Intuition The big pictureA ratio is just a comparison of sizes by division , not by subtraction. When we say the ratio of apples to oranges is 3 : 2 3:2 3 : 2 not mean "3 more apples" — we mean "for every 3 apples there are 2 oranges". The relationship is multiplicative . Everything in this topic (equivalent ratios, dividing in a ratio) flows from one single idea: a ratio only tells you the relative size, so you can scale both parts by the same factor without changing the ratio.
A ratio a : b a:b a : b same kind (same units) by division. It represents the fraction a b \dfrac{a}{b} b a
a a a b b b terms of the ratio.A ratio has no units (units cancel).
Order matters: 3 : 2 ≠ 2 : 3 3:2 \neq 2:3 3 : 2 = 2 : 3
WHY same kind? Because dividing 3 kg 3\text{ kg} 3 kg 2 kg 2\text{ kg} 2 kg 1.5 1.5 1.5
A proportion is a statement that two ratios are equal : a : b = c : d a:b = c:d a : b = c : d a a a b b b c c c d d d a , d a,d a , d extremes and b , c b,c b , c means .
Since a : b a:b a : b means the fraction a b \dfrac{a}{b} b a both terms by the same non-zero number k k k k a k b = a b \dfrac{ka}{kb}=\dfrac{a}{b} k b k a = b a a : b a:b a : b k a : k b ka:kb k a : k b equivalent ratios .
Worked example Simplifying18 : 24 18:24 18 : 24 Step 1 — find gcd. gcd ( 18 , 24 ) = 6 \gcd(18,24)=6 g cd( 18 , 24 ) = 6 Why? We want the largest number dividing both so the result can't be reduced further.
Step 2 — divide both terms. 18 ÷ 6 : 24 ÷ 6 = 3 : 4 18\div6 : 24\div6 = 3:4 18 ÷ 6 : 24 ÷ 6 = 3 : 4 Check: 18 24 = 0.75 = 3 4 \dfrac{18}{24}=0.75=\dfrac34 24 18 = 0.75 = 4 3
4 : 6 4:6 4 : 6 10 : 15 10:15 10 : 15 Cross-multiply to test a : b = c : d ⟺ a d = b c a:b=c:d \iff ad=bc a : b = c : d ⟺ a d = b c Why cross-multiply? Because a b = c d \dfrac{a}{b}=\dfrac{c}{d} b a = d c b d bd b d a d = b c ad=bc a d = b c 4 × 15 = 60 4\times15 = 60 4 × 15 = 60 6 × 10 = 60 6\times10 = 60 6 × 10 = 60 yes, equivalent. ✓
x : 12 = 5 : 4 x:12 = 5:4 x : 12 = 5 : 4 4 x = 12 × 5 = 60 4x = 12\times5 = 60 4 x = 12 × 5 = 60 x = 15 x = 15 x = 15 Why this step? a d = b c ad=bc a d = b c
Intuition Think in "parts"If you split ₹ 100 ₹100 ₹100 2 : 3 2:3 2 : 3 equal parts : A gets 2 parts, B gets 3 parts, so there are 2 + 3 = 5 2+3=5 2 + 3 = 5 100 5 = ₹ 20 \dfrac{100}{5}=₹20 5 100 = ₹20
₹ 560 ₹560 ₹560 3 : 5 3:5 3 : 5 Step 1 — total parts. 3 + 5 = 8 3+5=8 3 + 5 = 8 Why? We need the size of one part.
Step 2 — one part. x = 560 8 = ₹ 70 x = \dfrac{560}{8}=₹70 x = 8 560 = ₹70 Step 3 — shares. P = 3 × 70 = ₹ 210 =3\times70=₹210 = 3 × 70 = ₹210 = 5 × 70 = ₹ 350 =5\times70=₹350 = 5 × 70 = ₹350 Check: 210 + 350 = 560 210+350=560 210 + 350 = 560 210 : 350 = 3 : 5 210:350 = 3:5 210 : 350 = 3 : 5
Worked example Three-way split —₹ 1200 ₹1200 ₹1200 1 : 2 : 3 1:2:3 1 : 2 : 3 Step 1 total parts = 1 + 2 + 3 = 6 =1+2+3=6 = 1 + 2 + 3 = 6 Step 2 one part = 1200 / 6 = ₹ 200 =1200/6=₹200 = 1200/6 = ₹200 Step 3 shares = 200 , 400 , 600 = 200, 400, 600 = 200 , 400 , 600 Why extend so easily? The "parts" idea generalises to any number of terms; you just add all terms.
Check: 200 + 400 + 600 = 1200 200+400+600=1200 200 + 400 + 600 = 1200
Recall Forecast before reading the answerQ: A rope of length 84 84 84 4 : 3 4:3 4 : 3 Verify: total parts = 7 =7 = 7 = 84 / 7 = 12 =84/7=12 = 84/7 = 12 = 4 × 12 = 48 =4\times12=48 = 4 × 12 = 48 48 48 48
Common mistake Mistake 1: Adding instead of scalingThe wrong idea: "3 : 2 3:2 3 : 2 4 : 3 4:3 4 : 3 3 + 1 = 4 , 2 + 1 = 3 3+1=4,\ 2+1=3 3 + 1 = 4 , 2 + 1 = 3 Why it's wrong: ratios are multiplicative. 3 2 = 1.5 \dfrac32=1.5 2 3 = 1.5 4 3 = 1.33 \dfrac43=1.33 3 4 = 1.33 Fix: to make an equivalent ratio, multiply/divide both terms by the same number, never add.
Common mistake Mistake 2: Dividing total by only one termThe wrong idea: For ₹ 100 ₹100 ₹100 2 : 3 2:3 2 : 3 = 100 / 2 =100/2 = 100/2 100 / 3 100/3 100/3 Why it feels right: you see a 2 and a 3 and want to divide.
Fix: divide the total by the sum of parts ( a + b ) (a+b) ( a + b )
Common mistake Mistake 3: Forgetting ordera : b ≠ b : a a:b \ne b:a a : b = b : a Always label which term belongs to whom.
Mnemonic Remember dividing-in-ratio"Add the parts, split the total, multiply back."
Add (a + b a+b a + b T ÷ ( a + b ) T \div (a+b) T ÷ ( a + b ) × \times ×
Recall Feynman: explain to a 12-year-oldImagine sharing a pizza with your friend so that you get 2 slices for every 3 slices your friend gets. First cut the pizza into 2 + 3 = 5 2+3=5 2 + 3 = 5 pattern of sharing, not the exact numbers, so you can use it whether the pizza is small or huge. If the pizza had 10 slices instead, you'd double everything: you take 4, friend takes 6 — still the same 2 : 3 2:3 2 : 3
What does a ratio a : b a:b a : b Two quantities of the same kind, by division — it equals the fraction
a / b a/b a / b .
How do you make an equivalent ratio? Multiply (or divide) BOTH terms by the same non-zero number.
How do you simplify a ratio to lowest terms? Divide both terms by their
gcd \gcd g cd.
Test whether a : b = c : d a:b=c:d a : b = c : d Cross-multiply: they're equal iff
a d = b c ad=bc a d = b c (product of extremes = product of means).
To divide total T T T a : b a:b a : b x = T / ( a + b ) x=T/(a+b) x = T / ( a + b ) .
Formula for A's share when dividing T T T a : b a:b a : b A = a a + b T A = \frac{a}{a+b}\,T A = a + b a T .
Why must ratio terms have the same units? So units cancel and the ratio is a pure comparison number.
Common error when making equivalent ratios? Adding the same number to both terms instead of multiplying.
Divide ₹ 560 ₹560 ₹560 3 : 5 3:5 3 : 5 One part
= 70 =70 = 70 , Q
= 5 × 70 = ₹ 350 =5\times70=₹350 = 5 × 70 = ₹350 .
In proportion a : b = c : d a:b=c:d a : b = c : d The inner terms
b b b and
c c c .
Fractions and simplification — a ratio is a fraction in disguise.Highest Common Factor (GCD) — used to reduce ratios.Percentages — a percentage is a ratio out of 100.Unitary method — "value of one part" is the unitary idea.Direct and inverse proportion — proportions where ratios stay constant / reciprocal.Similar figures — corresponding sides are in equal ratios.
share = part fraction times total
Intuition Hinglish mein samjhoDekho, ratio ka matlab sirf comparison hai — do cheezon ko divide karke compare karna, subtract karke nahi. Jab hum bolte hain apples aur oranges ka ratio 3 : 2 3:2 3 : 2 multiplicative rishta hai. Isiliye agar tum dono terms ko same number se multiply ya divide karo, ratio same rehta hai. Yehi equivalent ratio ka funda hai: 2 : 3 = 4 : 6 = 6 : 9 2:3 = 4:6 = 6:9 2 : 3 = 4 : 6 = 6 : 9 2 / 3 2/3 2/3
Dividing in a ratio ka trick bilkul simple hai — "parts" mein socho. Maan lo ₹ 560 ₹560 ₹560 3 : 5 3:5 3 : 5 = 3 + 5 = 8 =3+5=8 = 3 + 5 = 8 = 560 / 8 = ₹ 70 = 560/8 = ₹70 = 560/8 = ₹70 3 × 70 = ₹ 210 3\times70=₹210 3 × 70 = ₹210 5 × 70 = ₹ 350 5\times70=₹350 5 × 70 = ₹350 add the parts, split the total, multiply back yaad rakho.
Sabse badi galti students yeh karte hain ki equivalent ratio banane ke liye dono terms mein same number add kar dete hain (jaise 3 : 2 3:2 3 : 2 4 : 3 4:3 4 : 3 3 / 2 = 1.5 3/2 = 1.5 3/2 = 1.5 4 / 3 = 1.33 4/3 = 1.33 4/3 = 1.33 sum of parts ( a + b ) (a+b) ( a + b )
