Aptitude Formulas for Placement Tests: 12 Topics With Worked Examples

Most quantitative aptitude sections in campus tests draw from the same dozen topics. For each one below you get the formulas that do the work, a fully worked example, and the shortcut that saves time in the exam. Every answer is worked out step by step, so you can check your own method against it.

Two habits help more than any formula. First, read what the question actually asks for: profit or profit percent, the time taken together or by one person. Second, before you calculate, guess roughly what the answer should be, so an option that is far off can be ruled out at once.

Percentages and money

1. Percentages

Formulas

• Percentage change = (new − old) ÷ old × 100
• Two changes one after another, a% then b%: net change = a + b + (a × b) ÷ 100. Use a minus sign for a decrease.
• If A is r% more than B, then B is r ÷ (100 + r) × 100 percent less than A.

Worked example 1. A price rises by 20% and then falls by 20%. What is the overall change?

Net = 20 + (−20) + (20 × −20) ÷ 100 = 0 − 4 = −4%, a 4% fall. Check with numbers: 100 → 120 → 96.

Worked example 2. A's salary is 25% more than B's. By what percent is B's salary less than A's?

25 ÷ 125 × 100 = 20%. Check: B = 100, A = 125; B is 25 less than 125, which is 20% of 125.

Shortcut. Turn common percentages into fractions: 12.5% = 1/8, 16.67% = 1/6, 20% = 1/5, 25% = 1/4, 33.33% = 1/3. "Increase by 1/8" means multiply by 9/8.

More practice: percentages.

2. Profit, loss and discount

Formulas

• Profit % = (SP − CP) ÷ CP × 100. Profit and loss percent are always on the cost price.
• SP = CP × (100 + profit%) ÷ 100
• Discount is on the marked price: SP = MP × (100 − discount%) ÷ 100

Worked example 1. A shopkeeper marks goods 40% above cost and gives a 10% discount. What is his profit percent?

Take CP = 100. MP = 140. SP = 140 × 0.9 = 126. Profit = 26%.

Worked example 2. A dealer claims to sell at cost price but gives 900 g instead of 1 kg. What is his gain percent?

He receives the price of 1000 g but gives goods costing 900 g. Gain = 100 ÷ 900 × 100 = 11.11% (11 1/9 %).

Shortcut. Two successive discounts of a% and b% equal one discount of a + b − (a × b) ÷ 100. So 20% and 10% together are 28%, not 30%.

More practice: profit and loss.

3. Simple and compound interest

Formulas

• Simple interest: SI = P × R × T ÷ 100
• Compound interest (compounded yearly): amount A = P × (1 + R/100)^T, and CI = A − P
• For 2 years, CI − SI = P × (R/100)²

Worked example. ₹10,000 at 10% a year for 2 years. Find SI, CI and the difference.

SI = 10,000 × 10 × 2 ÷ 100 = ₹2,000. A = 10,000 × 1.1 × 1.1 = 12,100, so CI = ₹2,100. Difference = ₹100. Check with the formula: 10,000 × (0.1)² = 100.

Shortcut. For half-yearly compounding, halve the rate and double the number of periods: 10% a year for 1 year compounded half-yearly is 5% for 2 periods.

More practice: simple and compound interest.

Ratios and averages

4. Ratio and proportion

Formulas

• To combine A : B and B : C, make B the same in both, then read off A : B : C.
• To divide an amount in the ratio a : b : c, one part = amount ÷ (a + b + c).

Worked example. A : B = 2 : 3 and B : C = 4 : 5. Divide ₹1,050 among A, B and C.

Make B equal: multiply the first ratio by 4 and the second by 3. A : B = 8 : 12, B : C = 12 : 15, so A : B : C = 8 : 12 : 15. Total parts = 35, so one part = 1,050 ÷ 35 = 30. A = ₹240, B = ₹360, C = ₹450. Check: 240 + 360 + 450 = 1,050.

Shortcut. If an option is not a multiple of a share's ratio number, it cannot be the answer.

More practice: ratio and proportion.

5. Averages

Formulas

• Average = sum ÷ count, so sum = average × count.
• When something joins or leaves a group, work with the sums before and after, not the averages.

Worked example 1. The average of 5 numbers is 40. One number is removed and the average of the rest is 38. What was removed?

Sum before = 5 × 40 = 200. Sum after = 4 × 38 = 152. Removed = 200 − 152 = 48.

Worked example 2. The average age of 30 students is 15. When the teacher is included, the average goes up by 1. How old is the teacher?

Sum of 31 people = 31 × 16 = 496. Sum of students = 30 × 15 = 450. Teacher = 46.

Shortcut. In example 2, the teacher is 15 years plus 1 extra year for each of the 31 people: 15 + 31 = 46.

More practice: averages.

6. Mixtures and alligation

Formulas

• Alligation: quantity of cheaper ÷ quantity of dearer = (dearer price − mean price) ÷ (mean price − cheaper price).
• Repeated replacement: if x litres are taken out of V litres and replaced with water, n times, the pure liquid left = V × (1 − x/V)^n.

Worked example 1. Rice at ₹40/kg is mixed with rice at ₹60/kg to get a mixture worth ₹52/kg. In what ratio?

Cheaper : dearer = (60 − 52) : (52 − 40) = 8 : 12 = 2 : 3. Check: (2 × 40 + 3 × 60) ÷ 5 = 260 ÷ 5 = 52.

Worked example 2. A container has 40 litres of milk. 4 litres are taken out and replaced with water, and this is done twice. How much milk is left?

40 × (1 − 4/40)² = 40 × 0.9 × 0.9 = 32.4 litres.

More practice: ratio and proportion.

Work and time

7. Time and work

Formulas

• If A finishes a job in a days, A does 1/a of it per day.
• Together, A and B take ab ÷ (a + b) days.
• LCM method: take the total work as the LCM of the days, then count units of work per day.

Worked example 1. A can finish a job in 12 days and B in 18 days. How long do they take together?

Total work = LCM(12, 18) = 36 units. A does 3 units/day, B does 2, together 5. Time = 36 ÷ 5 = 7.2 days. Check: 12 × 18 ÷ 30 = 7.2.

Worked example 2. A and B together take 10 days; A alone takes 15. How long does B take alone?

B's rate = 1/10 − 1/15 = 3/30 − 2/30 = 1/30, so 30 days.

More practice: time and work.

8. Pipes and cisterns

Formulas

The same as time and work, except that an emptying pipe or a leak does negative work.

Worked example. Pipe A fills a tank in 6 hours and pipe B in 8 hours. A leak empties the full tank in 12 hours. With all three open, how long does the tank take to fill?

Tank = LCM(6, 8, 12) = 24 units. A fills 4 units/hour, B fills 3, the leak removes 2. Net = 5 units/hour. Time = 24 ÷ 5 = 4.8 hours = 4 hours 48 minutes.

Shortcut. Always check that the net rate is positive. If the leak is faster than the pipes, the tank never fills, and that is sometimes the intended answer.

More practice: time and work.

Speed and distance

9. Time, speed and distance

Formulas

• Distance = speed × time.
• km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
• Average speed over equal distances at x and y = 2xy ÷ (x + y). It is not (x + y) ÷ 2.
• Moving towards each other, speeds add; moving the same way, subtract the smaller from the larger.

Worked example 1. A person drives to a town at 40 km/h and back at 60 km/h. What is the average speed?

2 × 40 × 60 ÷ (40 + 60) = 4,800 ÷ 100 = 48 km/h. Check with a 120 km trip: 3 hours there plus 2 hours back is 240 km in 5 hours, which is 48 km/h.

Worked example 2. Two cars start 300 km apart and drive towards each other at 70 km/h and 80 km/h. When do they meet?

Closing speed = 150 km/h. Time = 300 ÷ 150 = 2 hours.

More practice: time, speed and distance.

10. Trains

Formulas

• To pass a pole or a person standing still, a train covers its own length.
• To pass a platform or a bridge, it covers its own length plus the platform's.
• Two trains passing each other cover the sum of their lengths at their relative speed.

Worked example 1. A 240 m train runs at 72 km/h. How long does it take to pass a pole, and to pass a 360 m platform?

72 km/h = 72 × 5/18 = 20 m/s. Pole: 240 ÷ 20 = 12 seconds. Platform: (240 + 360) ÷ 20 = 30 seconds.

Worked example 2. Trains 150 m and 100 m long run towards each other at 60 km/h and 30 km/h. How long do they take to pass each other completely?

Relative speed = 90 km/h = 25 m/s. Distance = 150 + 100 = 250 m. Time = 250 ÷ 25 = 10 seconds.

More practice: time, speed and distance.

11. Boats and streams

Formulas

• Downstream speed = boat + stream; upstream speed = boat − stream.
• Boat speed in still water = (downstream + upstream) ÷ 2; stream speed = (downstream − upstream) ÷ 2.

Worked example. A boat goes 30 km downstream in 2 hours and 18 km upstream in 2 hours. Find the speed of the boat in still water and the speed of the stream.

Downstream = 30 ÷ 2 = 15 km/h; upstream = 18 ÷ 2 = 9 km/h. Boat = (15 + 9) ÷ 2 = 12 km/h; stream = (15 − 9) ÷ 2 = 3 km/h.

More practice: time, speed and distance.

Counting and chance

12. Permutations, combinations and probability

Formulas

• Arrangements, where order matters: nPr = n! ÷ (n − r)!
• Selections, where order does not matter: nCr = n! ÷ (r! × (n − r)!)
• Arrangements of n letters with repeats (p of one letter, q of another): n! ÷ (p! × q!)
• Probability = favourable outcomes ÷ total outcomes. "At least one" = 1 − P(none).

Worked example 1. In how many ways can the letters of LEVEL be arranged?

5 letters, with L twice and E twice: 5! ÷ (2! × 2!) = 120 ÷ 4 = 30.

Worked example 2. A committee of 3 is chosen from 5 men and 4 women. In how many ways can it include at least one woman?

All committees: 9C3 = 84. All-men committees: 5C3 = 10. At least one woman = 84 − 10 = 74.

Worked example 3. Two dice are thrown. What is the probability that the total is 7?

Favourable: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), which is 6 outcomes out of 36. Probability = 1/6.

Shortcut. "At least one" questions are almost always faster as "everything minus none", exactly as in example 2.

More practice: permutation and combination and probability.

How to use this list

1. Learn the formulas for one section a day, then solve 15–20 questions on that topic with a timer running.
2. Note every question you get wrong and the reason (formula, calculation, or misreading). Most marks are lost to misreading.
3. In the last week before a test, do mixed sets under time pressure, because the real test does not say which topic a question is from.

When you are ready to time yourself, aptitude practice has questions on these topics.

FAQ