Comparing Quantities — Study Notes

1. Introduction

In our daily life we are constantly comparing quantities. We compare prices of goods, savings in a bank, marks in examinations, heights of friends and the growth of investments. Comparison can be done in different ways: by subtraction, by division, or by using ratios and percentages. Percentages are perhaps the most common way of comparing quantities, and they appear everywhere - from discounts in shops to interest in banks to scores in examinations.

In this chapter we will learn about ratios and percentages, how to increase and decrease quantities by a given percentage, how to find profit and loss in transactions, how to compute simple interest and compound interest, and how to work with the concepts of cost price, selling price, discount and marked price. These are practical skills that every student will use throughout their life.

2. Ratios and Percentages

Ratio

A ratio is a comparison of two quantities of the same kind by division. For example, the ratio of 20 boys to 30 girls is 20:30 = 2:3. A ratio has no units, and both quantities must be in the same units.

Percentage

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is %. For example, 25% means 25 out of 100, that is, 25/100 = 1/4.

To convert a ratio to a percentage, convert the ratio to a fraction and then multiply by 100. For example, the ratio 2:3 gives the fraction 2/3, which is 2/3 x 100 = 66.67%.

To find a percentage of a quantity:

$$\text{Percentage value} = \frac{\text{Percentage}}{100} \times \text{Quantity}$$

For example, 20% of 250 = 20/100 x 250 = 50.

3. Increase and Decrease by a Percentage

When a quantity increases by a percentage, we find the increase and add it to the original quantity. When it decreases, we find the decrease and subtract it.

Percentage Increase

$$\text{New value} = \text{Original} \times \left(1 + \frac{\text{Percentage}}{100}\right)$$

For example, if the population of a town is 20,000 and it increases by 5%, the new population = 20000 x (1 + 5/100) = 20000 x 1.05 = 21,000.

Percentage Decrease

$$\text{New value} = \text{Original} \times \left(1 - \frac{\text{Percentage}}{100}\right)$$

4. Profit, Loss and Discount

Cost Price and Selling Price

The amount for which an article is bought is called its cost price (CP), and the amount for which it is sold is called its selling price (SP).

Profit and Loss

• If SP > CP, there is a profit: Profit = SP - CP.
• If SP < CP, there is a loss: Loss = CP - SP.
• Profit or loss percentage is always calculated on the cost price: $$\text{Profit} \% = \frac{\text{Profit}}{\text{CP}} \times 100$$ $$\text{Loss} \% = \frac{\text{Loss}}{\text{CP}} \times 100$$

Marked Price and Discount

The price printed on an article is called its marked price (MP). The reduction given on the marked price is called the discount. Usually the discount is given as a percentage of the marked price. $$\text{Selling Price} = \text{Marked Price} - \text{Discount}$$

Sales Tax, VAT and GST

The government collects a tax on the sale of goods. This tax is calculated as a percentage of the price of the goods and is added to the bill. The final amount a customer pays includes this tax.

5. Simple Interest

When money is borrowed, the borrower has to pay some extra money to the lender. This extra money is called the interest. The amount borrowed is called the principal, and the principal plus interest is called the amount.

$$\text{Simple Interest (SI)} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} = \frac{PRT}{100}$$

$$\text{Amount} = \text{Principal} + \text{Simple Interest}$$

For example, if 5000 rupees is deposited at 10% per annum for 2 years, then SI = (5000 x 10 x 2)/100 = 1000 rupees, and the amount = 5000 + 1000 = 6000 rupees.

6. Compound Interest

In compound interest, the interest earned in each period is added to the principal, and the next period's interest is calculated on this new amount. This means interest is calculated on interest too.

$$\text{Amount} = \text{Principal} \times \left(1 + \frac{\text{Rate}}{100}\right)^{\text{Time}}$$

$$\text{Compound Interest} = \text{Amount} - \text{Principal}$$

Compound Interest with Different Compounding Periods

When interest is compounded half-yearly, the rate is halved and the time is doubled. For example, 10% per annum compounded half-yearly for 2 years means the rate becomes 5% and the time becomes 4 half-years.

Applications of Compound Interest Formula

The compound interest formula can also be used to find population growth, price appreciation or depreciation, and growth of microorganisms.

Quick Revision Tables

Table 1: Key Formulas

Table 2: Difference between Simple and Compound Interest

Mind Map

Important Diagrams (SVG)

Diagram 1: Simple Interest vs Compound Interest Growth

Diagram 2: The Transaction Flow - Profit and Loss

Common Mistakes

1. Students calculate profit or loss percentage on the selling price instead of the cost price. The denominator must always be CP.
2. When comparing two quantities by ratio, students forget that the units must be the same. For example, 2 m to 50 cm must be written as 200:50 = 4:1.
3. Students add the percentage directly to the quantity instead of computing the percentage of it. For example, increasing 200 by 10% is not 200 + 10, but 200 + 20 = 220.
4. In simple interest problems, students forget to convert time into years when it is given in months.
5. Students confuse simple interest with compound interest and use the wrong formula. Simple interest uses PRT/100, while compound uses P(1 + R/100)^T.
6. While converting a ratio into a percentage, students multiply the ratio terms directly by 100 instead of first converting the ratio to a fraction of the whole.
7. In discount problems, students calculate discount on the cost price instead of the marked price.

Exam Tips

1. Always state whether the answer is profit or loss and mention the percentage sign correctly in the final answer.
2. For compound interest questions, first write the formula A = P(1 + R/100)^T and then substitute, so you earn method marks.
3. Practise converting fractions and ratios to percentages and vice versa; these appear in many questions.
4. Remember that profit and loss percentage is always based on cost price, while discount is based on marked price.
5. For word problems on population growth, identify that the compound interest formula is used.
6. When interest is compounded half-yearly, halve the rate and double the time before substituting.
7. Learn to quickly convert percentages to fractions (50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10) to speed up calculations.

Conclusion

Comparing quantities is one of the most practical chapters in Class 8 mathematics. We learnt how to express comparison using ratios and percentages, how to increase and decrease quantities by a given percentage, and how to analyse transactions involving cost price, selling price, profit, loss and discount. We also studied the two important concepts of interest - simple interest, which is calculated only on the principal, and compound interest, which is calculated on the principal as well as the accumulated interest, and which grows faster. These ideas are used constantly in banking, shopping, business and daily life. Understanding this chapter thoroughly will help students manage money wisely and perform well in examinations.