Equivalent fractions, simplifying fractions — Hinglish notes

1.1.13 · Maths › Arithmetic & Number Systems

WHAT is a fraction, really?

WHY $b \neq = 0$ ? Tum ek whole ko $0$ equal parts mein nahi kaat sakte — "zero pieces mein se har ek kitna bada hai?" ka koi jawab nahi hota. Zero se division undefined hai.

WHY equivalent fractions exist

Figure — Equivalent fractions, simplifying fractions

HOW to simplify a fraction

HOW (method 1 — GCD se divide karo, sabse fast):

$g cd (a, b)$ nikalo.
Dono ko ussi se divide karo.

HOW (method 2 — common factors ek ek karke hatao): Top aur bottom ko kisi bhi shared factor (2, 3, 5, …) se tab tak divide karte raho jab tak koi na bache. Thoda slow hai lekin GCD ki zarurat nahi padti.

Checking two fractions are equivalent (cross-multiplication)

Steel-manned mistakes

Flashcards

Fraction

\frac{a}{b}

ka matlab kya hai?

Ek whole ko

b

equal parts mein kaato aur

a

parts lo.

Denominator non-zero kyun hona chahiye?

Tum ek whole ko 0 equal parts mein nahi kaat sakte; 0 se division undefined hai.

Equivalent fractions kya hote hain?

Alag-alag fractions jo same value rakhte hain, jaise

\frac{1}{2} = \frac{2}{4} = \frac{50}{100}

Equivalent fraction banane ka rule?

Numerator AUR denominator dono ko same non-zero number se multiply (ya divide) karo.

Top aur bottom ko

k

se multiply karne se value kyon nahi badlti?

Kyunki

\frac{k}{k} = 1

aur 1 se multiply karne se kuch nahi badlta.

Fraction lowest terms mein kab hota hai?

Jab

g cd

(numerator, denominator)

= 1

ho.

Simplify karne ka sabse fast tarika?

Numerator aur denominator dono ko unke GCD se divide karo.

Cross-multiplication test for

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

Sahi hai iff

a \times d = b \times c

Top aur bottom mein same number add kyon nahi kar sakte?

Adding se ratio badal jaata hai; sirf ×/÷ by

\frac{k}{k} = 1

se value preserve hoti hai.

\frac{3}{5}

ko denominator 20 ke saath likho.

Top aur bottom dono ko 4 se multiply karo →

\frac{12}{20}

\frac{45}{60}

simplify karo.

\frac{3}{4}

(÷5 phir ÷3, ya ÷15).

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

Ek chocolate bar imagine karo. Agar tum use 2 bade chunks mein todo aur 1 khao, tumne aadha khaya. Ab wahi bar 4 pieces mein todo aur 2 khao — tumne bilkul wahi chocolate khayi! Toh "2 mein se 1" aur "4 mein se 2" same amount hai, bas chote pieces se count kiya gaya. Simplifying ka matlab hai ulta jaana: sabse bade possible pieces use karna taaki numbers jitne chote ho sake. Aur golden rule: top number ke saath jo bhi karo, bottom ke saath bhi karo — lekin sirf times ya divide, kabhi plus nahi, warna tum jo chocolate ki baat kar rahe ho woh badal jaati hai.

Connections

Factors, multiples and GCD/HCF — simplifying ke liye GCD chahiye.
Prime factorisation — GCD nikalne ka sabse fast reliable tarika.
Adding and subtracting fractions — pehle denominators equal banane padte hain equivalent fractions use karke.
Ratios and proportion — cross-multiplication wahi tool hai.
Decimals and percentages — $\frac{50}{100} = \frac{1}{2} = 0.5 = 50%$ sab equivalent hain.
Number Systems – rational numbers — har rational number lowest terms mein ek fraction hota hai.