FREE PREVIEW Week 2 4 lessons

Quantitative Aptitude: Arithmetic I

Percentage, profit, loss and discount, simple and compound interest, and ratio, proportion, partnership, mixtures and alligation, with step-by-step solved examples.

Lessons in this module

Percentage

Lesson • 5 quiz questions

Read notes ↓

Profit, Loss and Discount

Lesson • 5 quiz questions

Read notes ↓

Simple Interest and Compound Interest

Lesson • 5 quiz questions

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Ratio, Proportion, Partnership, Mixtures and Alligation

Lesson • 5 quiz questions

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Lesson notes

1. Percentage

Percentage is the base of many other chapters: profit and loss, interest, data interpretation and more. 'Per cent' means 'out of 100'. If you learn the fraction values of common percentages, you can calculate very fast without a pen.

Basic ideas and formulas

x% of y = (x ÷ 100) × y. Also, x% of y = y% of x. Example: 8% of 50 = 50% of 8 = 4.
Percentage change = (Change ÷ Original value) × 100. Always divide by the ORIGINAL (old) value.
Increase by r%: new value = old × (100 + r) ÷ 100. Decrease by r%: new value = old × (100 − r) ÷ 100.
If A is r% more than B, then B is less than A by [r ÷ (100 + r)] × 100 %. Example: A is 25% more than B → B is 25/125 × 100 = 20% less than A.
Use fraction values: 12.5% = 1/8, 16.67% = 1/6, 20% = 1/5, 25% = 1/4, 33.33% = 1/3, 37.5% = 3/8, 62.5% = 5/8.

Successive change and population

Two successive changes of a% and b% give a net change of a + b + (a × b ÷ 100) %. Use a minus sign for a decrease.
Worked example: price rises 20% and then falls 20%. Net = 20 − 20 + (20 × −20 ÷ 100) = −4%. So there is a 4% decrease, not zero.
Consumption rule: if price rises by r%, to keep spending the same, consumption must fall by [r ÷ (100 + r)] × 100 %. Example: price up 25% → cut 25/125 = 20%.
Population growing at r% per year for n years: P × (1 + r/100)ⁿ. Example: 50,000 at 10% for 2 years = 50,000 × 1.1 × 1.1 = 60,500.
Trap: a 10% rise for 2 years is not 20%. It is 10 + 10 + 1 = 21%.

Exam-type worked example: marks

Question: A student scores 35% and fails by 24 marks. The pass mark is 41%. Find the maximum marks.
Step 1: The gap between pass mark and his score = 41% − 35% = 6%.
Step 2: This 6% equals 24 marks. So 1% = 4 marks.
Step 3: Maximum marks = 100% = 400. Pass marks = 41% of 400 = 164.

Quick facts

x% of y: x × y ÷ 100 (= y% of x)
% change: Change ÷ Original × 100
Successive a% and b%: a + b + ab/100
A is r% more than B: B is r/(100 + r) × 100 % less than A
Price up r%: Cut consumption by r/(100 + r) × 100 %
Growth for n years: P × (1 + r/100)ⁿ

💡 Learn the fraction table (1/2 to 1/12 as percentages) by heart. It turns most percentage questions into simple multiplication.

📝 5-question quiz on this lesson: enrol free to take it and track your score.

2. Profit, Loss and Discount

Profit and loss questions are about buying and selling. Discount questions add one more price: the marked price printed on the item. Keep these three prices clear: Cost Price (CP), Selling Price (SP) and Marked Price (MP).

Key formulas

Profit = SP − CP. Loss = CP − SP. Profit % and Loss % are always calculated on CP, unless the question says otherwise.
SP = CP × (100 + Profit%) ÷ 100. SP = CP × (100 − Loss%) ÷ 100.
Discount = MP − SP. Discount % is always calculated on MP. SP = MP × (100 − Discount%) ÷ 100.
Successive discounts of a% and b% = a + b − (a × b ÷ 100) % in total. Example: 20% and 10% → 20 + 10 − 2 = 28%.
If the CP of x articles = SP of y articles, then Profit % = (x − y) ÷ y × 100. Example: CP of 15 = SP of 12 → 3/12 × 100 = 25%.

Worked examples

Example 1: An item is bought for ₹400 and sold for ₹460. Profit = ₹60. Profit % = 60 ÷ 400 × 100 = 15%.
Example 2: Selling an article for ₹720 gives a 10% loss. Find the SP for a 15% gain. Step 1: 90% of CP = 720 → CP = 800. Step 2: SP = 115% of 800 = ₹920.
Example 3: MP is ₹800, discount 15%, and CP is ₹600. Step 1: SP = 85% of 800 = ₹680. Step 2: Profit = 80. Step 3: Profit % = 80 ÷ 600 × 100 = 13.33%.
Example 4: MP ₹1,200 with successive discounts of 20% and 10%. SP = 1,200 × 0.8 × 0.9 = ₹864.

Special cases and traps

Two items sold at the SAME selling price, one at x% profit and the other at x% loss: there is always a net LOSS of x² ÷ 100 %. Example: 10% each → 1% loss.
Dishonest dealer: if a shopkeeper gives 900 g instead of 1 kg at the cost price, gain % = 100 ÷ 900 × 100 = 11.11%. Formula: error ÷ (true weight − error) × 100.
Trap: profit % is on CP, but discount % is on MP. Do not mix the two bases.
Trap: successive discounts of 20% and 10% are not 30%. They are 28%.

Quick facts

Profit %: (SP − CP) ÷ CP × 100
SP from CP: CP × (100 ± P%) ÷ 100
Discount %: (MP − SP) ÷ MP × 100
Successive discounts: a + b − ab/100
Same SP, x% gain and x% loss: Net loss = x²/100 %
CP of x = SP of y: Profit % = (x − y)/y × 100

💡 Think of percentages as multipliers: 15% profit means × 1.15, 20% discount means × 0.80. Chains of changes become one quick multiplication.

📝 5-question quiz on this lesson: enrol free to take it and track your score.

3. Simple Interest and Compound Interest

Interest is the extra money paid for using borrowed money. Simple Interest (SI) is calculated only on the original amount. Compound Interest (CI) is calculated on the amount plus the interest already added. Bank exams especially like questions on the difference between CI and SI.

Simple interest

SI = (P × R × T) ÷ 100, where P = principal, R = rate % per year, T = time in years. Amount A = P + SI.
Worked example: SI on ₹8,000 at 7.5% per year for 3 years = 8,000 × 7.5 × 3 ÷ 100 = ₹1,800. Amount = ₹9,800.
A sum doubles in T years at SI: then SI = P, so R = 100 ÷ T. Example: doubles in 8 years → R = 12.5%.
A sum becomes n times in T years: R = (n − 1) × 100 ÷ T.
Trap: if time is in months, divide by 12. 9 months = 9/12 = 0.75 year.

Compound interest

Amount A = P × (1 + R/100)ⁿ. CI = A − P.
Worked example: P = ₹5,000, R = 10%, n = 2 years. A = 5,000 × 1.1 × 1.1 = ₹6,050. CI = ₹1,050. SI for the same = ₹1,000.
Half-yearly compounding: use rate R/2 and time 2n. Quarterly: rate R/4 and time 4n.
Worked example: ₹20,000 at 10% per year, compounded half-yearly for 1 year. Rate = 5% per half-year, 2 periods. A = 20,000 × 1.05 × 1.05 = ₹22,050. CI = ₹2,050.
Effective rate for 2 years at R% = R + R + R²/100. Example: 10% → 21% over 2 years.

CI − SI difference shortcuts

For 2 years: CI − SI = P × (R/100)². Check with the example above: 5,000 × (0.1)² = ₹50 = 1,050 − 1,000.
For 3 years: CI − SI = P × (R/100)² × (3 + R/100).
Worked example: the difference between CI and SI for 2 years at 8% is ₹96. Then P × 0.08 × 0.08 = 96 → P × 0.0064 = 96 → P = ₹15,000.
Trap: in the first year, CI and SI are equal (for yearly compounding). The difference starts only from the second year.

Quick facts

Simple interest: SI = PRT/100
Compound amount: A = P(1 + R/100)ⁿ
Half-yearly: Rate R/2, time 2n
Doubles in T years (SI): R = 100/T
CI − SI, 2 years: P(R/100)²
CI − SI, 3 years: P(R/100)²(3 + R/100)

💡 Remember 10% CI values: 2 years = 21%, 3 years = 33.1%. Many exam questions use 10% because the numbers are easy.

📝 5-question quiz on this lesson: enrol free to take it and track your score.

4. Ratio, Proportion, Partnership, Mixtures and Alligation

A ratio compares two quantities of the same kind. Ratio ideas are used in partnership (sharing profit), mixtures (milk and water), and alligation (mixing two prices). This lesson brings all of them together because they use the same basic thinking.

Ratio and proportion

a : b means a ÷ b. Dividing a total in the ratio a : b gives a/(a + b) and b/(a + b) of the total. Example: ₹2,700 in 4 : 5 → 4/9 × 2,700 = 1,200 and 5/9 × 2,700 = 1,500.
Joining ratios: if a : b = 2 : 3 and b : c = 4 : 5, make b equal: a : b = 8 : 12 and b : c = 12 : 15. So a : b : c = 8 : 12 : 15.
Proportion: a : b = c : d means a × d = b × c. Fourth proportional of a, b, c = (b × c) ÷ a.
Mean proportional of a and b = √(a × b). Example: mean proportional of 9 and 25 = √225 = 15. Third proportional of a and b = b² ÷ a.

Partnership

Profit is shared in the ratio of (capital × time) for each partner.
Worked example: A invests ₹30,000 for 12 months. B invests ₹40,000 for 6 months. Ratio = 30,000 × 12 : 40,000 × 6 = 3,60,000 : 2,40,000 = 3 : 2.
If the total profit is ₹25,000, A gets 3/5 × 25,000 = ₹15,000 and B gets ₹10,000.
If all partners invest for the same time, the profit ratio is simply the capital ratio.
Trap: a working partner may first get a salary from the profit. Subtract it before sharing the rest, if the question says so.

Mixtures and alligation

Alligation rule: Cheaper quantity : Dearer quantity = (Dearer price − Mean price) : (Mean price − Cheaper price).
Worked example: rice at ₹40/kg and ₹55/kg is mixed to get a mixture worth ₹45/kg. Ratio = (55 − 45) : (45 − 40) = 10 : 5 = 2 : 1. So 2 kg of the cheaper rice for every 1 kg of the dearer rice.
Repeated replacement: a container has x litres of milk. y litres are taken out and replaced with water, n times. Milk left = x × (1 − y/x)ⁿ.
Worked example: 40 litres of milk, 4 litres replaced with water twice. Milk left = 40 × (36/40)² = 40 × 0.81 = 32.4 litres.
Trap: in alligation, the mean price must lie between the two prices. If it does not, check the question again.

Quick facts

Divide T in a : b: aT/(a + b) and bT/(a + b)
Mean proportional: √(ab)
Fourth proportional of a, b, c: bc/a
Partnership: Profit ratio = Capital × Time
Alligation: Cheaper : Dearer = (D − M) : (M − C)
Replacement n times: x(1 − y/x)ⁿ

💡 Draw the alligation cross on rough paper: two prices on top, the mean in the middle, and the differences below. It takes 10 seconds and avoids mistakes.

📝 5-question quiz on this lesson: enrol free to take it and track your score.

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