Comparing Quantities

Essential Terms & Definitions

Ratio

A comparison of two quantities of the exact same unit by division, denoted as a:b.

Percentage

A fraction with a denominator of 100; representing parts per hundred, denoted by the % symbol.

Discount

The reduction offered on the Marked Price (MP) of goods. Formula: Discount = Marked Price - Sale Price.

Cost Price (CP)

The total amount paid to buy or manufacture a product, including any overhead expenses like transport and repairs.

Selling Price (SP)

The final price at which a product is sold to the consumer.

Goods and Services Tax (GST)

A unified indirect tax levied on the supply of goods and services by the government, calculated on the selling price.

Compound Interest (CI)

Interest calculated on the initial principal and also on the accumulated interest of previous periods.

Understanding Ratios, Percentages, and Financial Math

In everyday transactions, we frequently compare quantities. Ratios show how many times one number contains another, requiring both quantities to be in the same units (e.g., comparing grams with grams, not grams with kilograms). Percentages offer a uniform scale of comparison out of 100, which is extremely useful in determining score ratios, population growths, or financial shifts. When dealing with business mathematics, we focus heavily on Profit and Loss calculated against the Cost Price (CP). Discount represents a deduction on the listed Marked Price (MP), which helps businesses clear inventory. Lastly, taxes such as GST are calculated on the ultimate selling price of the item and always added to the final bill amount. Memorizing how these values scale is crucial for CBSE exam speed.

Comparison of Profit, Loss, and Discount Formulas

Step-by-Step Process: Compounding Interest Half-Yearly

1. Identify Given Parameters — Write down the Principal (P), Annual Rate of Interest (R), and Time Period in years (T or n).
2. Modify the Interest Rate — Divide the annual rate of interest (R) by 2 since interest is calculated twice a year. New Rate (R') = R / 2.
3. Modify the Time Period — Multiply the number of years (n) by 2 to get the total compounding periods. New Time (n') = 2n.
4. Calculate the Amount — Apply the compound interest formula: Amount (A) = P * (1 + R' / 100)^n'.
5. Calculate Compound Interest — Subtract the principal from the calculated amount: CI = Amount (A) - Principal (P).

Solved Mini-Examples for Exam Practice

• {"title":"Example 1: Finding Percentage Increase","problem":"The price of a motorcycle increased from Rs. 80,000 to Rs. 88,000. Find the percentage increase.","solution":"1. Find the absolute increase: Rs. 88,000 - Rs. 80,000 = Rs. 8,000.\n2. Percentage Increase = (Increase / Original Price) 100.\n3. Percentage Increase = (8,000 / 80,000) 100 = 10%."}
• {"title":"Example 2: Discount and Tax Calculation","problem":"An item marked at Rs. 1,200 is sold at a 10% discount. If a GST of 5% is charged on the discounted price, find the final billing amount.","solution":"1. Calculate Discount: 10% of Rs. 1,200 = (10/100) 1,200 = Rs. 120.\n2. Sale Price (before tax) = Rs. 1,200 - Rs. 120 = Rs. 1,080.\n3. Calculate GST: 5% of Rs. 1,080 = (5/100) 1,080 = Rs. 54.\n4. Final Billing Amount = Rs. 1,080 + Rs. 54 = Rs. 1,134."}
• {"title":"Example 3: Compound Interest (Annual)","problem":"Find the Compound Interest on Rs. 10,000 for 2 years at 10% per annum compounded annually.","solution":"1. Identify: P = Rs. 10,000, R = 10%, n = 2.\n2. Use Amount Formula: A = P (1 + R/100)^n = 10,000 (1 + 10/100)^2.\n3. A = 10,000 (1.1)^2 = 10,000 1.21 = Rs. 12,100.\n4. Calculate Compound Interest: CI = A - P = 12,100 - 10,000 = Rs. 2,100."}

Key Points and Must-Remember Formulas

• Ratio is a comparison of two quantities of the same units. Ensure units are matched before starting.
• Percentage means per hundred. Multiply a fraction or decimal by 100 to change it to a percentage.
• Profit % and Loss % are always calculated on the Cost Price (CP) unless explicitly stated otherwise.
• Discount = Marked Price (MP) - Selling Price (SP). Discount % is always computed on the MP.
• Tax (VAT or GST) is computed on the actual Selling Price (SP) of the item, not on the original Marked Price.
• Compound Interest Formula: Amount (A) = P * (1 + R/100)^n.
• If interest is compounded half-yearly, the rate is halved (R/2) and the time periods are doubled (2n).
• If the interest rate is R1% for the first year and R2% for the second year, the total amount is A = P (1 + R1/100) (1 + R2/100).

Exam Trap Alerts & Cues

Beware of the Unit Trap: Always double-check if ratio elements have the same units. For example, comparing 5 m to 10 km requires converting km to m first (5 m : 10,000 m = 1 : 2000). Compounding Mistake: In half-yearly compounding questions, students often forget to divide the rate by 2 and multiply the time by 2. This is a common step error. Formula Note: For Step-by-Step marks in CBSE papers, always state your formula clearly ($A = P(1+R/100)^n$) before substituting values. Even if your calculation goes wrong, you will receive step-marking for writing the correct formula and values!

Practice Questions with Solutions

• What is the discount percentage of a book marked at Rs. 500 and sold for Rs. 450? Discount = MP - SP = 500 - 450 = Rs. 50. Discount % = (Discount/MP) 100 = (50 / 500) 100 = 10%.
• If an item is purchased for Rs. 200 and sold for Rs. 240, find the profit percentage. Profit = SP - CP = 240 - 200 = Rs. 40. Profit % = (Profit/CP) 100 = (40 / 200) 100 = 20%.
• A table is bought for Rs. 5,000 and 8% GST is added. What is the buyer's final cost? GST = 8% of Rs. 5,000 = (8/100) * 5000 = Rs. 400. Total buying cost = Cost + GST = 5,000 + 400 = Rs. 5,400.
• How do you calculate compound interest compounded half-yearly for 1.5 years at 10% per annum? Adjust values: New rate R' = 10% / 2 = 5% per half-year. Number of half-years (n') = 1.5 * 2 = 3. Use formula A = P(1 + 5/100)^3, then calculate CI = A - P.

Frequently Asked Questions