Lesson 10.3: Numerical Ability (Percentages, Ratios, Averages, Profit-Loss, SI/CI)

Numerical Ability is a compulsory part of General Aptitude in GATE and PSU exams. It tests basic arithmetic, quantitative reasoning, and problem-solving skills.

🔹 1. Introduction

• Definition: Numerical Ability measures proficiency in solving numerical problems using mathematical concepts.

• Applications: Aptitude tests in GATE, PSU exams, interviews, and competitive exams.

• Key Concepts: Percentages, ratios, averages, profit & loss, simple interest (SI), compound interest (CI), time-speed-distance, work & time.

🔹 2. Percentages

• Definition: Part of a quantity expressed as a fraction of 100.

Percentage=PartWhole×100Percentage = \frac{\text{Part}}{\text{Whole}} \times 100Percentage=WholePart​×100

• Applications: Profit/Loss calculations, discounts, interest calculations, data interpretation.

• Example: If a product costs ₹200 and sells for ₹250, profit% = 50200×100=25%\frac{50}{200} \times 100 = 25\%20050​×100=25%

🔹 3. Ratios & Proportions

• Ratio: Comparison of two quantities.

Ratio=Quantity 1Quantity 2\text{Ratio} = \frac{\text{Quantity 1}}{\text{Quantity 2}}Ratio=Quantity 2Quantity 1​

• Proportion: Equality of two ratios.

• Applications: Mixture problems, allocation, scaling, recipe adjustments.

• Example: If the ratio of boys to girls is 3:2 in a class of 25 students, boys = 15, girls = 10.

🔹 4. Averages

• Definition: Sum of quantities divided by their number.

Average=Sum of quantitiesNumber of quantitiesAverage = \frac{\text{Sum of quantities}}{\text{Number of quantities}}Average=Number of quantitiesSum of quantities​

• Applications: Exam scores, production statistics, financial calculations.

• Example: Average of 5 numbers: 10, 15, 20, 25, 30 → 10+15+20+25+305=20\frac{10+15+20+25+30}{5} = 20510+15+20+25+30​=20

🔹 5. Profit & Loss

• Formulas:

$$Profit=SellingPrice-CostPrice$$ $$Loss=CostPrice-SellingPrice$$ Profit%=ProfitCost Price×100,Loss%=LossCost Price×100Profit\% = \frac{Profit}{Cost\ Price} \times 100, \quad Loss\% = \frac{Loss}{Cost\ Price} \times 100Profit%=Cost PriceProfit​×100,Loss%=Cost PriceLoss​×100

• Applications: Trade, sales, discounts, cost analysis

• Example: Cost price = ₹400, selling price = ₹500 → Profit = ₹100, Profit% = 25%

🔹 6. Simple Interest (SI) & Compound Interest (CI)

• Simple Interest:

SI=P×R×T100SI = \frac{P \times R \times T}{100}SI=100P×R×T​

where P = Principal, R = Rate%, T = Time in years

• Compound Interest:

CI=P(1+R100)T−PCI = P \left(1 + \frac{R}{100}\right)^T – PCI=P(1+100R​)T−P

• Applications: Banking, finance, investment problems

• Example (SI): P = ₹1000, R = 5%, T = 2 years → SI = $1000\times 5\times 2/100=₹100$

• Example (CI): P = ₹1000, R = 10%, T = 2 years → CI = 1000(1+0.1)2−1000=₹2101000(1+0.1)^2 – 1000 = ₹2101000(1+0.1)2−1000=₹210

🔹 7. Solved Examples (PYQ Style)

1. A shopkeeper buys a product for ₹800 and sells at ₹1000 → Profit% = ?

2. Average marks of 5 students: 40, 50, 60, 70, x. If average = 55 → x = ?

3. Find CI on ₹5000 at 8% p.a. for 3 years.

4. Ratio of two numbers = 3:4, sum = 28 → Find numbers.

5. Discount of 20% on marked price of ₹500 → Selling price?

🔹 8. Practice Exercises

1. Calculate percentages for given marks in exams

2. Solve ratio and proportion problems in real-life scenarios

3. Compute averages for datasets of 5–10 numbers

4. Solve profit & loss problems with cost price, selling price, and discounts

5. Solve SI & CI problems with varying principal, rate, and time

🔹 9. Summary

• Numerical Ability: Tests arithmetic, quantitative reasoning, and problem-solving skills

• Key Topics: Percentages, Ratios, Averages, Profit & Loss, SI/CI

• Strategies: Learn formulas, practice shortcuts, solve previous GATE/PSU questions

• Exam Tip: Focus on speed and accuracy; most questions are calculation-based