Working with Fractions – Class 7 Maths Chapter 8 Notes & NCERT PDF

Introduction

The chapter opens with Aaron, who walks 3 km in 1 hour, and his slow pet tortoise, which walks only ¼ km in an hour. Finding distances for whole or fractional times leads us to multiplying fractions. Then, by turning division into multiplication, we learn to divide fractions — using methods first written in general form by Brahmagupta in 628 CE.

Key Ideas

8.1 Multiplication of Fractions

Whole number × fraction: the tortoise walks ¼ km per hour, so in 3 hours: 3 × ¼ = ¼ + ¼ + ¼ = ¾ km.

Fraction × whole number: Aaron walks 3 km per hour. In ⅕ hour he walks 3 km divided into 5 equal parts = 3/5 km, so ⅕ × 3 = 3/5. In ⅖ hour: divide 3 by 5 (the multiplier's denominator) → 3/5, then multiply by 2 (its numerator) → 6/5 km.

Fraction × fraction using a unit square: the tortoise in ½ hour walks ½ × ¼. Divide the unit square into 4, take one part, then halve it: the whole is now 8 parts, one shaded → 1/8. For ¾ × ⅖: divide ⅖ into 4 parts — the square has 5 rows × 4 columns = 20 parts, 2 shaded → 2/20; multiply by 3 → 6/20 = 3/10.

Area link: the area of a rectangle with fractional sides equals the product of its sides.

For 1/12 × 1/18, the square is divided into 18 rows and 12 columns = 216 parts, so the product is 1/216. In general:

Brahmagupta's formula (628 CE): a/b × c/d = (a × c)/(b × d)

Example: 5/12 × 7/18 = 35/216. A whole number is just a fraction with denominator 1: 3 × ¾ = 3/1 × 3/4 = 9/4.

Cancelling common factors first saves work: 12/7 × 5/24 → cancel 12 with 24 → 1/7 × 5/2 = 5/14; 14/15 × 25/42 → cancel 14 with 42, 5 with 15 → 5/9.

Is the product always bigger? No!

Multiplying by a number between 0 and 1 makes the other number smaller; multiplying by a number greater than 1 makes it bigger.

Order doesn't matter: a/b × c/d = c/d × a/b — just as a rectangle's area stays the same when length and breadth swap.

8.2 Division of Fractions

Every division is a hidden multiplication: 12 ÷ 4 = ? means 4 × ? = 12.

The reciprocal of a/b is b/a; a fraction times its reciprocal is 1.

To divide: multiply the dividend by the reciprocal of the divisor: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c).

Quotient vs dividend: 6 ÷ 3 = 2 (smaller), but 6 ÷ ¼ = 24 and 1/8 ÷ ¼ = ½ (bigger!).

Dividing by a number between 0 and 1 makes the quotient greater than the dividend; dividing by a number greater than 1 makes it smaller.

8.3 Problems Involving Fractions (from Indian texts)

A Dramma-tic Donation (Bhāskara II, Līlāvatī, 1150 CE)

A miser gave a beggar ½ of ⅔ of ¾ of ⅕ of 1/16 of ¼ of a dramma (silver coin). Multiply: ½ × ⅔ × ¾ × ⅕ × 1/16 × ¼ = 1/1280 dramma. Since 1 dramma = 1280 cowrie shells, the beggar got just one cowrie — Bhāskara's joke about a miser!

With 1 gold dinar = 12 drammas, 1 dramma = 4 panas and 1 pana = 30 cowries: 1 pana = 1/12 × ¼ = 1/48 dinar, and 1 cowrie = 1/30 pana = 1/1440 dinar.

Worked Examples

Example 1 — 5/4 × 3/2. Divide 3/2 into 4 parts: 2 rows × 4 columns = 8 parts, 3 shaded → 3/8. Multiply by 5 → 15/8.

Example 2 — Somu's land. The government takes ⅙; remaining = 5/6. Krishna gets half of it = 5/12; Bora gets ⅓ of it = 5/18. Somu keeps 5/6 − 5/12 − 5/18 = 5/36 of the original land.

Example 3 — Area of a rectangle 3¾ ft × 9⅗ ft. 15/4 × 48/5 = 36 sq ft.

Example 4 — Tsewang's saplings. Four saplings ¾ m apart have 3 gaps: 3 × ¾ = 9/4 = 2¼ m.

Example 5 — Sridharacharya's puzzle (Pāṭīgaṇita). 1 ÷ 1/6 + 1 ÷ 1/10 + 1 ÷ 1/13 + 1 ÷ 1/9 + 1 ÷ 1/2 = 6 + 10 + 13 + 9 + 2 = 40.

Example 6 — A car runs 16 km per litre. On 2¾ litres: 16 × 11/4 = 44 km.

Figure it Out — Hints

Maths Words

Historical & Indian Connections

Common Mistakes

Maths in Daily Life

Scaling recipes (¾ of a recipe), sharing land or food, finding fuel needed for a trip, and working out cost for part of an hour all use fraction multiplication and division.