IMO › Class 8 › Comparing Quantities

Comparing Quantities

Increase & Decrease Percent

What is Percentage?

Percentage means "per hundred" (from Latin "per centum"). A percentage is a fraction with denominator 100. The symbol % represents "out of 100".

Key Conversions:

Convert FromToMethodExample
FractionPercentageMultiply by 100%\(\frac{3}{4} = 0.75 \times 100\% = 75\%\)
PercentageFractionDivide by 100%\(40\% = \frac{40}{100} = \frac{2}{5}\)
DecimalPercentageMultiply by 100%\(0.65 \times 100\% = 65\%\)
PercentageDecimalDivide by 100\(25\% = 25 \div 100 = 0.25\)

Percentage of a Quantity:

To find \(x\%\) of a quantity \(Q\), calculate: \(\frac{x}{100} \times Q\)

Example: 20% of 250 = \(\frac{20}{100} \times 250 = 0.2 \times 250 = 50\)

What is Profit and Loss?

In business, items are bought at Cost Price (CP) and sold at Selling Price (SP).

  • Profit occurs when SP > CP
  • Loss occurs when SP < CP

Important Formulas:

ConceptFormula
ProfitSP - CP
LossCP - SP
Profit Percentage\(\frac{\text{Profit}}{CP} \times 100\%\)
Loss Percentage\(\frac{\text{Loss}}{CP} \times 100\%\)
SP when Profit% given\(CP \times (1 + \frac{\text{Profit%}}{100})\)
SP when Loss% given\(CP \times (1 - \frac{\text{Loss%}}{100})\)
CP when Profit% given\(\frac{SP \times 100}{100 + \text{Profit%}}\)
CP when Loss% given\(\frac{SP \times 100}{100 - \text{Loss%}}\)

Real-life Example: If you buy a pen for ₹10 (CP) and sell it for ₹15 (SP), your profit is ₹5, and profit percentage = \(\frac{5}{10} \times 100\% = 50\%\).

Percentage, Profit and Loss100%75%75 out of 100Profit & Loss FormulasProfit = SP − CP (if SP > CP)Loss = CP − SP (if CP > SP)Profit % = (Profit/CP) × 100Loss % = (Loss/CP) × 100SP = CP × (1 + P/100)SP = CP × (1 − L/100)Example: CP = ₹800, SP = ₹920 → Profit = ₹120 → Profit% = (120/800)×100 = 15%Conversion: x% = x/100, e.g. 35% = 35/100 = 0.35 = 7/20
Example 1: Increase 50 by 20%.
20% of 50 = 10, so 50 + 10 = 60.
Example 2: Decrease 80 by 25%.
25% of 80 = 20, so 80 − 20 = 60.
Example 3:

Convert \(\frac{3}{8}\) to a percentage. Also, find 35% of 240.

  • Step 1: Fraction to percentage: Multiply by 100%
  • Step 2: \(\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%\)
  • Step 3: 35% of 240 = \(\frac{35}{100} \times 240\)
  • Step 4: \(= 0.35 \times 240 = 84\)

Answer: $\frac{3}{8} = 37.5\%$; 35% of 240 = 84

Example 4:

A shopkeeper buys a watch for ₹800 and sells it for ₹960. Find the profit percentage.

  • Step 1: CP = ₹800, SP = ₹960
  • Step 2: Profit = SP - CP = 960 - 800 = ₹160
  • Step 3: Profit% = \(\frac{\text{Profit}}{CP} \times 100\% = \frac{160}{800} \times 100\%\)
  • Step 4: \(= 0.2 \times 100\% = 20\%\)

Answer: Profit percentage = 20%

Example 5:

A dealer sold a bicycle at a loss of 12%. If the selling price was ₹2640, find the cost price.

  • Step 1: Loss% = 12%, SP = ₹2640
  • Step 2: Loss% means SP = CP × (1 - \(\frac{12}{100}\)) = CP × 0.88
  • Step 3: \(2640 = CP \times 0.88\)
  • Step 4: \(CP = \frac{2640}{0.88} = \frac{2640 \times 100}{88} = \frac{264000}{88}\)
  • Step 5: \(CP = 3000\)

Answer: Cost price = ₹3000

Quick recap
  • Increase: add the percent of the number.
  • Decrease: subtract the percent of the number.
  • Percentage means "out of 100" (per hundred)
  • To convert fraction to %: multiply by 100%
  • Profit = SP - CP (when SP > CP)
  • Loss = CP - SP (when CP > SP)
  • Profit% = (Profit/CP) × 100%
  • Loss% = (Loss/CP) × 100%
  • Always calculate percentages relative to Cost Price
✓ Quick check
Increase 200 by 10%.
10% of 200 = 20, so 200 + 20 = 220.
Decrease 100 by 40%.
40% of 100 = 40, so 100 − 40 = 60.

Discount, Marked Price & GST

What is Discount?

Discount is a reduction in the selling price of an item. It is usually expressed as a percentage of the Marked Price (MP) or List Price.

Important Terms:

  • Marked Price (MP): The original price printed on the item
  • Discount: The amount subtracted from MP
  • Selling Price (SP): The actual price paid after discount

Formulas:

ConceptFormula
Discount AmountMP × \(\frac{\text{Discount%}}{100}\)
Selling Price after DiscountMP - Discount = MP × \((1 - \frac{\text{Discount%}}{100})\)
Discount Percentage\(\frac{\text{Discount}}{MP} \times 100\%\)

What is Simple Interest?

Simple Interest (SI) is the extra money paid for borrowing money or earned on savings. It is calculated only on the original principal amount.

Key Terms:

  • Principal (P): The initial amount borrowed or invested
  • Rate (R): Interest rate per year (as percentage)
  • Time (T): Time period in years
  • Simple Interest (SI): Interest earned/paid
  • Amount (A): Principal + Interest

Simple Interest Formulas:

ConceptFormula
Simple Interest\(SI = \frac{P \times R \times T}{100}\)
Amount\(A = P + SI = P(1 + \frac{RT}{100})\)
Principal\(P = \frac{SI \times 100}{R \times T}\)
Rate\(R = \frac{SI \times 100}{P \times T}\)
Time\(T = \frac{SI \times 100}{P \times R}\)

Real-life Applications:

  • Bank loans and mortgages
  • Savings accounts interest
  • Festival sales discounts
  • "Buy one get one" offers
Discount and Simple InterestDISCOUNTMarked Price (MP)₹2,000Discount: 20% = ₹400Sale Price: ₹1,600Discount = MP × Rate/100SP = MP − DiscountSIMPLE INTERESTSI = P × R × T 100P = Principal (₹)R = Rate per annum (%)T = Time (years)A = Amount = P + SIe.g. SI = 5000×8×3/100 = ₹1200
Example 1: MP = ₹500 with a 10% discount. Find the SP.
10% of 500 = ₹50 off, so SP = ₹450.
Example 2: Find 18% GST on ₹200.
18% of 200 = ₹36.
Example 3:

A shop offers a 15% discount on a ₹1200 watch. Find the discount amount and the selling price.

  • Step 1: Discount = 15% of MP = \(\frac{15}{100} \times 1200\)
  • Step 2: Discount = \(0.15 \times 1200 = ₹180\)
  • Step 3: SP = MP - Discount = \(1200 - 180 = ₹1020\)

Answer: Discount = ₹180, Selling Price = ₹1020

Example 4:

Find the simple interest on ₹5000 for 3 years at 8% per annum. Also find the amount.

  • Step 1: P = ₹5000, R = 8%, T = 3 years
  • Step 2: \(SI = \frac{P \times R \times T}{100} = \frac{5000 \times 8 \times 3}{100}\)
  • Step 3: \(SI = \frac{120000}{100} = ₹1200\)
  • Step 4: Amount = P + SI = \(5000 + 1200 = ₹6200\)

Answer: Simple Interest = ₹1200, Amount = ₹6200

Example 5:

A television is marked at ₹25,000. It is sold at a discount of 12%. If the shopkeeper still makes a profit of 10%, find the cost price of the television.

  • Step 1: MP = ₹25,000, Discount = 12%
  • Step 2: SP = MP × \((1 - \frac{12}{100}) = 25000 \times 0.88 = ₹22,000\)
  • Step 3: Profit = 10%, so SP = CP × \((1 + \frac{10}{100}) = CP \times 1.1\)
  • Step 4: \(22000 = CP \times 1.1\)
  • Step 5: \(CP = \frac{22000}{1.1} = ₹20,000\)

Answer: Cost Price = ₹20,000

Quick recap
  • SP = MP − discount.
  • GST is added on top of the price.
  • Discount = MP × Discount% | SP after discount = MP - Discount
  • Simple Interest \(= \frac{P \times R \times T}{100}\)
  • Amount = Principal + Simple Interest
  • Discount and profit percentage are calculated on different bases (MP vs CP)
  • Simple interest is linear (same interest each year)
  • Time must be in years; convert months to years (divide by 12)
✓ Quick check
MP = ₹800 with a 25% discount. What is the SP?
25% of 800 = ₹200 off, so SP = ₹600.
What is 12% GST on ₹100?
12% of 100 = ₹12.

Compound Interest & CI vs SI

With compound interest, interest is added to the principal each period, so the next period's interest is larger. On ₹1000 at 10% for 2 years: year 1 gives ₹100 (₹1100), year 2 gives ₹110 (₹1210), so CI = ₹210.

This is ₹10 more than the simple interest of ₹200 — the extra is interest on the first year's interest.

Example 1: Find the CI and amount on ₹1000 at 10% for 2 years.
Year 1: 1100; year 2: 1210. CI = ₹210, amount = ₹1210.
Example 2: By how much does CI exceed SI here?
₹210 − ₹200 = ₹10.
Quick recap
  • CI adds interest to the principal each period.
  • Over 2 years, CI exceeds SI by interest on the first year's interest.
✓ Quick check
What is the amount on ₹2000 at 10% compound interest for 2 years?
Year 1: 2200, year 2: 2420.
For 1 year, the compound interest on ₹500 at 10% is ___ ?
For 1 year CI = SI = ₹50.
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