The table to memorise before anything else
Almost every percentage, profit, interest and data-interpretation question hides a fraction. If you can convert instantly in both directions, you stop dividing by hand. Learn this table cold - it is the single highest-return thing on this page.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% |
| 1/3 | 33.33% | 1/10 | 10% |
| 1/4 | 25% | 1/11 | 9.09% |
| 1/5 | 20% | 1/12 | 8.33% |
| 1/6 | 16.67% | 1/16 | 6.25% |
| 1/7 | 14.29% | 1/20 | 5% |
| 1/8 | 12.5% | 1/25 | 4% |
- ✓Successive percentage change: net % = a + b + ab/100. A 20% rise then a 20% fall is 20 − 20 − 400/100 = −4%, not zero.
- ✓Percentage to multiplier: a 15% increase is ×1.15, a 15% decrease is ×0.85. Chain multipliers instead of computing each step.
- ✓'x% of y' equals 'y% of x' - 18% of 50 is easier read as 50% of 18 = 9.
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Percentages, profit & loss
This is the base every other chapter builds on, and it is where the successive-change formula earns its keep.
A student scores 480 marks out of 600. What is the percentage?
80%. 480/600 = 4/5, and 1/5 = 20%, so 4/5 = 80% - no division needed if the fraction table is in memory.
The price of a book increases by 20% and then decreases by 20%. What is the net change?
A 4% decrease. Using a + b + ab/100: 20 − 20 − (20 × 20)/100 = −4%. Equal-and-opposite percentage changes never cancel; the net loss is always (x²/100)%.
An article bought for ₹400 is sold for ₹500. Find the profit percentage.
25%. Profit is ₹100 on a cost of ₹400, and 100/400 = 1/4 = 25%. Profit percentage is always on cost price unless the question says otherwise.
A shopkeeper marks his goods 40% above cost and offers a 25% discount. What is his profit percentage?
5%. Chain the multipliers: 1.40 × 0.75 = 1.05, so the selling price is 105% of cost. Marked price problems are far faster as multipliers than as rupee amounts.
If A's salary is 25% more than B's, by what percentage is B's salary less than A's?
20%. Take B = 100, so A = 125. The difference of 25 is now measured against A: 25/125 = 1/5 = 20%. The base changes when the comparison reverses - this is the most-tested trap in the chapter.
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Related reading
Ratio, proportion and averages
Divide ₹1,200 among three people in the ratio 2 : 3 : 7.
₹200, ₹300 and ₹700. Total parts = 12, so one part = 1200/12 = ₹100. Always find the value of one part first; it turns every ratio question into multiplication.
The average of 10 numbers is 25. If one number, 30, is replaced by 50, what is the new average?
27. The sum was 250 and rises by 20, giving 270 over 10 numbers. Work with the change in the total, never recompute the whole sum.
The average age of 5 students is 16 years. A sixth student joins and the average becomes 17. How old is the new student?
22 years. The new student must supply their own 17 plus 1 extra year for each of the 5 existing students: 17 + 5 = 22. This 'surplus' method solves most average-change questions in one line.
Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.
18 and 30. Let them be 3x and 5x. Then (3x + 9)/(5x + 9) = 3/4, so 12x + 36 = 15x + 27, giving 3x = 9 and x = 3.
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Time & work, pipes and cisterns
Never work in fractions of a day. Assume the job is the LCM of the given times, convert each worker into units per day, and add.
A can finish a job in 12 days and B in 18 days. Working together, how long will they take?
7.2 days. Take the job as 36 units (LCM of 12 and 18): A does 3 units/day, B does 2, so together 5 units/day, and 36/5 = 7.2 days.
A and B together finish a job in 8 days, and A alone takes 12 days. How long does B take alone?
24 days. B's rate = 1/8 − 1/12 = (3 − 2)/24 = 1/24, so 24 days. Rates subtract cleanly; times never do.
A pipe fills a tank in 6 hours and a drain empties it in 9 hours. With both open, how long to fill?
18 hours. Take the tank as 18 units: inflow 3/hour, outflow 2/hour, net 1 unit/hour. Outflow is simply a negative worker.
If 12 workers build a wall in 10 days, how many days will 15 workers take?
8 days. Workers and time are inversely proportional: 12 × 10 = 15 × d, so d = 8. Check the direction of the proportion before you multiply - this is where most errors happen.
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Time, speed and distance
Two conversions carry this chapter: multiply km/h by 5/18 for m/s, and remember that a train 'crossing a pole' travels its own length while 'crossing a platform' travels length plus platform.
A 150 m long train running at 54 km/h crosses a pole. How long does it take?
10 seconds. 54 × 5/18 = 15 m/s, and 150/15 = 10 s.
Two trains 120 m and 180 m long move in opposite directions at 40 km/h and 50 km/h. How long do they take to cross each other?
12 seconds. Opposite directions means speeds add: 90 km/h = 25 m/s. The distance is the sum of the lengths, 300 m, so 300/25 = 12 s.
A man covers half a journey at 40 km/h and the other half at 60 km/h. What is his average speed?
48 km/h - not 50. For equal distances the average is the harmonic mean, 2ab/(a + b) = (2 × 40 × 60)/100 = 48 km/h. Averaging speeds directly is the classic wrong answer that examiners plant in the options.
A boat travels 24 km downstream in 3 hours and returns in 4 hours. Find the speed of the stream.
1 km/h. Downstream speed = 8 km/h, upstream = 6 km/h. Boat speed = (8 + 6)/2 = 7 and stream = (8 − 6)/2 = 1 km/h.
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Simple and compound interest
Find the simple interest on ₹8,000 at 5% per annum for 3 years.
₹1,200. SI = PRT/100 = (8000 × 5 × 3)/100. With SI, each year contributes the same amount - here ₹400 a year.
Find the compound interest on ₹10,000 at 10% per annum for 2 years.
₹2,100. Reuse the successive-change formula: 10 + 10 + 100/100 = 21%, so the interest is 21% of ₹10,000. For two years this is always faster than expanding (1 + r)ⁿ.
What is the difference between CI and SI on ₹5,000 at 8% for 2 years?
₹32. For two years the difference is always P(r/100)² = 5000 × 0.0064 = ₹32. Memorise this one-line result; it appears every season.
In how many years will a sum double at 12.5% simple interest?
8 years. Doubling needs 100% interest, and 100/12.5 = 8. Note 12.5% = 1/8, another payoff from the fraction table.
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Related reading
Number systems, permutations and probability
What is the unit digit of 7⁹⁵?
3. The unit digits of powers of 7 cycle 7, 9, 3, 1 with period 4. 95 ÷ 4 leaves remainder 3, so the answer is the third in the cycle.
Find the HCF and LCM of 36 and 48.
HCF 12, LCM 144. Sanity check with HCF × LCM = product of the numbers: 12 × 144 = 1,728 = 36 × 48.
In how many ways can the letters of the word LEVEL be arranged?
30. Five letters with L twice and E twice: 5!/(2! × 2!) = 120/4 = 30. Repeated letters always divide by the factorial of their count.
Two dice are thrown. What is the probability that the sum is 8?
5/36. The favourable outcomes are (2,6), (3,5), (4,4), (5,3) and (6,2) out of 36 equally likely pairs. List outcomes for small sums rather than reaching for a formula.
A bag has 4 red and 6 blue balls. Two are drawn at random. What is the probability both are red?
2/15. C(4,2)/C(10,2) = 6/45 = 2/15.
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How to practise these so the speed sticks
Reading solutions builds recognition; solving under a clock builds speed. The gap between the two is exactly what a 35-minute sectional cutoff measures. So use this page twice: once to learn the shortcut for each family, then repeatedly as a self-test where you cover the answer and give yourself 45 seconds.
Then move to mixed sets. Chapter-wise practice tells you a question is a time-and-work question before you read it, which is the one advantage the real test never gives you. A mixed adaptive quiz forces the identification step - noticing which family a question belongs to - and that is the skill that actually transfers to test day.
- ✓Two timed sets of 15 questions a day beats one untimed set of 50.
- ✓Keep an error log with one line per miss: the family, the trap, the faster method.
- ✓Re-solve every wrong question 48 hours later, from scratch, without looking.
- ✓Once a chapter feels reliable, stop drilling it alone and only meet it inside mixed sets.
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