Numerical Ability: Percentages, Ratios & Averages

The objective paper includes "General Mental Ability" — simple maths and reasoning that many aspirants ignore and then lose easy marks on. The good news: the maths never goes beyond Class-10 level, and a handful of techniques answer most questions in under a minute. This chapter covers the number-based half.

Percentages — the Master Key

"Per cent" simply means per hundred. So 25% = 25/100 = 1/4. The fastest solvers convert percentages to fractions: 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 33⅓% = 1/3. To find x% of N, multiply: x% of N = (x × N) / 100.

The fraction–percentage table (memorise this!)
Fraction	Percentage
1/2	50%
1/3	33⅓%
1/4	25%
1/5	20%
1/8	12.5%
1/10	10%

⚠️ Exam trap

Successive changes do not simply add. Two discounts of 10% and 10% are NOT 20%: ₹100 → ₹90 → ₹81, i.e. 19% overall. Formula for two successive changes of a% and b%: net = a + b + (a×b)/100 (use minus signs for decreases).

🧠 Memory trick

Reverse trick: x% of y = y% of x. So 8% of 50 feels hard, but 50% of 8 = 4 — same answer, instant.

Ratio, Proportion & Averages

Ratio: to divide an amount in the ratio a : b, the parts are a/(a+b) and b/(a+b) of the total. Example: divide 64 in the ratio 3 : 5 → 8 equal parts of 8 each → 24 and 40.
Average = Sum ÷ Count. Example: average of 12, 18, 24, 30, 36 = 120 ÷ 5 = 24. Useful reverse form: Sum = Average × Count.
Equally spaced numbers: the average is simply the middle value (or the average of the two middle values). In the example above the middle number is 24 — no addition needed!

💡 Think of it like this

Think of an average as the "equal sharing" number: if 5 friends pool all their money and share it equally, each gets the average. That instantly explains why one very rich friend (an outlier) pulls the average up.

Direct & Inverse Proportion, and the Chain Rule

Two quantities are in direct proportion when they rise and fall together, keeping their ratio constant (e.g. cost and quantity: more items cost proportionally more). They are in inverse proportion when one rises as the other falls, keeping their product constant (e.g. speed and time for a fixed distance, or number of workers and days for a fixed job).

Direct proportion worked example: if 6 notebooks cost ₹90, then 10 notebooks cost — set up the ratio 6/90 = 10/x → x = 90 × 10/6 = ₹150.
Inverse proportion worked example: if 8 workers finish a job in 15 days, how many days will 12 workers take? Product stays constant: 8 × 15 = 12 × x → x = 120/12 = 10 days (more workers, fewer days — makes sense).
The Chain Rule (compound proportion) handles problems with three or more varying quantities at once. Classic example: If 6 men working 8 hours a day can build a wall in 10 days, how many men are needed to build it in 8 days working 6 hours a day? Set it up so each new ratio is direct or inverse as the situation demands: men are inversely proportional to days and to hours/day (more hours or more days needs fewer men). So: Men₂ = Men₁ × (Hours₁/Hours₂) × (Days₁/Days₂) = 6 × (8/6) × (10/8) = 6 × 8 × 10 / (6 × 8) = 10 men.

⚠️ Exam trap

The most common chain-rule mistake is writing every ratio as "new/old" without checking whether the relationship is direct or inverse for that specific pair. Always ask: "if this quantity increases, should the answer increase or decrease?" — that tells you whether to flip the fraction.

Speed–Time & Work — Two Formulas Rule All

Speed = Distance ÷ Time. Example: 240 km in 4 hours → 240 ÷ 4 = 60 km/h. Conversion: km/h → m/s multiply by 5/18; m/s → km/h multiply by 18/5.
Simple Interest = (P × R × T) / 100. Example: ₹5,000 at 8% for 3 years → (5000 × 8 × 3)/100 = ₹1,200.
Work: if A finishes a job in 10 days and B in 15 days, in one day they do 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 → together they finish in 6 days.

🎯 Why it matters

Mental-ability questions are the most score-friendly in the whole paper: no facts to forget, no current affairs to update — just the same dozen patterns, year after year. Fifteen minutes of practice a day converts this section into guaranteed marks.

Key takeaways

x% of N = (x × N)/100; learn the fraction table (25% = 1/4, 20% = 1/5, 12.5% = 1/8). x% of y = y% of x.
Successive changes: net = a + b + ab/100. Two 10% discounts = 19%, not 20%.
Ratio a : b over total T → parts aT/(a+b) and bT/(a+b). 64 in 3 : 5 → 24 and 40.
Average = Sum ÷ Count; for equally spaced numbers, average = middle value.
Speed = Distance ÷ Time (km/h ↔ m/s via 5/18); Simple Interest = PRT/100.
Work: add the per-day fractions — A(10 days) + B(15 days) → 1/10 + 1/15 = 1/6 → 6 days together.
Direct proportion: ratio stays constant (cost & quantity). Inverse proportion: product stays constant (speed & time; workers & days, for a fixed job). Chain rule (compound proportion): chain each new quantity as direct or inverse based on the specific relationship, not by rote.

Practice questions (11)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

What is 25% of 480?
1. 100
2. 110
3. 120
4. 140
Show answer
Answer: C — 120
25% = 1/4, so the calculation is 480 ÷ 4 = 120 — one division instead of a multiplication by 0.25. Learning the percentage-to-fraction pairs (25% = 1/4, 12.5% = 1/8, 33⅓% = 1/3) turns most percentage questions into a single step.
Divide ₹64 between two people in the ratio 3 : 5. The larger share is:
1. ₹24
2. ₹32
3. ₹40
4. ₹48
Show answer
Answer: C — ₹40
3 + 5 = 8 parts; each part = 64/8 = 8. Larger share = 5 × 8 = ₹40.
The average of 12, 18, 24, 30 and 36 is:
1. 22
2. 24
3. 26
4. 28
Show answer
Answer: B — 24
These are equally spaced (gap 6), so the average is the middle value: 24. (Check: sum 120 ÷ 5 = 24.)
A train covers 240 km in 4 hours. Its speed is:
1. 50 km/h
2. 55 km/h
3. 60 km/h
4. 65 km/h
Show answer
Answer: C — 60 km/h
Speed = Distance ÷ Time = 240 ÷ 4 = 60 km/h.
The simple interest on ₹5,000 at 8% per annum for 3 years is:
1. ₹1,000
2. ₹1,200
3. ₹1,250
4. ₹1,500
Show answer
Answer: B — ₹1,200
SI = PRT/100 = (5000 × 8 × 3)/100 = ₹1,200.
A number increased by 20% becomes 240. The original number is:
1. 192
2. 200
3. 210
4. 220
Show answer
Answer: B — 200
If the original is x, then 1.2x = 240 → x = 200. (Trap: 240 − 20% of 240 = 192 is wrong, because the 20% was taken on the original, not on 240.)
Two successive discounts of 10% and 10% are equivalent to a single discount of:
1. 20%
2. 19%
3. 18%
4. 21%
Show answer
Answer: B — 19%
₹100 → ₹90 → ₹81, so the total discount is ₹19 = 19%. Formula: −10 − 10 + (10×10)/100 = −19%.
A can finish a job in 10 days and B in 15 days. Working together they finish it in:
1. 5 days
2. 6 days
3. 7.5 days
4. 12.5 days
Show answer
Answer: B — 6 days
Per day: 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 of the job → 6 days together.
If 8 workers can finish a job in 15 days, how many days will 12 workers take to finish the same job?
1. 8 days
2. 10 days
3. 20 days
4. 22.5 days
Show answer
Answer: B — 10 days
Workers and days are inversely proportional (product constant): 8 × 15 = 12 × x → x = 120/12 = 10 days. More workers means fewer days.
If 6 notebooks cost ₹90, what is the cost of 10 notebooks (assuming direct proportion)?
1. ₹120
2. ₹135
3. ₹150
4. ₹160
Show answer
Answer: C — ₹150
Cost and quantity are directly proportional: 6/90 = 10/x → x = 90 × 10/6 = ₹150.
If 6 men working 8 hours a day can build a wall in 10 days, how many men are needed to build it in 8 days working 6 hours a day?
1. 8
2. 9
3. 10
4. 12
Show answer
Answer: C — 10
Using the chain rule: Men₂ = 6 × (8/6) × (10/8) = 10 men. Since fewer hours per day and fewer days both require more men (inverse relationships), the answer is higher than the original 6.