SATHEE: Ratio Proportion Tricks

Ratio Proportion Tricks – Quick Revision

Ratio = a : b ⇒ fraction = a/b ; always cancel to lowest form.
Proportion ⇒ two ratios set equal: a : b :: c : d ⇒ a·d = b·c (cross-product).
Compounded ratio of a : b and c : d is a·c : b·d.
Duplicate ratio of a : b = a² : b²; sub-duplicate = √a : √b.
Triplicate & sub-triplicate follow cubes & cube-roots respectively.
Invertendo – invert both sides: a/b = c/d ⇒ b/a = d/c.
Alternendo – swap means: a/b = c/d ⇒ a/c = b/d.
Componendo – add 1 to both sides: a/b = c/d ⇒ (a+b)/b = (c+d)/d.
Dividendo – subtract 1: (a–b)/b = (c–d)/d.
Componendo-Dividendo – (a+b)/(a–b) = (c+d)/(c–d); fastest shortcut for 2-variable equations.
If A : B = x : y and B : C = y : z ⇒ A : B : C = x : y : z (chain-link).
To divide ₹X in ratio m : n : p, 1st part = X·m/(m+n+p).
Adding/subtracting same number to both terms changes ratio; use short-cut: (a±k)/(b±k) = new ratio.
Fourth proportional to a, b, c is d satisfying a : b = c : d.
Mean proportional b/w a & c = √(a·c).
Third proportional to a, b is c such that a : b = b : c.
Percentage change in product A·B if A↑x% & B↓y% ≈ (x–y –xy/100)%.
In map-scale problems, 1 : r ⇒ 1 cm = r cm; convert units first.
Partnership profit shares ∝ (capital × time); express time in same units.
Never assume integer values unless specified—always take LCM-friendly numbers.

Formula/Rule	Application
Cross-product check	a:b::c:d ⇔ ad = bc
Compounded ratio	(a:b)(c:d)(e:f) = ace : bdf
Componendo-Dividendo	If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
Dividing quantity Q in m:n	1st share = Q·m/(m+n)
Chain rule (3 ratios)	A:B = x:y & B:C = y:z ⇒ A:B:C = x:y:z
Partnership share	(Capital × Months) ratio = profit ratio
Fourth proportional	a:b = c:x ⇒ x = bc/a
Mean proportional	x = √(ab)
Third proportional	a:b = b:x ⇒ x = b²/a
% change in ratio	New – Old / Old × 100

C-D Magic: Componendo-Dividendo → “Cross-Add, Cross-Subtract, Then Cross-Divide.”
Proportion = Product: “P-P handshake” – outer product = inner product.
Chain-Link: “B repeats – drop it; A C E … keep firsts.”
Partnership: “CM = PM” (Capital × Months = Profit Money).
LCM hack: Ratio sums fear LCM – take LCM of denominators to clear fractions instantly.

Mistake	Correct Approach
Forgetting to reduce ratio to lowest terms	Always divide by HCF
Cross-multiplying without checking 4 terms	Ensure a,b,c,d are in proportion first
Applying C-D on 3 variables	Use C-D only for 2-variable ratio equality
Taking map-scale in different units	Convert both sides to same unit (cm/km)
Using arithmetic mean instead of geometric mean for proportional	Mean proportional needs √(ab)

First 5 sec: spot keywords “divides in ratio”, “share”, “proportional” & tag the correct formula.
C-D shortcut saves 90 sec – use when same (ax±b)/(cx±d) form appears.
Take 100 as base value for % based ratio Qs – keeps numbers integer.
Write extreme left & extreme right products (ad) above/below to avoid cross-product error.
If options are far apart, estimate with nearest simple ratio instead of exact calc.

1. If 3 : 7 is equivalent to (3+x) : (7+x), find x.

⇒ 3(7+x)=7(3+x) ⇒ 21+3x=21+7x ⇒ x=0 Ans: 0

2. Divide ₹2,640 in ratio 5 : 3 : 2; largest share is?

Largest = 2640×5/10 = ₹1,320 Ans: 1320

3. Mean proportional between 18 & 8 is?

√(18×8)=√144=12 Ans: 12

4. If A:B=2:3, B:C=4:5, find A:C.

A:C = (2×4):(3×5)=8:15 Ans: 8:15

5. Fourth proportional to 5,8,15 is?

5:8=15:x ⇒ x=(8×15)/5=24 Ans: 24