Episode 8 — Aptitude and Reasoning / 8.5 — Ratio and Proportion
8.5 Quick Revision -- Ratio and Proportion
Use this sheet for last-minute revision before exams. All formulas, diagrams, and shortcuts in one place.
1. Core Definitions
Ratio of a to b = a : b = a/b (same units, no units in result)
Proportion = a : b :: c : d (four terms, cross product equal)
2. Master Formula Table
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| Concept | Formula / Rule |
+-------------------------------------------+----------------------------------+
| Simplify a : b | Divide both by HCF(a, b) |
| Fraction ratio (a/b) : (c/d) | Multiply by LCM(b, d) |
| Decimal ratio | Multiply by 10^n to clear |
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| Fourth Proportional to a, b, c | x = bc / a |
| Third Proportional to a, b | x = b^2 / a |
| Mean Proportional of a, b | x = sqrt(a * b) |
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| Cross Product Rule | a:b :: c:d => ad = bc |
| Componendo | (a+b)/b = (c+d)/d |
| Dividendo | (a-b)/b = (c-d)/d |
| Componendo-Dividendo | (a+b)/(a-b) = (c+d)/(c-d) |
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| Compound Ratio of a:b and c:d | ac : bd |
| Duplicate Ratio of a : b | a^2 : b^2 |
| Triplicate Ratio of a : b | a^3 : b^3 |
| Sub-duplicate Ratio of a : b | sqrt(a) : sqrt(b) |
| Sub-triplicate Ratio of a : b | cbrt(a) : cbrt(b) |
| Reciprocal Ratio of a : b | b : a |
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| Direct Proportion | x1/y1 = x2/y2 |
| Inverse Proportion | x1 * y1 = x2 * y2 |
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| Divide Q in ratio a : b | Qa/(a+b) and Qb/(a+b) |
| Divide Q in ratio a : b : c | Qa/(a+b+c), Qb/(a+b+c), etc. |
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| Partnership (same time) | Profit ratio = Capital ratio |
| Partnership (different time) | Ratio = C1*T1 : C2*T2 |
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| Alligation | Cheaper:Dearer = (D-M):(M-C) |
| Replacement (n times, x litres from C) | Left = C * (1 - x/C)^n |
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3. Alligation Diagram (Memorize This Shape)
Cheaper (C) Dearer (D)
\ /
\ /
Mean (M)
/ \
/ \
(D - M) (M - C)
Cheaper : Dearer = (D - M) : (M - C)
When to use: Any problem involving mixing two things at different rates/prices/concentrations to get a target rate/price/concentration.
4. Componendo-Dividendo (Speed Formula)
If (a + b) / (a - b) = p / q
Then a / b = (p + q) / (p - q)
When to use: Anytime you see a fraction with sum in numerator and difference in denominator (or vice versa).
5. The k-Multiplier Method
If A : B = a : b
=> Let A = ak, B = bk
=> Use any additional condition to find k
=> Substitute back
This converts every ratio problem into simple algebra. Use it always.
6. Combining Ratios -- Quick Steps
Given A:B and B:C, find A:B:C:
1. Note the value of B in each ratio.
2. Find LCM of those two B-values.
3. Scale both ratios so B = LCM.
4. Write A:B:C.
Shortcut for A:C directly:
A/C = (A/B) x (B/C)
7. Age Ratio Formula
Present ratio = a : b
Ratio after t years = c : d
Let present ages = ak, bk.
Direct formula for k:
k = t(c - d) / (ad - bc)
For "t years ago": use -t instead of t.
8. Income-Expenditure Framework
Income ratio = a : b => Incomes = ax, bx
Expenditure ratio = c : d => Expenditures = cy, dy
Each saves S:
ax - cy = S
bx - dy = S
Solve these two equations for x and y.
Quick relation: (a-b)x = (c-d)y
9. Percentage-Ratio Quick Reference
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| % change | Ratio |
+------------------+----------+
| 10% increase | 11 : 10 |
| 20% increase | 6 : 5 |
| 25% increase | 5 : 4 |
| 33.33% increase | 4 : 3 |
| 50% increase | 3 : 2 |
| 100% increase | 2 : 1 |
| 10% decrease | 9 : 10 |
| 20% decrease | 4 : 5 |
| 25% decrease | 3 : 4 |
| 33.33% decrease | 2 : 3 |
| 50% decrease | 1 : 2 |
+------------------+----------+
10. Variation Formulas
Direct: y = kx (y increases as x increases)
Inverse: y = k/x (y decreases as x increases)
Joint: z = kxy (z depends on both x and y)
Combined: z = kx/y (direct with x, inverse with y)
Step 1: Write the formula.
Step 2: Use given values to find k.
Step 3: Use k to find the unknown.
11. Speed-Time-Distance Ratio Relationships
Same distance: Speed ratio = a : b => Time ratio = b : a
Same time: Speed ratio = a : b => Distance ratio = a : b
Same speed: Time ratio = a : b => Distance ratio = a : b
12. Common Exam Traps -- Avoid These
1. Different units: Convert BEFORE forming the ratio.
Example: 2 kg : 500 g = 2000 : 500 = 4 : 1 (NOT 2 : 500)
2. Order matters: "Ratio of A to B" = A/B, not B/A.
3. Adding/subtracting from ratios changes them:
If A:B = 3:4, then (A+5):(B+5) is NOT 3:4.
4. Alligation: The mean must be BETWEEN the two values.
5. Partnership: Don't forget to multiply capital BY time.
6. Fraction ratios: Clear the fractions first using LCM.
13. Problem Type Recognition Guide
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| You see this in the problem... | Use this method... |
+--------------------------------------+-------------------------------+
| "Divide Rs X in ratio a:b" | Division formula |
| "Ratio becomes p:q after adding x" | k-multiplier + algebra |
| "Ages in ratio, after n years..." | Age ratio formula |
| "Income ratio, expenditure ratio" | Two-equation framework |
| "Mix two things at different rates" | Alligation cross method |
| "A invests X for T months" | Partnership: Capital x Time |
| "(a+b)/(a-b) = p/q" | Componendo-Dividendo |
| "x varies directly/inversely as y" | Variation formula |
| "Speed ratio a:b, find time ratio" | Inverse relationship |
| "2A = 3B = 4C, find A:B:C" | Set equal to k, express each |
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14. One-Minute Drills (Practice These Mentally)
1. Simplify 72 : 108 => 2 : 3
2. (3/5) : (2/7) => 21 : 10
3. Mean proportional of 3, 27 => 9
4. Third proportional of 6, 12 => 24
5. 4th proportional to 2,5,6 => 15
6. Divide 450 in 2:3 => 180, 270
7. Compound ratio 3:4 and 2:5 => 6 : 20 = 3 : 10
8. Duplicate ratio of 5:7 => 25 : 49
9. If 2A = 3B, then A:B => 3 : 2
10. Alligation: 20,40 for mean 25 => 15:5 = 3:1
15. Last-Minute Reminders
- In proportion
a:b::c:d, always check with cross multiplication:ad = bc. - For combining multiple ratios, work two at a time from left to right.
- In partnership problems, if a partner withdraws money, treat the before and after as two separate investments.
- Alligation works for ANY mixing problem: prices, concentrations, percentages, marks, speeds (for average speed).
- When the question says "what must be added," the answer is always positive. If your algebra gives a negative, recheck the setup.