Percentage and ratio are two sides of the same coin. You use them every day — a bank clerk comparing loan amounts, a farmer splitting produce, a cashier calculating discounts. In the exam, they appear in almost every DI set and standalone arithmetic question.
Percentage is simply a fraction with denominator 100. When you say "40% of a salary", you mean 40/100 of that salary. The word itself comes from "per centum" — per hundred. Think of it as a universal translator: it lets you compare quantities that have different bases. Can't directly compare 18 out of 24 with 25 out of 40? Convert both to percentages (75% vs 62.5%) and the comparison is instant.
Here's a grounding analogy: imagine a village well that holds 500 litres. If 100 litres are drawn, that's 20% of the well's capacity. If a second well holds 200 litres and 60 litres are drawn, that's 30%. Percentages let you say the second well is being used more intensively — even though the raw numbers are smaller.
Ratio expresses how two or more quantities relate to each other in multiplicative terms. A ratio of 3:5 between wheat and rice in a bag means for every 3 parts wheat, there are 5 parts rice — total 8 parts. The actual quantities could be 30 kg and 50 kg, or 300 kg and 500 kg. The ratio stays the same.
Proportion is when two ratios are equal: a:b = c:d, which gives you the cross-multiplication rule a × d = b × c. This is the engine behind most mixture, partnership, and sharing problems.
The connection between the two: a ratio a:b is equivalent to saying A is a/(a+b) × 100% of the total and B is b/(a+b) × 100% of the total. Lock this translation in your head — it's the bridge you'll cross dozens of times in this exam.
A ratio a:b is written in simplest form by dividing both terms by their HCF. Always simplify before doing anything else.
Combining two separate ratios into one combined ratio is the most tested ratio skill. If A:B = 2:3 and B:C = 4:5, you need A:B:C. The method: make the common term (B) the same value in both ratios.
A:B = 2:3 → multiply by 4 → 8:12B:C = 4:5 → multiply by 3 → 12:15A:B:C = 8:12:15This LCM technique works for three or more linked ratios too. Chain them the same way.
Finding actual quantities from a ratio: If two numbers are in ratio m:n and their sum is S, then:
m/(m+n) × Sn/(m+n) × SNo need to set up x and mx, nx every time — just use the fraction directly. Saves one algebraic step per problem.
Converting between fraction and percentage:
3/4 = 75%.37.5% = 3/8.Memorise the fraction-percent equivalents for common values: 1/8 = 12.5%, 1/6 = 16.67%, 1/5 = 20%, 1/4 = 25%, 1/3 = 33.33%, 2/3 = 66.67%, 3/4 = 75%. These save you from division in 80% of percentage problems.
Percentage increase and decrease:
r%: multiply by (1 + r/100) or equivalently (100+r)/100.r%: multiply by (1 - r/100) or (100-r)/100.Successive percentage changes: If a value increases by a% then decreases by b%, the net change is not simply (a - b)%. The correct formula for net percentage effect:
For two increases: a + b + ab/100. For two decreases: -a - b + ab/100.
Price-consumption inverse relationship: This is tested every cycle. If price increases by r%, consumption must decrease to keep expenditure constant. The required reduction:
For a 25% price rise: reduction = 25/125 × 100 = 20%. Memorise this formula, don't re-derive under time pressure.
Percentage chain problems (A relates to B, B relates to C): Work step-by-step. Convert each percentage to a multiplier and chain them. If B is 25% less than C and A is 20% more than B:
0.75 × C1.20 × B = 1.20 × 0.75 × C = 0.90 × C
So A is 10% less than C.When quantities have different weights (like subjects with different credit hours), the simple average of percentages is wrong. You need weighted average:
This shows up in DI questions disguised as "overall performance" or "combined score" problems.
When a substance is added to a mixture, only the added component changes. Set up the equation with the ratio condition on the new total. Cross-multiply to solve for the unknown variable x. This is the template for all alligation-type problems at this level.
When price rises by r%, consumption falls by r/(100+r) × 100% to keep spending constant. Standard derivation takes 45 seconds (setting up P×C = new P × new C, solving). This formula gives the answer in under 10 seconds.
Micro-example: Price rises 25%. Reduction = 25/125 × 100 = 20%. Done in one step vs four algebraic steps.
Speed: standard method ~45s, formula ~8s.
Given ratio m:n and sum S, find smaller number directly as m/(m+n) × S without introducing variable x.
Micro-example: Ratio 5:8, sum 156. Smaller = 5/13 × 156 = 5 × 12 = 60. You bypass writing "5x + 8x = 156, x = 12, 5x = 60" and go straight to the answer in one mental multiplication.
Speed: standard method 4 steps, shortcut 2 steps.
To combine A:B and B:C into A:B:C, take LCM of the two values of B and scale both ratios up. No algebra required.
Micro-example: A:B = 3:5, B:C = 2:7. LCM(5,2)=10. A:B becomes 6:10, B:C becomes 10:35. So A:B:C = 6:10:35. This takes under 20 seconds versus setting up simultaneous equations (~60 seconds).
Speed: equation method ~60s, LCM method ~15s.
For two consecutive percentage changes of +a% then +b% (or with signs for decrease), net effect = a + b + ab/100.
Micro-example: Population grows 10% then 10%. Net = 10 + 10 + (10×10)/100 = 20 + 1 = 21%. So 50,000 × 1.21 = 60,500. You skip two multiplications and get the net multiplier directly.
Speed: two-step multiplication ~30s, formula ~10s.
Memorise: 1/8=12.5%, 1/6≈16.67%, 1/5=20%, 1/4=25%, 1/3≈33.33%, 3/8=37.5%, 2/5=40%, 1/2=50%, 3/5=60%, 2/3≈66.67%, 3/4=75%, 4/5=80%, 5/6≈83.33%.
Micro-example: "What is 37.5% of 480?" Recognize 37.5% = 3/8. Then 3/8 × 480 = 3 × 60 = 180. Versus long multiplication of 480 × 0.375.
Speed: decimal multiplication ~25s, fraction lookup ~8s.
When you see a percentage or ratio question in the exam hall, run this decision tree:
Step 1 — Identify question type:
Step 2 — Check if a fraction equivalent exists before doing any decimal arithmetic. If the percentage is 12.5%, 25%, 33.33%, 37.5%, 66.67%, 75%, or 87.5%, switch to fraction immediately.
Step 3 — Avoid unnecessary variables. In most ratio problems, you don't need to solve for x if you can work with the ratio fraction directly. Introduce x only when you need an actual quantity and no direct shortcut applies.
Step 4 — Sanity check: Does your answer match the expected range? If the smaller of two numbers in ratio 5:8 comes out larger than the sum, something is wrong. One-second logic check before marking.
Why this question: Tests the core combined-ratio skill — the most frequently examined ratio concept across all banking clericals.
Solving path: A:B = 2:3, B:C = 4:5. B appears as 3 in the first and 4 in the second. LCM(3,4) = 12. Scale A:B by 4 → 8:12. Scale B:C by 3 → 12:15. Read off: A:B:C = 8:12:15. Elapsed time: under 15 seconds using the LCM method.
Why this question: The ratio-to-actual-value conversion. Simple template that students sometimes over-complicate.
Solving path: Total parts = 5 + 8 = 13. Sum = 156, so one part = 156/13 = 12. Smaller number = 5 × 12 = 60. Alternatively, smaller = 5/13 × 156 = 60 directly. Don't write out 5x + 8x = 156 unless you're stuck — the fraction route is faster.
Why this question: Weighted average disguised as a "percentage" problem. Students who take simple average (85+90+80)/3 get 85%, which is wrong and is a trap option.
Solving path: Weights are 2, 3, 1. Total weight = 6. Numerator = (85×2) + (90×3) + (80×1) = 170 + 270 + 80 = 520. Weighted average = 520/6 = 86.67%. The simple average trap (85%) is option B — don't fall for it.
Why this question: Classic price-consumption problem. Every IBPS RRB batch sees at least one of these.
Solving path: Price up 25% means r = 25. Required consumption reduction = 25/(100+25) × 100 = 25/125 × 100 = 20%. If you don't have the formula memorised, derive: original expenditure P×C = new expenditure 1.25P × new_C → new_C = C/1.25 = 0.8C → reduction = 20%.
Why this question: Mixture-with-addition — tests whether you can set up a cross-multiplication correctly when only one component changes.
Solving path: Let milk = 4x, water = 3x initially. After adding 14 litres water: milk stays 4x, water becomes 3x + 14. New ratio condition: 4x/(3x+14) = 4/5. Cross multiply: 20x = 12x + 56. So 8x = 56, x = 7. Initial milk = 4 × 7 = 28 litres. Note: the milk quantity never changed — only water did. Many students add to both sides by mistake.
Taking simple average instead of weighted average. When subjects or categories have different weights, the straight average of percentages is wrong. Always check if weights are equal before averaging.
Ignoring the direction of a percentage relationship. "A is 20% more than B" means A = 1.2B. It does NOT mean B = 0.8A. These give different values of B. The base matters — percentage comparisons are not symmetric.
Adding percentage changes linearly for successive changes. Two consecutive 10% increases are not a 20% increase — they are 21% (the cross-term ab/100 is missing). This error surfaces in compound interest, population growth, and depreciation questions.
Not simplifying ratios before combining. If A:B = 6:9, simplify to 2:3 before using it in a combination. Combining unsimplified ratios leads to inflated LCMs and arithmetic errors.
Setting up the mixture equation incorrectly when only one component is added. When water is added to a milk-water mixture, milk stays constant. Students sometimes increase both milk and water by the added amount. Only the component actually added changes.
Using the wrong base for percentage change. "Reduced by 20% from new price" and "reduced by 20% from original price" give different answers. Identify what the percentage is being applied to before calculating.