Comparing Quantities

7.1 Recalling Ratios and Percentages

Ratios and percentages offer essential tools for making comparisons between quantities. A ratio compares two quantities, usually in simplest form. For example, if there are 20 apples and 5 oranges, the ratio of oranges to apples is 5:20, which simplifies to 1:4. In percentage, this is represented as follows:

• Percentage of Oranges = (Number of Oranges / Total Fruits) * 100 = (5 / 25) * 100 = 20%.
• Conversely, the percentage of Apples is 100% - 20% = 80%.

Example 1: Class Picnic Problem

This example illustrates applying ratios and percentages in a real-world scenario involving students planning a picnic.

1. Finding Ratio: To find the total number of boys, let the total students be x. Given that 60% of x are girls (which amounts to 18 girls), we derive:

   60% of x = 18 (60/100) * x = 18 → x = 30 (total students). Boys = Total students - Girls = 30 - 18 = 12. Hence, the ratio of girls (18) to boys (12) is 3:2.

2. Cost per Head: The transportation cost for 55 km, charged at ₹12/km, totals ₹1320. Total expenses are refreshment charges (₹4280) plus transportation, yielding ₹5600. Hence, each of the 32 persons pays: Cost per head = Total Expenses / Total Persons = 5600 / 32 = ₹175.

3. Distance Percentage: The first stop at 22 km represents 40% of the total 55 km distance since: (22 / 55) * 100 = 40%. Remaining distance percentage = 100% - 40% = 60%.

7.2 Finding Discounts

Discounts are reductions from the marked price (MP) of an item, typically calculated as: Discount = Marked Price - Sale Price. To find the discount percentage, use: Discount % = (Discount / Marked Price) * 100. For items with a fixed discount percentage (e.g., 20%), one can quickly calculate the sale price from the marked price.

Example on Discount Calculation

For an item marked at ₹840 sold for ₹714, the discount is: Discount = 840 - 714 = ₹126, leading to: Discount % = (126 / 840) * 100 ≈ 15%.

7.3 Sales Tax/Value Added Tax/Goods and Services Tax

Sales tax (ST) is a government levy on sales added to the item's price while paid by the customer. Today's rates might also include GST, which simplifies tax administration for goods/services. To compute ST: ST = (Sales Tax Rate / 100) * Bill Amount.

7.4 Compound Interest

Interest is a key concept in finance, with two main types: simple and compound interest. Simple Interest (SI) is calculated on initial principal only, whereas Compound Interest (CI) considers the interest on both the principal and the previously accumulated interest.

Example on CI Calculation

For example, on ₹20,000 at 8% for two years:

1. 1st Year Interest: 20000 * 8/100 = ₹1600.
   • Total after Year 1 = 20,000 + 1600 = 21,600.
2. 2nd Year Interest: 21600 * 8/100 = ₹1728.
   • Total amount after 2 years = 21600 + 1728 = ₹23,328.

Thus, CI = Total Amount - Principal = ₹3328.

7.5 Deducing a Formula for CI

Using the derived formula, we can quickly calculate CI: A = P(1 + R/100)^n, where A is the amount after n years, P is principal, and R is the rate of interest.

7.6 Applications of Compound Interest Formula

Compound interest can be applied in various contexts such as population growth and depreciation of assets. For example, calculating estimated year-end populations based on growth rates or determining depreciation over time.

Summary Points

• Ratios provide a comparative perspective of quantities, expressed in simplest terms.
• Percentages aid in understanding proportions of quantities within a whole.
• Discounts reduce the selling price of goods, enhancing sales appeal.
• Sales Tax (ST) & GST are critical components of pricing in commerce.
• Simple Interest (SI) is static upon the principal, whereas Compound Interest (CI) is dynamic and grows annually.
• The CI formula simplifies future value calculation based on principal and interest rates over time.
• Applications extend to real-life scenarios involving finance, budgeting, and resource management.

1. Ratios help compare two quantities in simplest form.
2. Percentages express a portion of a whole out of 100.
3. Discounts are percentages deducted from the marked price.
4. Compound Interest accounts for interest on previously earned interest.
5. Simple Interest is based solely on principal amount without considering prior interest.
6. Sales Tax (ST) is calculated on the selling price and collected for government.
7. GST simplifies indirect taxation on goods and services.
8. Understanding percentages is fundamental to effective financial literacy.