Comparing quantities involves using ratios, percentages, profit, loss, and simple interest to understand how two or more things relate to each other. These concepts are used extensively in everyday life — from shopping to banking.
Ratios:
A ratio compares two quantities of the same kind. The ratio of a to b is written as a:b or a/b. Ratios should be expressed in simplest form.
Percentages:
A percentage means "out of 100". To convert a fraction to a percentage, multiply by 100.
To convert a percentage to a fraction, divide by 100.
To find x% of a quantity: (x/100) x quantity.
Percentage Increase and Decrease:
Percentage change = (Change / Original) x 100
Percentage increase = (Increase / Original value) x 100
Percentage decrease = (Decrease / Original value) x 100
- Profit and Loss:
- Profit = Selling Price (SP) - Cost Price (CP), when SP > CP
- Loss = Cost Price - Selling Price, when CP > SP
- Profit% = (Profit / CP) x 100
- Loss% = (Loss / CP) x 100
- SP = CP + Profit = CP x (1 + Profit%/100)
- SP = CP - Loss = CP x (1 - Loss%/100)
Simple Interest:
SI = (Principal x Rate x Time) / 100 = PRT/100
Amount = Principal + SI = P + PRT/100 = P(1 + RT/100)
Where P = Principal, R = Rate per annum (%), T = Time in years.
Convert 3/4 to a percentage.
(3/4) x 100 = 75%.
A shirt costs Rs 500. It is sold at Rs 600. Find the profit%.
Profit = 600 - 500 = Rs 100. Profit% = (100/500) x 100 = 20%.
A toy is bought for Rs 400 and sold at a loss of 15%. Find the SP.
SP = 400 x (1 - 15/100) = 400 x 0.85 = Rs 340.
Find the simple interest on Rs 2000 at 5% per annum for 3 years.
SI = (2000 x 5 x 3)/100 = 30000/100 = Rs 300.
A number is increased by 20% and then decreased by 20%. Is the final number equal to the original?
Increase: 100 becomes 120. Decrease 20% of 120: 120 - 24 = 96. Not equal — the final number is less.
A shopkeeper sells 12 items for the cost of 10. Find profit%.
CP of 12 = 12 units, SP of 12 = cost of 10 = 10 units... let CP per item = 1. CP of 12 = 12. SP = 12 (sells at price that would buy 10 originally... wait, re-reading: sells 12 items at price of 10 items). So SP of 12 = 10 x (CP per item). Hmm, standard problem: buys 10, gets 12 — gain of 2 items. Profit = 2, CP = 10. Profit% = 20%.
Priya borrowed Rs 5000 at 8% per annum. What is the amount to be returned after 2 years?
SI = (5000 x 8 x 2)/100 = 800. Amount = 5000 + 800 = Rs 5800.
- Key Formulas:
- Profit% = (Profit/CP) x 100
- Loss% = (Loss/CP) x 100
- SI = PRT/100
- Amount = P + SI
Common mistakes
Always calculate profit% or loss% on the COST PRICE, not the selling price. In simple interest, time must be in years — if given in months, convert by dividing by 12.
Summary
Percentages, profit-loss, and simple interest are real-world applications of ratios and fractions. Knowing the base (original value or cost price) is the key to computing percentage changes correctly.