Addition, subtraction, multiplication, division of fractions

WHY do fractions behave this way?

WHAT is a fraction? A number written $\frac{a}{b}$ (with $b\neq 0$ ) where $a$ is the numerator (how many parts) and $b$ is the denominator (size of each part = how many parts make one whole).

Golden rule (the source of everything): multiplying top and bottom by the same nonzero number does not change the value: $\frac{a}{b} = \frac{a\cdot k}{b\cdot k}\qquad(k\neq 0)$

1. Addition & Subtraction — same-size pieces first

HOW (derivation from first principles): Start with $\frac{a}{b} + \frac{c}{d}$ . Use the golden rule on each to force a shared bottom of $bd$ : $\frac{a}{b}=\frac{a\cdot d}{b\cdot d},\qquad \frac{c}{d}=\frac{c\cdot b}{d\cdot b}=\frac{bc}{bd}$ Now both have denominator $bd$ , so the pieces match and we can count: $\boxed{\;\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\;}$ Subtraction is identical with a minus: $\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}$ .

2. Multiplication — a fraction of a fraction

HOW (derivation): Split a unit square into $b$ columns and $d$ rows → $bd$ tiny cells, each of area $\frac{1}{bd}$ . Taking $a$ columns and $c$ rows selects $a\cdot c$ cells: $\boxed{\;\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\;}$ No common denominator needed — you multiply straight across.

3. Division — flip and multiply

HOW (derivation, two ways): Way 1 — undo multiplication. We want $x$ with $x\cdot\frac{c}{d}=\frac{a}{b}$ . Multiply both sides by $\frac{d}{c}$ (the reciprocal, which turns $\frac{c}{d}$ into $1$ ): $x=\frac{a}{b}\cdot\frac{d}{c}$ Way 2 — clear the denominators. Multiply top and bottom of the complex fraction by $\frac{d}{c}$ : $\frac{\ \frac{a}{b}\ }{\ \frac{c}{d}\ }=\frac{\frac{a}{b}\cdot\frac{d}{c}}{\frac{c}{d}\cdot\frac{d}{c}}=\frac{\frac{a}{b}\cdot\frac{d}{c}}{1}$ Either way: $\boxed{\;\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc}\;}$

Figure — Addition, subtraction, multiplication, division of fractions

Common Mistakes (Steel-manned)

Active Recall

Recall Try before revealing

Why do you need a common denominator for $+$ but not for $\times$ ?
Derive $\frac{a}{b}\div\frac{c}{d}$ from " $x\cdot\frac{c}{d}=\frac{a}{b}$ ".
Compute $\frac56-\frac34$ using the LCM.

Answers: 1) Addition counts equal-size pieces, so pieces must match; multiplication takes a fraction-of-a-fraction (area), no matching needed. 2) Multiply both sides by reciprocal $\frac{d}{c}\Rightarrow x=\frac{a}{b}\cdot\frac{d}{c}$ . 3) $\frac{10-9}{12}=\frac1{12}$ .

Recall Feynman: explain to a 12-year-old

Imagine pizzas cut into slices. To add pizza, the slices must be the same size — so first re-cut both pizzas into equal slices (common denominator), then just count the slices. To find half of a third of a pizza, you slice it twice — that's multiplying, and you multiply the "cut" numbers straight across. To divide "how many quarter-pizzas fit in half a pizza?", you just flip the little one and multiply — the answer (2) is bigger because tiny pieces fit lots of times.

Flashcards

What must you do before adding two fractions?

Rewrite them with a common denominator so the pieces are the same size, then add numerators.

Formula for

\frac{a}{b}+\frac{c}{d}

\frac{ad+bc}{bd}

Why can you multiply fractions straight across but not add them?

Multiplying takes a fraction of a fraction (an area), needing no equal-size pieces; adding counts pieces, which must be equal-sized.

Formula for

\frac{a}{b}\times\frac{c}{d}

\frac{ac}{bd}

How do you divide

\frac{a}{b}\div\frac{c}{d}

Multiply by the reciprocal:

\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc}

(Keep-Change-Flip).

Why does the divisor flip in division?

Because the reciprocal

\frac{d}{c}

turns

\frac{c}{d}

into 1, undoing the multiplication.

What is the reciprocal of

\frac{2}{5}

\frac{5}{2}

(since

\frac25\times\frac52=1

Compute

\frac{2}{3}+\frac{1}{4}

\frac{8+3}{12}=\frac{11}{12}

Compute

\frac{3}{4}\div\frac{2}{5}

\frac{3}{4}\times\frac{5}{2}=\frac{15}{8}

Why is

\frac{a}{b}=\frac{ak}{bk}

Cutting each piece into

k

smaller pieces gives

k\times

as many pieces of

k\times

smaller size — the total amount is unchanged.

When multiplying, why cancel before multiplying?

It keeps numbers small and is just the golden rule (

\frac{ak}{bk}=\frac ab

) applied early.

Connections

Fractions - meaning and equivalent fractions
LCM and HCF — used to find the smallest common denominator
Reciprocals and multiplicative inverse
Decimals and fractions
Ratios and proportion
Rational numbers

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, fraction $\frac{a}{b}$ ka matlab hai ek cheez ko $b$ equal tukdon me kaato aur unme se $a$ tukde le lo. Ab sabse important baat: jodne aur ghatane (addition/subtraction) ke liye tukde same size ke hone chahiye. Isiliye pehle common denominator banate hain — dono fractions ko aisa likhte hain ki denominator same ho, phir sirf upar wale (numerator) add/subtract karte hain. Jaise $\frac23+\frac14$ : dono ko $12$ pe le aao ( $\frac{8}{12}+\frac{3}{12}$ ), phir $=\frac{11}{12}$ . Yaad rakho — kabhi bhi $\frac{a+c}{b+d}$ mat karna, wo galat hai!

Multiplication sabse easy hai: seedha straight across multiply karo, upar-upar aur neeche-neeche. $\frac23\times\frac45=\frac{8}{15}$ . Kyunki multiply karna matlab "fraction ka fraction" nikalna — jaise ek rectangle me overlap area. Yahan common denominator ki zaroorat hi nahi. Aur ek trick: multiply karne se pehle cancel kar lo (jaise $4$ aur $8$ ), numbers chhote reh jaate hain.

Division me bas ek jaadu — Keep, Change, Flip. Pehla fraction same rakho, ÷ ko × me badlo, aur doosre ko ulta (reciprocal) kar do. $\frac34\div\frac25=\frac34\times\frac52=\frac{15}{8}$ . Iska logic: division poochta hai "kitne $\frac25$ ek $\frac34$ me samate hain?" — chhoti cheez se divide karoge to answer bada aayega, isiliye flip hota hai. Ye chapter poore maths ki neenv hai — algebra, ratio, decimals sab yahin se aate hain, isliye ratta nahi, samajh ke pakka karo.

Addition, subtraction, multiplication, division of fractions

WHY do fractions behave this way?

1. Addition & Subtraction — same-size pieces first

2. Multiplication — a fraction of a fraction

3. Division — flip and multiply

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Arithmetic & Number Systems

Connections