3}{4}$ km.
  • Problem: A tortoise walks $\frac{2}{5}$ km in 1 hour. How far does it walk in $\frac{3}{4}$ of an hour?
  • Solution: $\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$ km.
  • Internet Costs:
  • Problem: 1 hour of internet costs ₹8. Find the cost of $1\frac{1}{4}$ hours.
  • Solution: $1\frac{1}{4} = \frac{5}{4}$. Cost $= \frac{5}{4} \times 8 = 5 \times 2 = ₹10$.
  • Milk and Canal calculations:
  • Problem: Tenzin drinks $\frac{1}{2}$ glass of milk daily. How many glasses in a week and in January?
  • Solution: In a week (7 days): $\frac{7}{2} = 3\frac{1}{2}$ glasses. In January (31 days): $\frac{31}{2} = 15\frac{1}{2}$ glasses.
  • Problem: A canal segment of 1 km is completed in 8 days. What length is completed in 1 day? What length is completed in a 5-day week?
  • Solution: In 1 day: $\frac{1}{8}$ km. In 5 days: $\frac{5}{8}$ km.
  • Problem: Three families share 5 litres of oil weekly. Weekly oil per family? Oil per family in 4 weeks?
  • Solution: In 1 week: $\frac{5}{3}$ litres. In 4 weeks: $4 \times \frac{5}{3} = \frac{20}{3} = 6\frac{2}{3}$ litres.
  • Problem: The moon sets $\frac{5}{6}$ hour later each day. If it sets on Monday at 10 pm, when does it set on Thursday?
  • Solution: In 3 days (Monday to Thursday), delay is $3 \times \frac{5}{6} = \frac{15}{6} = 2\frac{1}{2}$ hours. Moon sets at $10\text{ pm} + 2\frac{1}{2}\text{ hours} = 12:30\text{ am}$.
  • Apavartana (Cancellation) Simplification:
  • Problem: Multiply $\frac{12}{7} \times \frac{5}{24}$ and $\frac{14}{15} \times \frac{25}{42}$.
  • Solution:
    • $\frac{12}{7} \times \frac{5}{24} = \frac{1}{7} \times \frac{5}{2} = \frac{5}{14}$ (canceling 12).
    • $\frac{14}{15} \times \frac{25}{42} = \frac{1}{3} \times \frac{5}{3} = \frac{5}{9}$ (canceling 14 and 5).
  • Water Tank Tap:
  • Problem: Tap fills $\frac{7}{10}$ of a tank in 1 hour. How much is filled in $\frac{1}{3}$ hour, $\frac{2}{3}$ hour, $\frac{3}{4}$ hour, $\frac{7}{10}$ hour, and how long to fill it completely?
  • Solution:
    • $\frac{1}{3}$ hour: $\frac{1}{3} \times \frac{7}{10} = \frac{7}{30}$ of the tank.
    • $\frac{2}{3}$ hour: $\frac{2}{3} \times \frac{7}{10} = \frac{14}{30} = \frac{7}{15}$ of the tank.
    • $\frac{3}{4}$ hour: $\frac{3}{4} \times \frac{7}{10} = \frac{21}{40}$ of the tank.
    • $\frac{7}{10}$ hour: $\frac{7}{10} \times \frac{7}{10} = \frac{49}{100}$ of the tank.
    • Full tank: $1 \div \frac{7}{10} = \frac{10}{7} = 1\frac{3}{7}$ hours.
  • Somu's Land Sharing:
  • Problem: Government takes $\frac{1}{6}$ of Somu's land. Remaining is $1 - \frac{1}{6} = \frac{5}{6}$. She gives half of the remaining to daughter Krishna, $\frac{1}{3}$ of the remaining to son Bora, and keeps the rest. Find shares.
  • Solution:
    • Krishna's share: $\frac{1}{2} \times \frac{5}{6} = \frac{5}{12}$ of the original land.
    • Bora's share: $\frac{1}{3} \times \frac{5}{6} = \frac{5}{18}$ of the original land.
    • Somu's share: $\frac{5}{6} - \left(\frac{5}{12} + \frac{5}{18}\right) = \frac{30 - (15 + 10)}{36} = \frac{5}{36}$ of the original land.
  • Rectangle Area:
  • Problem: Find the area of a rectangle with sides $3\frac{3}{4}$ ft and $9\frac{3}{5}$ ft.
  • Solution: $\frac{15}{4} \times \frac{48}{5} = \frac{3 \times 12}{1 \times 1} = 36\text{ sq ft}$.
  • Tsewang's saplings:
  • Problem: Distance between adjacent saplings in a row of 4 is $\frac{3}{4}$ m. Find the distance from the first to the last sapling.
  • Solution: 3 gaps $\times \frac{3}{4}$ m $= \frac{9}{4} = 2\frac{1}{4}$ m.
  • Weight Comparison:
  • Problem: Which is heavier: $\frac{12}{15}$ of 500 grams or $\frac{3}{20}$ of 4 kg?
  • Solution:
    • $\frac{12}{15} \times 500 = \frac{4}{5} \times 500 = 400$ grams.
    • $\frac{3}{20} \times 4000 = 3 \times 200 = 600$ grams.
    • Since $600\text{ g} > 400\text{ g}$, the latter is heavier.
  • Basic Division Problems:
  • $3 \div \frac{7}{9} = \frac{27}{7}$
  • $\frac{14}{4} \div 2 = \frac{7}{4}$
  • $\frac{2}{3} \div \frac{2}{3} = 1$
  • $\frac{14}{6} \div \frac{7}{3} = 1$
  • $\frac{4}{3} \div \frac{3}{4} = \frac{16}{9}$
  • $\frac{7}{4} \div \frac{1}{7} = \frac{49}{4}$
  • $\frac{8}{2} \div \frac{4}{15} = 15$
  • $\frac{1}{5} \div \frac{1}{9} = \frac{9}{5}$
  • $\frac{1}{6} \div \frac{11}{12} = \frac{2}{11}$
  • $3\frac{2}{3} \div 1\frac{3}{8} = \frac{11}{3} \div \frac{11}{8} = \frac{8}{3} = 2\frac{2}{3}$
  • Baudhāyana Śhulbasūtra Brick Puzzle:
  • Problem: Cover an area of $7\frac{1}{2}$ square units with square bricks of side $\frac{1}{5}$ unit. Find brick count.
  • Solution: Area of brick $= \frac{1}{5} \times \frac{1}{5} = \frac{1}{25}$ sq units. Bricks needed $= \frac{15}{2} \div \frac{1}{25} = \frac{15 \times 25}{2} = \frac{375}{2} = 187\frac{1}{2}$ bricks.
  • Chaturveda Pṛithūdakasvāmī Fountain Cistern:
  • Problem: Fountain 1 fills in 1 day. Fountain 2 in $\frac{1}{2}$ day. Fountain 3 in $\frac{1}{4}$ day. Fountain 4 in $\frac{1}{5}$ day. If flowing together, how long to fill cistern?
  • Solution: Daily fills: Fountain 1 (1), Fountain 2 (2), Fountain 3 (4), Fountain 4 (5). Total daily fills $= 1 + 2 + 4 + 5 = 12$ times. Cistern is filled in $\frac{1}{12}$ of a day.
  • Līlāvatī Miser Dramma Riddle:
  • Problem: A miser gives a beggar $\frac{1}{5} \times \frac{1}{16} \times \frac{1}{4} \times \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}$ of a dramma. Find value in cowrie shells if $1\text{ dramma} = 1280\text{ cowries}$.
  • Solution: Product $= \frac{1 \times 1 \times 1 \times 1 \times 2 \times 3}{5 \times 16 \times 4 \times 2 \times 3 \times 4} = \frac{6}{7680} = \frac{1}{1280}\text{ dramma}$. This is exactly 1 cowrie shell.
  • Flight Time Saving:
  • Problem: Train takes $5\frac{1}{6}$ hours. Plane takes $\frac{1}{2}$ hour. Saving?
  • Solution: $5\frac{1}{6} - \frac{1}{2} = \frac{31}{6} - \frac{3}{6} = \frac{28}{6} = \frac{14}{3} = 4\frac{2}{3}$ hours saved.
  • Mira's Novel Reading:
  • Problem: Novel has 400 pages. Mira reads $\frac{1}{5}$ yesterday and $\frac{3}{10}$ today. Pages left?
  • Solution: Read $= \left(\frac{1}{5} + \frac{3}{10}\right) \times 400 = \frac{5}{10} \times 400 = 200$ pages. Pages left $= 400 - 200 = 200$ pages.
  • Patiganita Reciprocal Sum:
  • Problem: Evaluate $1 \div \frac{1}{6} + 1 \div \frac{1}{10} + 1 \div \frac{1}{13} + 1 \div \frac{1}{9} + 1 \div \frac{1}{2}$.
  • Solution: Sum $= 6 + 10 + 13 + 9 + 2 = 40$.
  • Cake Sharing:
  • Problem: Cousins eat $\frac{4}{5}$ of cake. Remaining $\frac{1}{5}$ cake shared by 3 friends. Share per friend?
  • Solution: $\frac{1}{5} \div 3 = \frac{1}{15}$ of the cake.
  • Ant Colony Split Exits:
  • Solution: Fraction of ants reaching Mango Tree source $= \frac{29}{32}$; Sugarcane source $= \frac{3}{32}$.

Practical Activities & Experiments

  • Area shading visual models: Shading rows and columns on a grid inside a unit square to demonstrate fraction multiplication.
  • Ant Split simulations: Tracing split paths on tree diagrams to calculate probability pathways using fractions.
  • Cistern simulations: Measuring rates of flow for vessels, calculating cumulative rates, and verifying combined completion times.
Previous Chapter 8 of 8