Comparing Quantities

Introduction

Comparing quantities is a fundamental concept in mathematics that helps us understand relationships between different values in everyday life. This chapter focuses on tools like ratios, proportions, percentages, discounts, profit and loss, taxes, and interest to compare quantities effectively. These concepts are essential for real-world applications such as shopping, banking, and business calculations. By mastering these, students can solve problems involving increases, decreases, and financial transactions. The chapter builds on basic fractions and decimals, extending them to practical scenarios.

Ratios express the comparison of two quantities of the same kind, written as a:b or a/b. Proportions state that two ratios are equal, like a/b = c/d. Percentages represent quantities as parts per hundred, making comparisons straightforward. For instance, converting fractions to percentages involves multiplying by 100 and adding the % sign. Understanding these basics allows us to tackle more complex topics like profit percentages or compound interest.

Ratios and Proportions

In Comparing Quantities A ratio compares two quantities by division, showing how many times one is contained in the other. For example, if there are 3 red balls and 5 blue balls, the ratio is 3:5. Ratios are simplified by dividing both terms by their greatest common divisor (GCD). Equivalent ratios are those that represent the same relationship, like 3:5 and 6:10.

Proportions occur when two ratios are equal. If a/b = c/d, then ad = bc (cross-multiplication). This is useful in solving problems like finding missing values. Example: If 2 kg of sugar costs Rs 80, how much does 5 kg cost? Set up 2/80 = 5/x, so x = (5*80)/2 = Rs 200.

Unitary method is another approach: find the value for one unit and multiply. For the above, cost per kg = 80/2 = Rs 40, so for 5 kg = 40*5 = Rs 200. These methods help in comparing quantities in recipes, maps, or speeds.

Percentages

n Comparing Quantities Percentage means “per hundred,” denoted by %. To convert a fraction to a percentage: (fraction) * 100%. Example: 3/4 = (3/4)*100 = 75%. Conversely, percentage to fraction: divide by 100. 25% = 25/100 = 1/4.

Decimals to percentages: multiply by 100. 0.75 = 75%. Percentages to decimals: divide by 100. 50% = 0.50.

Finding a percentage of a number: (percentage/100) * number. Example: 20% of 500 = (20/100)*500 = 100.

Percentages are used to compare parts of wholes, like exam scores or population growth. They simplify comparisons across different totals. For instance, if Class A has 40/50 students passing (80%) and Class B has 36/45 (80%), both have the same pass rate despite different numbers.

Finding Increase or Decrease Percent

Increase percent = (increase / original value) * 100%. Example: If a price rises from Rs 100 to Rs 120, increase = 20, percent = (20/100)*100 = 20%.

Decrease percent = (decrease / original value) * 100%. Example: Price falls from Rs 100 to Rs 80, decrease = 20, percent = 20%.

New value after increase: original * (1 + rate/100). Example: Rs 15000 with 5% increase = 15000 * (1 + 5/100) = 15000 * 1.05 = Rs 15750.

After decrease: original * (1 – rate/100). Rs 15000 with 5% decrease = 15000 * 0.95 = Rs 14250.

These calculations are vital for understanding inflation, salary hikes, or depreciation. For successive changes, apply sequentially. Example: 10% increase followed by 10% decrease on Rs 100: First 110, then 110*0.9 = Rs 99 (net 1% loss).

Finding Discounts

In Comparing Quantities Discount is the reduction on the marked price (MP) to arrive at the selling price (SP). Discount = MP – SP.

If given as percentage: Discount = (discount % / 100) * MP. Example: Toy MP Rs 500, 5% discount = (5/100)*500 = Rs 25. SP = 500 – 25 = Rs 475.

Successive discounts: Apply one after another. Example: 20% then 10% on Rs 100: First discount 20, new price 80; then 10% of 80 = 8, final SP = 72.

Estimation: Round prices for quick calculations. For Rs 535 shirt at Rs 495 SP, discount ≈ Rs 40.

Discounts encourage sales and are common in retail. Overhead expenses like transportation add to cost but discounts subtract from MP.

Prices Related to Buying and Selling (Profit and Loss)

Cost Price (CP): Buying price plus overheads (repairs, transport). Selling Price (SP): Price at which sold.

Profit: SP > CP, Profit = SP – CP. Profit % = (profit / CP) * 100%.

Loss: SP < CP, Loss = CP – SP. Loss % = (loss / CP) * 100%.

Example: TV CP Rs 12000, SP Rs 13500. Profit = 1500, % = (1500/12000)*100 = 12.5%.

Laptop CP Rs 20000, SP Rs 18000. Loss = 2000, % = 10%.

For multiple items: Overall profit/loss = total SP – total CP.

No profit/loss if SP = CP. These concepts apply to business, helping calculate viability.

Taxes

Sales Tax: Government charge on SP, added to bill. Sales Tax = (tax % / 100) * bill amount.

Value Added Tax (VAT): Tax on value added at each stage, included in price.

Example: Watch SP Rs 1200, 8% VAT = (8/100)*1200 = Rs 96. Total bill = 1296.

Taxes fund public services. In bills, they appear separately or included (e.g., GST in India).

Simple and Compound Interest

Interest is extra money on loans or investments.

Simple Interest (SI): On original principal only. SI = (P * R * T) / 100, where P=principal, R=rate %, T=time (years).

Amount (A) = P + SI.

Example: P=Rs 1000, R=5%, T=2 years. SI= (100052)/100 = Rs 100. A=1100.

Compound Interest (CI): Interest on principal plus previous interest.

Annually: A = P (1 + R/100)^n, CI = A – P.

Example: P=1000, R=5%, n=2. A=1000*(1.05)^2 = 1000*1.1025 = Rs 1102.5. CI=102.5.

Half-yearly: Rate halves, time doubles. A = P (1 + (R/2)/100)^{2n}.

For different rates: Multiply factors sequentially.

Applications: Bank savings, loans, population growth (positive R), depreciation (negative R).

Example: Population 10000 grows 5% annually for 2 years: 10000*(1.05)^2 = 11025.

Some realistic examples of Comparing Quantities..

1. Ratios: Comparing Quantities

Example: In a classroom, there are 24 boys and 16 girls. Find the ratio of boys to girls and simplify it.

Solution: Ratio of boys to girls = 24:16. Divide both by their GCD (8) = 3:2.
Application: This ratio helps in organizing teams for a school event, ensuring balanced participation.

Example: A recipe requires 2 cups of flour and 3 cups of water. What is the ratio of flour to water?

Solution: Ratio = 2:3 (already simplified).
Application: If you scale the recipe for 4 cups of flour, the ratio helps calculate water needed (6 cups).

2. Proportions: Comparing Quantities

Example: If 4 movie tickets cost Rs 1200, how much will 7 tickets cost?

Solution: Set up proportion: 4/1200 = 7/x. Cross-multiply: 4x = 1200 * 7, x = (1200 * 7)/4 = Rs 2100.
Application: Useful for budgeting group outings.

Example (Unitary Method): A car travels 240 km in 4 hours. How far will it travel in 6 hours at the same speed?

Solution: Distance per hour = 240/4 = 60 km. For 6 hours = 60 * 6 = 360 km.
Application: Helps plan travel distances based on time.

3. Percentages: Comparing Quantities

Example: In a test, Priya scored 45 out of 60 marks. What is her percentage score?

Solution: Percentage = (45/60) * 100 = 75%.
Application: Used to compare performance across different tests with varying total marks.

Example: A shop has 200 toys, and 30% are soft toys. How many soft toys are there?

Solution: Number of soft toys = (30/100) * 200 = 60.
Application: Helps in inventory management for specific categories.

4. Increase or Decrease Percent: Comparing Quantities

Example: The price of a smartphone increased from Rs 15000 to Rs 16500. What is the percentage increase?

Solution: Increase = 16500 – 15000 = 1500. Percent = (1500/15000) * 100 = 10%.
Application: Useful for tracking price hikes during inflation.

Example: A jacket’s price dropped from Rs 2000 to Rs 1800. Find the percentage decrease.

Solution: Decrease = 2000 – 1800 = 200. Percent = (200/2000) * 100 = 10%.
Application: Helps shoppers assess sale benefits.

Example (New Value): A salary of Rs 25000 increases by 8%. What is the new salary?

Solution: New salary = 25000 * (1 + 8/100) = 25000 * 1.08 = Rs 27000.
Application: Useful for employees calculating raises.

5. Discounts: Comparing Quantities

Example: A pair of shoes has a marked price of Rs 2500 with a 15% discount. Find the selling price.

Solution: Discount = (15/100) * 2500 = Rs 375. SP = 2500 – 375 = Rs 2125.
Application: Helps shoppers calculate final prices during sales.

Example (Successive Discounts): A laptop MP Rs 40000 gets a 20% discount, then an additional 10% off. Find the final SP.

Solution: First discount = 20% of 40000 = Rs 8000, new price = 32000. Second discount = 10% of 32000 = 3200, final SP = 32000 – 3200 = Rs 28800.
Application: Common in festive sales with multiple offers.

6. Profit and Loss: Comparing Quantities

Example: A shopkeeper buys a chair for Rs 1200 and sells it for Rs 1500. Find the profit and profit percentage.

Solution: Profit = 1500 – 1200 = Rs 300. Profit % = (300/1200) * 100 = 25%.
Application: Helps businesses evaluate product profitability.

Example: A bicycle CP Rs 5000 is sold for Rs 4500. Find the loss and loss percentage.

Solution: Loss = 5000 – 4500 = Rs 500. Loss % = (500/5000) * 100 = 10%.
Application: Useful for assessing losses on unsold inventory.

Example (Multiple Items): A vendor buys 10 pens at Rs 20 each and sells 8 at Rs 30 each, 2 at Rs 25 each. Find overall profit/loss.

Solution: Total CP = 10 * 20 = Rs 200. Total SP = (8 * 30) + (2 * 25) = 240 + 50 = Rs 290. Profit = 290 – 200 = Rs 90.
Application: Tracks profit across varied pricing.

7. Taxes: Comparing Quantities

Example: A restaurant bill is Rs 1500 with a 5% GST. Calculate the total amount paid.

Solution: GST = (5/100) * 1500 = Rs 75. Total = 1500 + 75 = Rs 1575.
Application: Helps customers understand final bills.

Example: A phone costs Rs 20000 (including 12% VAT). Find the base price before VAT.

Solution: Let base price = x. Then x * (1 + 12/100) = 20000. So, x * 1.12 = 20000, x = 20000/1.12 ≈ Rs 17857.14.
Application: Useful for comparing pre-tax prices.

8. Simple and Compound Interest: Comparing Quantities

Example (Simple Interest): A loan of Rs 5000 is taken at 6% per annum for 3 years. Find the simple interest and amount.

Solution: SI = (5000 * 6 * 3)/100 = Rs 900. Amount = 5000 + 900 = Rs 5900.
Application: Used for short-term loans like personal loans.

Example (Compound Interest): Rs 10000 is invested at 5% per annum compounded annually for 2 years. Find the CI and amount.

Solution: A = 10000 * (1 + 5/100)^2 = 10000 * (1.05)^2 = 10000 * 1.1025 = Rs 11025. CI = 11025 – 10000 = Rs 1025.
Application: Common in savings accounts or fixed deposits.

Example (Half-Yearly CI): Rs 8000 at 10% per annum compounded half-yearly for 1 year. Find the amount.

Solution: Rate = 10/2 = 5%, time = 1 * 2 = 2 periods. A = 8000 * (1 + 5/100)^2 = 8000 * (1.05)^2 = 8000 * 1.1025 = Rs 8820.
Application: Used in investments with frequent compounding.

Example (Depreciation): A car worth Rs 500000 depreciates at 10% per annum for 2 years. Find its value.

Solution: A = 500000 * (1 – 10/100)^2 = 500000 * (0.9)^2 = 500000 * 0.81 = Rs 405000.
Application: Helps estimate resale value of assets.

Conclusion

This chapter equips students with skills to compare quantities using mathematical tools, fostering logical thinking for financial decisions. Practice examples to reinforce concepts like converting units or calculating CI for real-life problems.

Download pdf notes and Sulutions of the Chapter:

Chapter 8 Comparing Quantities

Chapter 9: Algebraic Expressions and Identities

8th Maths

Chapter 8: Comparing Quantities