Ratios are everywhere in CAT. Master the fraction mindset and proportion shortcuts to solve questions in under 60 seconds.
A ratio is simply a way to compare two quantities.
When you say "for every 2 boys there are 3 girls in the class," you are using a ratio. The ratio is 2:3 (read as "2 is to 3").
A proportion says that two ratios are equal. If 2 boys to 3 girls is the same relationship in two different classes, that is a proportion.
These two ideas — ratio and proportion — are the foundation of almost every CAT Arithmetic problem. Percentages, profit & loss, time & work, mixtures — all of them are secretly ratio problems in disguise.
Ratio & Proportion directly appears in 3–5 CAT questions every year. More importantly, it appears indirectly in almost every other topic. A student who thinks in ratios solves problems 2× faster than one who uses formulas.
| Topic | How Ratio Appears |
|---|---|
| Percentages | Percent = Part/Whole × 100 |
| Profit & Loss | Profit Ratio = Profit/Cost |
| Time & Work | Efficiency Ratio |
| Mixtures | Concentration Ratio |
| Speed & Distance | Speed Ratio = Distance/Time |
A ratio a:b means for every a units of the first quantity, there are b units of the second.
Key rule: Ratios only compare. They do NOT give actual values.
If the ratio of boys to girls is 3:4, there could be:
The actual values depend on the total or one specific value.
This is the most powerful idea in ratio problems.
If ratio is a:b, then actual values are ak and bk for some positive number k.
Example: Boys:Girls = 3:4 and total students = 35.
Boys = 3k, Girls = 4k. So 3k + 4k = 35 → 7k = 35 → k = 5.
Boys = 15, Girls = 20.
Always introduce k when you see a ratio and need actual values.
| Type | Meaning | Example |
|---|---|---|
| Duplicate Ratio | Square of each term | 3:4 → 9:16 |
| Sub-duplicate Ratio | Square root | 9:16 → 3:4 |
| Triplicate Ratio | Cube | 2:3 → 8:27 |
| Inverse Ratio | Flip | 3:4 → 4:3 |
| Compound Ratio | Multiply two ratios | (2:3) × (4:5) = 8:15 |
A proportion states: a:b = c:d, which means a/b = c/d.
This gives the cross-multiplication rule: a × d = b × c
The outer terms (a and d) are called extremes. The inner terms (b and c) are called means.
Property: Product of extremes = Product of means.
If 12:x = 4:7, find x.
Cross multiply: 12 × 7 = 4 × x → 84 = 4x → x = 21.
These are three important types that appear in exams.
Fourth Proportional: If a:b = c:x, find x.
x = (b × c) / a
Third Proportional: If a:b = b:x, find x.
x = b² / a
Mean Proportional: If a:x = x:b, find x.
x = √(ab)
Find the mean proportional between 4 and 25.
x = √(4 × 25) = √100 = 10.
Check: 4:10 = 10:25 ✓ (both equal 2:5)
When dividing quantity Q in ratio a:b:c:
Divide ₹1200 in ratio 3:4:5.
Total parts = 12.
To compare a:b and c:d, cross multiply.
If ad > bc, then a:b > c:d.
If ad < bc, then a:b < c:d.
Which is greater: 7:9 or 5:6?
Cross multiply: 7×6=42 vs 5×9=45.
Since 42 < 45, we get 7:9 < 5:6.
If a/b = c/d, then:
(a+b)/(a-b) = (c+d)/(c-d)
This is called Componendo-Dividendo and it lets you simplify complex fraction equations instantly.
If (x+3)/(x-3) = 5/3, find x.
By Componendo-Dividendo in reverse: x/3 = (5+3)/(5-3) = 8/2 = 4.
So x = 12.
Verification: (12+3)/(12-3) = 15/9 = 5/3 ✓
Direct Variation: If A increases, B increases proportionally.
A ∝ B means A = kB for some constant k.
Inverse Variation: If A increases, B decreases proportionally.
A ∝ 1/B means AB = k (constant).
Joint Variation: A ∝ BC means A = kBC.
If 8 workers can build a wall in 6 days, how many days for 12 workers?
Workers × Days = constant work → 8 × 6 = 12 × D → D = 4 days.
(This is inverse variation between workers and days.)
Partnership Problems:
When partners invest for different durations:
Profit ratio = (Investment × Time) for each partner.
Partner A invests ₹5000 for 8 months, B invests ₹6000 for 5 months.
Profit ratio = (5000×8) : (6000×5) = 40000 : 30000 = 4:3.
Mixing Ratios:
If mixture A has milk:water = 3:1 and mixture B has 5:3, and you mix them in 1:1 ratio:
Milk in A per litre = 3/4. Milk in B per litre = 5/8.
Combined milk = (3/4 + 5/8)/2 = (6/8 + 5/8)/2 = 11/16.
Water = 5/16. New ratio = 11:5.
| Term | Meaning |
|---|---|
| Ratio | Comparison of two quantities (a:b) |
| Proportion | Equality of two ratios (a:b = c:d) |
| Extremes | Outer terms of a proportion |
| Means | Inner terms of a proportion |
| Multiplier (k) | The scaling factor linking ratio to actuals |
| Direct Variation | As one increases, other increases |
| Inverse Variation | As one increases, other decreases |
| Componendo | a:b → (a+b):b |
| Dividendo | a:b → (a-b):b |
Think of ratio as a recipe. If a cookie recipe uses flour:sugar = 3:1, and you have 12 cups of flour, you need 4 cups of sugar. The ratio scales up and down but the proportion stays the same.
❌ Mistake 1: "Ratio 3:5 means 3 and 5 exactly."
✅ Correct: Ratio 3:5 means 3k and 5k. Find k from additional info.
❌ Mistake 2: Adding/subtracting ratios directly.
✅ Correct: Convert to fractions first: 3:4 = 0.75, not 3.
❌ Mistake 3: Forgetting to adjust for time in partnership.
✅ Correct: Profit ∝ Capital × Time, not just Capital.
Ratio compares two quantities using the format a:b. The key skill is introducing the multiplier k to find actual values. Proportion (a:b = c:d) gives the cross-multiplication rule ad = bc. Variation describes how quantities change together — directly or inversely. Componendo-Dividendo is the most powerful ratio shortcut for CAT algebra. Every arithmetic topic builds on ratio thinking.
Ratio & Proportion
├── Percentages (percent = ratio × 100)
├── Partnership Problems (profit ∝ capital × time)
├── Mixture & Alligation (ratio of components)
├── Time & Work (efficiency ratio)
└── Variation (direct, inverse, joint)
Topics covered