SATHEE: Ratio Master - Quick Revision

Ratio Master - Quick Revision

Ratio a:b is the same as ka:kb for any k≠0.
Always reduce a ratio to its lowest terms (HCF = 1).
Compounded ratio of a:b & c:d is ac:bd.
Duplicate ratio of a:b is a²:b²; triplicate is a³:b³.
Sub-duplicate ratio of a:b is √a:√b.
Reciprocal (inverse) ratio of a:b is b:a.
If A:B = 2:3 & B:C = 4:5, then A:B:C = 8:12:15.
In a proportion a:b = c:d, product of extremes = product of means (ad = bc).
Fourth proportional to a,b,c is (b×c)/a.
Third proportional to a,b is b²/a.
Mean proportional between a & b is √(ab).
Componendo: if a/b = c/d then (a+b)/b = (c+d)/d.
Dividendo: (a–b)/b = (c–d)/d.
Componendo-Dividendo: (a+b)/(a–b) = (c+d)/(c–d).
In a mixture, strength % = (part/whole) × 100.
Split ₹X in a:b:c → shares are Xa/(a+b+c), Xb/(a+b+c), Xc/(a+b+c).
Allegation: (Cheaper qty)/(Dearer qty) = (Dearer mean – Mean)/(Mean – Cheaper mean).
If incomes are 5x & 3x and expenses 4y & 2y, savings are 5x–4y & 3x–2y.
Speed ratio 3:4 ⇒ time ratio for same distance is 4:3.
“Cross-multiply first, simplify last” – saves 10 seconds per proportion.

Formula	Use
a:b = c:d ⇔ ad = bc	Test or create proportion
a:b:c = (a×HCF of b,c) : (b×HCF of a,c) : (c×HCF of a,b)	Chain ratio LCM trick
% of A in A:B mix = a/(a+b) × 100	Quick strength
(Amount × Ratio digit) ⁄ Sum of ratio digits	Share in partnership
C.P. in allegation = (D×m – C×m)/(C – D)	Cost mixing
If x:y becomes (x+k):(y+k) & new ratio = p:q, solve (x+k)/(y+k)=p/q	Age/dilution problems
A:B:C = (A/B)(B/C) : (B/C) : 1	Shortcut 3-variable link
Speed1:Speed2 = T2:T1 (distance constant)	Time-speed inverse
Mean ratio √(ab)	Geometric mean
k = (a – b)/(c – d) ⇒ a = kc + (b – kd)	Linear model constant

“CRAB-L” – Cross-multiply Ratio, Answer Becomes Legit (check proportion).
“D2T2” – Distance same, Speed ∝ 1/Time (write 2 below each).
“C-D brings CD” – Componendo-Dividendo always gives (a+b)/(a–b) form.
“Mean Root Beer” – Mean proportional = √ab (think root beer in middle).
“Allegation Smiley” – Draw two dots & a smile; cheaper left, dearer right, mean on chin – subtract diagonally for ratio.

Error	Correct
Writing 2:4 in answer (not reduced)	Always 1:2
Applying componendo on only one side	Do SAME operation to both sides
Taking third proportional as b/a	It’s b²/a
Using allegation on percentages without same base	Convert both to common % strength first
Confusing duplicate with square of entire ratio	Duplicate of a:b is a²:b², not (a:b)²

Show MCQs

If 3:7 = x:42, then x is
A) 14 B) 18 C) 21 D) 12
The triplicate ratio of 2:3 is
A) 4:9 B) 8:27 C) 6:9 D) 2:9
Mean proportional between 4 & 9 is
A) 6 B) 6.5 C) 5 D) 36
In what ratio must tea @₹40/kg be mixed with tea @₹65/kg to get ₹50/kg?
A) 3:2 B) 2:3 C) 3:1 D) 1:3
If A:B = 3:4 and B:C = 8:5, then A:B:C equals
A) 6:8:5 B) 3:4:5 C) 24:32:20 D) 6:8:10

Answers: 1-B, 2-B, 3-A, 4-A, 5-C