Addition, subtraction, multiplication, division of fractions

1.1.14 · Maths › Arithmetic & Number Systems

WHY do fractions behave this way?

WHAT is a fraction? Ek number jo $\frac{a}{b}$ likha jaata hai (jahan $b \neq = 0$ ) jisme $a$ numerator hai (kitne parts) aur $b$ denominator hai (har part ka size = kitne parts milke ek whole bante hain).

Golden rule (sab kuch isi se aata hai): upar aur neeche ko same nonzero number se multiply karne se value nahi badalti: $\frac{a}{b} = \frac{a \cdot k}{b \cdot k} (k \neq = 0)$

1. Addition & Subtraction — same-size pieces first

HOW (derivation from first principles): $\frac{a}{b} + \frac{c}{d}$ se shuru karo. Golden rule ko har ek par use karo taaki ek shared bottom $b d$ ban sake: $\frac{a}{b} = \frac{a \cdot d}{b \cdot d}, \frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{b c}{b d}$ Ab dono ka denominator $b d$ hai, toh pieces match karti hain aur hum count kar sakte hain: $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ Subtraction bilkul same hai ek minus ke saath: $\frac{a}{b} - \frac{c}{d} = \frac{a d - b c}{b d}$ .

2. Multiplication — ek fraction ka fraction

HOW (derivation): Ek unit square ko $b$ columns aur $d$ rows mein split karo → $b d$ tiny cells, har ek ka area $\frac{1}{b d}$ . $a$ columns aur $c$ rows lene se $a \cdot c$ cells select hoti hain: $\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ Common denominator ki zaroorat nahi — tum seedha across multiply karte ho.

3. Division — flip and multiply

HOW (derivation, do tarike): Tarika 1 — multiplication undo karo. Hum $x$ chahte hain jisme $x \cdot \frac{c}{d} = \frac{a}{b}$ ho. Dono sides ko $\frac{d}{c}$ (reciprocal, jo $\frac{c}{d}$ ko $1$ bana deta hai) se multiply karo: $x = \frac{a}{b} \cdot \frac{d}{c}$ Tarika 2 — denominators clear karo. Complex fraction ke top aur bottom ko $\frac{d}{c}$ se multiply karo: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b} \cdot \frac{d}{c}}{\frac{c}{d} \cdot \frac{d}{c}} = \frac{\frac{a}{b} \cdot \frac{d}{c}}{1}$ Kisi bhi tarike se: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}$

Figure — Addition, subtraction, multiplication, division of fractions

Common Mistakes (Steel-manned)

Active Recall

Recall Reveal se pehle try karo

$+$ ke liye common denominator kyun chahiye lekin $\times$ ke liye nahi?
" $x \cdot \frac{c}{d} = \frac{a}{b}$ " se $\frac{a}{b} \div \frac{c}{d}$ derive karo.
LCM use karke $\frac{5}{6} - \frac{3}{4}$ compute karo.

Answers: 1) Addition equal-size pieces count karta hai, isliye pieces match karni chahiye; multiplication fraction-of-a-fraction (area) leta hai, koi matching nahi chahiye. 2) Dono sides ko reciprocal $\frac{d}{c}$ se multiply karo $\Rightarrow x = \frac{a}{b} \cdot \frac{d}{c}$ . 3) $\frac{10 - 9}{12} = \frac{1}{12}$ .

Recall Feynman: ek 12-saal ke bacche ko explain karo

Pizzas imagine karo jo slices mein kaate hue hain. Add karne ke liye, slices same size ki honi chahiye — isliye pehle dono pizzas ko equal slices mein re-cut karo (common denominator), phir bas slices count karo. Ek-tihai pizza ka aadha nikalne ke liye, tum ise do baar kaatate ho — yahi multiply karna hai, aur tum "cut" numbers seedha across multiply karte ho. Divide karne ke liye — "aadhe pizza mein kitne quarter-pizza fit hote hain?" — tum bas chote wale ko flip karte ho aur multiply karte ho — answer (2) bada hai kyunki chote pieces bahut baar fit hote hain.

Flashcards

Do fractions add karne se pehle kya karna chahiye?

Unhe common denominator ke saath likhna chahiye taaki pieces same size ki hoon, phir numerators add karo.

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

ka formula kya hai?

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

Fractions ko straight across multiply kyun kar sakte hain lekin add nahi kar sakte?

Multiply karna ek fraction ka fraction leta hai (ek area), jisme equal-size pieces ki zaroorat nahi; addition pieces count karta hai, jo equal-size hone chahiye.

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

ka formula kya hai?

\frac{a c}{b d}

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

kaise karte hain?

Reciprocal se multiply karo:

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

(Keep-Change-Flip).

Division mein divisor kyun flip hota hai?

Kyunki reciprocal

\frac{d}{c}

\frac{c}{d}

1

bana deta hai, multiplication undo karta hai.

\frac{2}{5}

ka reciprocal kya hai?

\frac{5}{2}

(kyunki

\frac{2}{5} \times \frac{5}{2} = 1

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

compute karo.

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

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

compute karo.

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

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

kyun hai?

Har piece ko

k

chote pieces mein kaatne se

k \times

zyada pieces milti hain jo

k \times

choti hoti hain — total maatra unchanged rehti hai.

Multiply karte waqt, pehle cancel kyun karo?

Yeh numbers ko chota rakhta hai aur yeh bas golden rule (

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

) pehle apply karna hai.

Connections

Fractions - meaning and equivalent fractions
LCM and HCF — sabse chota common denominator dhundhne mein use hota hai
Reciprocals and multiplicative inverse
Decimals and fractions
Ratios and proportion
Rational numbers