Ratios and proportions are the silent backbone of aptitude. They show up directly in their own questions, and indirectly in mixtures, partnerships, time-work, age problems — basically everywhere. Getting comfortable with ratio manipulation is one of the highest-leverage things we can do for our score.
What Is a Ratio?
A ratio compares two quantities of the same kind. If A has 30 apples and B has 20 apples, the ratio A:B = 30:20 = 3:2.
In simple language, a ratio tells us “for every 3 apples A has, B has 2.”
Key points:
- A ratio has no units (it’s a pure comparison)
- a:b = ka:kb for any non-zero k (we can multiply/divide both sides)
- a:b is NOT the same as b:a (order matters!)
- a:b = a/b (we can treat it as a fraction)
Simplifying Ratios
Always reduce to lowest terms by dividing by HCF.
- 24:36 → divide by 12 → 2:3
- 1.5:2.5 → multiply by 2 → 3:5
- 1/3 : 1/4 → multiply by LCM(3,4) = 12 → 4:3
Ratios of fractions: To compare a/b : c/d, cross multiply → a×d : c×b
Types of Proportion
Direct Proportion
When one quantity increases, the other increases proportionally. If A ∝ B, then A/B = constant.
Example: If 5 pens cost Rs 40, how much do 8 pens cost? → 8 × (40/5) = Rs 64
Inverse Proportion
When one quantity increases, the other decreases proportionally. If A ∝ 1/B, then A × B = constant.
Example: If 6 workers can do a job in 10 days, how many days for 15 workers? → 6×10/15 = 4 days
Continued Proportion
a, b, c are in continued proportion if a:b = b:c, which means b² = ac. Here b is the mean proportional of a and c.
Key Operations on Ratios
Componendo-Dividendo in Action
This is extremely powerful for simplification problems.
Example: If (x + y)/(x - y) = 7/3, find x:y.
Using componendo-dividendo in reverse:
- (x+y)/(x-y) = 7/3
- By componendo-dividendo: [(x+y)+(x-y)] / [(x+y)-(x-y)] = (7+3)/(7-3)
- 2x/2y = 10/4
- x/y = 5/2
- x:y = 5:2
Dividing a Quantity in a Given Ratio
If we need to divide quantity Q in the ratio a:b:c:
- First part = Q × a/(a+b+c)
- Second part = Q × b/(a+b+c)
- Third part = Q × c/(a+b+c)
Example: Divide Rs 1200 among A, B, C in the ratio 2:3:5.
Sum of ratio = 10.
- A = 1200 × 2/10 = Rs 240
- B = 1200 × 3/10 = Rs 360
- C = 1200 × 5/10 = Rs 600
Combining Ratios
When we know A:B and B:C separately, we can find A:B:C.
Method: Make B common in both ratios.
Example: A:B = 2:3 and B:C = 4:5. Find A:B:C.
B is 3 in the first ratio and 4 in the second. LCM(3,4) = 12.
- A:B = 2:3 → multiply by 4 → 8:12
- B:C = 4:5 → multiply by 3 → 12:15
- A:B:C = 8:12:15
Income, Expenditure, and Savings Problems
These are classic ratio problems. The key relationship is:
Income - Expenditure = Savings
Example: A and B’s incomes are in ratio 5:4. Their expenditures are in ratio 3:2. If each saves Rs 2000, find their incomes.
Let incomes = 5x and 4x. Let expenditures = 3y and 2y.
- 5x - 3y = 2000 … (1)
- 4x - 2y = 2000 … (2)
From (2): 2x - y = 1000, so y = 2x - 1000. Sub in (1): 5x - 3(2x - 1000) = 2000 → 5x - 6x + 3000 = 2000 → -x = -1000 → x = 1000.
Incomes: A = 5000, B = 4000. A earns Rs 5000, B earns Rs 4000.
Worked Examples
Example 1: If a:b = 3:4, b:c = 5:7, and c:d = 2:3, find a:d.
a/b × b/c × c/d = a/d = 3/4 × 5/7 × 2/3 = 30/84 = 5/14. a:d = 5:14
Example 2: Rs 5600 is divided among A, B, and C such that A gets 2/3 of what B gets and B gets 1/4 of what C gets. Find each person’s share.
B = C/4, so B:C = 1:4. A = 2B/3, so A:B = 2:3. Combine: A:B = 2:3, B:C = 1:4 → make B common → A:B = 2:3, B:C = 3:12. A:B:C = 2:3:12
Sum = 17. A = 5600 × 2/17 ≈ Rs 658.82, B = 5600 × 3/17 ≈ Rs 988.24, C = 5600 × 12/17 ≈ Rs 3952.94.
Example 3: In a mixture of 60 liters, the ratio of milk to water is 2:1. How much water should be added to make the ratio 1:2?
Current: Milk = 40L, Water = 20L. After adding x liters of water: 40/(20+x) = 1/2 → 80 = 20 + x → x = 60 liters
Example 4: Two numbers are in the ratio 3:5. If 9 is added to each, the ratio becomes 3:4. Find the numbers.
Let numbers = 3x and 5x. (3x + 9)/(5x + 9) = 3/4 4(3x + 9) = 3(5x + 9) 12x + 36 = 15x + 27 3x = 9 → x = 3. Numbers: 9 and 15
Example 5: The mean proportional between two numbers is 12. If one number is 8, find the other.
Mean proportional: b² = ac → 12² = 8 × c → 144 = 8c → c = 18
Common Exam Patterns
- “Divide X in ratio a:b:c” → Each share = X × (own ratio part / total)
- “Incomes in ratio, expenditures in ratio, each saves Y” → Set up two equations, solve
- “A:B and B:C given, find A:B:C” → Make B common using LCM
- “If X is added to both, ratio becomes…” → Let quantities be ax and bx, set up equation
- “Componendo-dividendo” → If (a+b)/(a-b) = k, use C-D to find a:b directly
- “Mean proportional” → b = √(ac)
Practice Problems
Q1: A and B share a profit of Rs 9600. If A’s investment is 1.5 times B’s investment, find each person’s share.
Q2: The ratio of boys to girls in a class is 5:3. If 4 boys leave and 4 girls join, the ratio becomes 1:1. How many students were in the class originally?
Q3: If (3a + 5b)/(3a - 5b) = 5/1, find a:b.
Answers:
A1: A:B = 1.5:1 = 3:2. A’s share = 9600 × 3/5 = Rs 5760. B’s share = 9600 × 2/5 = Rs 3840.
A2: Let boys = 5x, girls = 3x. After changes: (5x-4)/(3x+4) = 1/1 → 5x-4 = 3x+4 → 2x = 8 → x = 4. Total originally = 5(4) + 3(4) = 32 students.
A3: By componendo-dividendo: [(3a+5b)+(3a-5b)] / [(3a+5b)-(3a-5b)] = (5+1)/(5-1) → 6a/10b = 6/4 → a/b = 60/40 = 3/2. a:b = 5:3.
Wait, let me redo: 6a/10b = 6/4 → a/b = (6 × 10)/(4 × 6) → that’s wrong. Let me be careful. 6a/10b = 6/4, so a/b = (6 × 10b)/(4 × 6a)… No. Cross multiply: 6a × 4 = 10b × 6 → 24a = 60b → a/b = 60/24 = 5/2. a:b = 5:2.