The67Sprint — LearningMap: Working with Fractions

✖️ Multiplying Fractions

Multiply numerator by numerator, denominator by denominator: a/b × c/d = (a×c)/(b×d)
Treat a whole number as a fraction over 1 before multiplying
Visualise it on a unit square: divide it into rows and columns matching the two denominators — the double-shaded overlap region is the product
Multiplying by a proper fraction (between 0 and 1) always shrinks the result — the product is smaller than both factors

➗ Dividing Fractions

The reciprocal of a/b is b/a — a fraction times its reciprocal always equals 1
Brahmagupta's Rule (628 CE): a/b ÷ c/d = a/b × d/c — keep, change, flip
Dividing by a proper fraction (less than 1) always grows the result — the quotient is bigger than the dividend
Example: dividing by 1/2 asks "how many halves fit in this whole?" — so the answer doubles

🇮🇳 Apavartana & Heritage

Apavartana is the ancient Indian practice of cancelling common factors before multiplying or dividing — fewer errors, smaller numbers
Brahmagupta systematised fraction division in his 628 CE text, the Brahmasphutasiddhanta
Always convert mixed numbers to improper fractions first — never multiply or divide them directly

Brahmagupta 628 CE Apavartana

📐 The Area Model, Step by Step

Draw a unit square and split it into rows equal to one denominator, columns equal to the other
Shade rows for the first fraction, columns for the second — the double-shaded overlap is the product
For 2/3 × 4/5: shade 2 of 3 rows and 4 of 5 columns — the overlap covers 8 of the 15 small boxes, giving 8/15
This is the same logic used to find "a piece of a piece" in recipes, crafts, or sharing problems

Memory Cheat Code

Keep-Change-Flip: Keep the first fraction, Change ÷ to ×, Flip the second fraction to its reciprocal.

Quick Recall — Sprint Facts

Multiply Rule Numerator × Numerator over Denominator × Denominator

Divide Rule Keep first fraction, change ÷ to ×, flip the second

Fraction × Reciprocal Always equals exactly 1

Multiplier < 1 Product is smaller than the original number

Divisor < 1 Quotient is bigger than the dividend

Always Simplify Apply apavartana — cancel common factors before and after

🔑 Key Terms

Apavartana

Ancient Indian method of reducing a fraction to lowest terms by cancelling common factors before calculating

Reciprocal

A fraction with numerator and denominator swapped — also called the multiplicative inverse

Dividend

The number or fraction being divided into parts

Divisor

The number or fraction by which another number is divided

Quotient

The result obtained by dividing one number by another

Improper Fraction

A fraction where the numerator is greater than the denominator — mixed numbers must be converted to this form first

Olympiad & Challenger Edge

Sridharacharya's Puzzle

A 9th-century sum from Patiganita: 1÷1/6 + 1÷1/10 + 1÷1/13 + 1÷1/9 + 1÷1/2. Flip each reciprocal to get 6+10+13+9+2 = 40.

The Shrinking Sequence

(1−1/2) × (1−1/3) × (1−1/4) ... × (1−1/n) simplifies to 1/2 × 2/3 × 3/4 ... — every term cancels the next, leaving just 1/n.

The Ant Colony Split

An ant colony splits repeatedly at forks in its path — the textbook problem tracks the fractions until 29/32 reach a mango tree and 3/32 reach a sugarcane field.

The Water Canal Team

A team builds 1 km of canal in 8 days — that's 1/8 km per day. Working 5 days a week, they complete 5/8 km every week.

❌ Missing units

Examiners deduct marks for a bare number. Write "32 bags" or "44 km" — never just "32" or "44".

❌ Straight-across division

Never divide numerator by numerator and denominator by denominator. Always keep, change, flip using Brahmagupta's Rule.

❌ Skipping mixed-number conversion

Convert mixed fractions like 3⅔ to improper form (11/3) before multiplying or dividing — never operate on them directly.

❌ Leaving answers unsimplified

720/20 must become 36. Examiners deduct half a mark for a final fraction that isn't reduced with apavartana.

🌙 The Moon's Delay

Safia sees the moon set at 10 PM Monday. It sets 5/6 hour later daily, so by Thursday: 3 × 5/6 = 2½ hours later — 12:30 AM.

🍞 Baking for a Bake Sale

5 kg of flour, 1/6 kg per loaf: 5 ÷ 1/6 = 30 loaves — dividing by a fraction less than 1 grows the answer.

🎨 Decorating School Bags

An 8 m roll of lace, 1/4 m per bag: 8 ÷ 1/4 = 32 bags decorated.

🚧 Building a Canal

A team finishes 1 km in 8 days — that's 1/8 km daily, or 5/8 km across a 5-day work week.

🚨 "Multiply always grows it"

False. Multiplying by a proper fraction shrinks the number — only multiplying by something greater than 1 grows it.

🚨 "Divide always shrinks it"

False. Dividing by a proper fraction grows the quotient — it's bigger than the original dividend.

🚨 The Linear Gap Trap

4 saplings in a row have only 3 gaps between them, not 4. Always multiply the distance by (n − 1), never by n.

🚨 Reciprocal changes the sign?

No — a reciprocal only flips numerator and denominator. The sign of the fraction never changes.

🚨 Flipping the wrong fraction

In a ÷ b, only b gets flipped to its reciprocal. Students often flip both fractions or flip a instead — keep the first one exactly as is.

Logic Chain — How Fraction Division Works

Step 1

Identify the divisor — the second fraction — which leads you to find its flipped version next

Step 2

Find the reciprocal by flipping its numerator and denominator, which turns division into multiplication

Step 3

Change the ÷ sign to ×, which lets you multiply straight across numerators and denominators

Step 4

Multiply across and apply apavartana, which gives you the fully simplified final answer

🦉

Mentor Byte

"Don't let them memorise Keep-Change-Flip as an empty rule. Ground it in scaling: show them 1/2 of 1/4 physically shrinking on a grid, then ask why dividing 5 kg of dough by 1/6 kg gives 30 loaves, not a fraction. Once they see they're counting how many small pieces fit into a whole, Brahmagupta's 628 CE rule stops being a trick and becomes plain sense — carry that pride into every calculation."

💛 Parent Tip

Next time you bake together, ask your child how many small portions of flour fit into the full bag — let them work out the division themselves before you check the arithmetic.