Ratio, Proportion, Percentage, Profit and Loss, Simple and Compound Interest

1. Ratio and Proportion

A ratio is a comparison of two quantities by division. It tells us how much of one thing there is compared to another. Ratios are usually written with a colon (:) or as a fraction. For example, if there are 2 apples and 3 oranges, the ratio of apples to oranges is 2:3 or 2/3.

Types of Ratios:

  • Continued Ratio: When three or more quantities are compared, it's called a continued ratio. For example, if the ratio of A's age to B's age is 2:3 and B's age to C's age is 3:4, then the continued ratio of A:B:C is 2:3:4.
  • Duplicate Ratio: The ratio obtained by squaring the terms of a given ratio. The duplicate ratio of a:b is a2:b2.
  • Sub-duplicate Ratio: The ratio obtained by taking the square root of the terms of a given ratio. The sub-duplicate ratio of a:b is √a:√b.
  • Triplicate Ratio: The ratio obtained by cubing the terms of a given ratio. The triplicate ratio of a:b is a3:b3.
  • Sub-triplicate Ratio: The ratio obtained by taking the cube root of the terms of a given ratio. The sub-triplicate ratio of a:b is 3√a:3√b.
  • Inverse Ratio: The ratio obtained by inverting the terms of a given ratio. The inverse ratio of a:b is b:a.

Proportion is a statement that two ratios are equal. If a:b and c:d are two ratios, they are in proportion if a:b = c:d. This can also be written as a/b = c/d. In a proportion a:b = c:d, 'a' and 'd' are called the 'extremes' and 'b' and 'c' are called the 'means'. The product of the extremes is equal to the product of the means (a * d = b * c).

Types of Proportion:

  • Direct Proportion: Two quantities are said to be in direct proportion if they increase or decrease together in the same ratio. For example, if the cost of 5 kg of sugar is ₹200, then the cost of 10 kg of sugar will be ₹400.
  • Inverse Proportion: Two quantities are said to be in inverse proportion if an increase in one causes a decrease in the other, and vice versa, in such a way that their product remains constant. For example, if 4 men can build a wall in 6 days, then 8 men will build the same wall in 3 days (4 * 6 = 24, 8 * 3 = 24).

Example Problems:

  1. If 3:5 is equal to x:20, find the value of x.

    Solution: Using the property of proportion, product of extremes = product of means. 3 * 20 = 5 * x 60 = 5x x = 60 / 5 x = 12

  2. The ratio of two numbers is 2:3. If the sum of the numbers is 45, find the numbers.

    Solution: Let the numbers be 2k and 3k. 2k + 3k = 45 5k = 45 k = 45 / 5 k = 9 The numbers are 2 * 9 = 18 and 3 * 9 = 27.

Shortcut: When dealing with ratios, if you need to combine them (like A:B and B:C), make the common term (B in this case) equal in both ratios by multiplying. For A:B = 2:3 and B:C = 4:5, multiply the first ratio by 4 and the second by 3 to get A:B = 8:12 and B:C = 12:15. Then A:B:C = 8:12:15.

2. Percentage

Percentage means "per hundred". It is a way of expressing a number as a fraction of 100. The symbol "%" is used to denote percentage. For example, 50% means 50 out of 100, which is equal to 50/100 or 1/2.

Converting Fractions and Decimals to Percentages:

  • Fraction to Percentage: Multiply the fraction by 100. Example: 3/4 = (3/4) * 100% = 75%
  • Decimal to Percentage: Multiply the decimal by 100. Example: 0.65 = 0.65 * 100% = 65%

Converting Percentages to Fractions and Decimals:

  • Percentage to Fraction: Divide the percentage by 100 and remove the % sign. Example: 40% = 40/100 = 2/5
  • Percentage to Decimal: Divide the percentage by 100. Example: 75% = 75/100 = 0.75

Calculating Percentage of a Quantity:

To find P% of a quantity Q, use the formula: (P/100) * Q.

Percentage Increase and Decrease:

  • Percentage Increase: ((New Value - Original Value) / Original Value) * 100%
  • Percentage Decrease: ((Original Value - New Value) / Original Value) * 100%

Example Problems:

  1. What is 20% of 500?

    Solution: (20/100) * 500 = 0.20 * 500 = 100.

  2. A number increased from 80 to 100. What is the percentage increase?

    Solution: Increase = 100 - 80 = 20. Percentage Increase = (20 / 80) * 100% = (1/4) * 100% = 25%.

  3. If a student scores 45 marks out of 60, what is their percentage score?

    Solution: (45 / 60) * 100% = (3/4) * 100% = 75%.

Shortcut: To quickly calculate percentage changes, remember common fractions: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%. Also, for successive percentage changes, if a quantity is increased by x% and then by y%, the net percentage change is (x + y + xy/100)%.

3. Profit and Loss

Profit and loss calculations are based on the cost price and selling price of an item.

  • Cost Price (CP): The price at which an item is bought.
  • Selling Price (SP): The price at which an item is sold.
  • Profit: Occurs when SP > CP. Profit = SP - CP.
  • Loss: Occurs when CP > SP. Loss = CP - SP.

Formulas:

  • Profit Percentage: (Profit / CP) * 100%
  • Loss Percentage: (Loss / CP) * 100%
  • If Profit% is P, then SP = CP * (100 + P) / 100
  • If Loss% is L, then SP = CP * (100 - L) / 100
  • If SP is known and Profit% is P, then CP = SP * 100 / (100 + P)
  • If SP is known and Loss% is L, then CP = SP * 100 / (100 - L)

Example Problems:

  1. A shopkeeper buys a toy for ₹200 and sells it for ₹250. Find the profit or loss and the percentage.

    Solution: CP = ₹200, SP = ₹250. Since SP > CP, there is a profit. Profit = SP - CP = ₹250 - ₹200 = ₹50. Profit Percentage = (Profit / CP) * 100% = (50 / 200) * 100% = (1/4) * 100% = 25%.

  2. An item is sold for ₹800 after a loss of 20%. What was the cost price?

    Solution: SP = ₹800, Loss% = 20%. CP = SP * 100 / (100 - Loss%) CP = 800 * 100 / (100 - 20) CP = 800 * 100 / 80 CP = 10 * 100 = ₹1000.

  3. A trader sells two articles for ₹5000 each. On one, he gains 10% and on the other, he loses 10%. What is the overall gain or loss?

    Solution: Article 1 (Gain): SP = ₹5000, Gain% = 10%. CP1 = 5000 * 100 / (100 + 10) = 5000 * 100 / 110 = ₹4545.45 (approx) Article 2 (Loss): SP = ₹5000, Loss% = 10%. CP2 = 5000 * 100 / (100 - 10) = 5000 * 100 / 90 = ₹5555.56 (approx) Total SP = 5000 + 5000 = ₹10000. Total CP = CP1 + CP2 = 4545.45 + 5555.56 = ₹10101.01 (approx) Since Total CP > Total SP, there is a loss. Total Loss = Total CP - Total SP = 10101.01 - 10000 = ₹101.01 (approx) Overall Loss Percentage = (Loss / Total CP) * 100% = (101.01 / 10101.01) * 100% ≈ 1%. Special Case: When two articles are sold at the same selling price, and there is a gain of x% on one and a loss of x% on the other, there is always an overall loss. The loss percentage is (x/10)2%. In this case, (10/10)2% = 12% = 1% loss.

Memorization Trick: Remember that profit/loss percentages are always calculated on the Cost Price (CP), not the Selling Price (SP), unless stated otherwise.

4. Simple Interest (SI)

Simple interest is the interest calculated on the principal amount only. It does not include the interest earned in previous periods. It's a fixed percentage of the principal amount for a given time period.

Key Terms:

  • Principal (P): The initial amount of money borrowed or invested.
  • Rate of Interest (R): The percentage at which interest is charged or earned per annum.
  • Time (T): The duration for which the money is borrowed or invested, usually in years.
  • Simple Interest (SI): The amount of interest earned or paid.
  • Amount (A): The total money after adding the interest to the principal (A = P + SI).

Formula for Simple Interest:

SI = (P * R * T) / 100

Formula for Amount:

A = P + SI A = P + (P * R * T) / 100 A = P * (1 + (R * T) / 100)

Example Problems:

  1. Calculate the simple interest on ₹5000 at a rate of 8% per annum for 3 years.

    Solution: P = ₹5000, R = 8%, T = 3 years. SI = (5000 * 8 * 3) / 100 SI = 50 * 8 * 3 SI = 400 * 3 = ₹1200. Amount (A) = P + SI = 5000 + 1200 = ₹6200.

  2. In what time will ₹10000 amount to ₹12000 at 5% simple interest per annum?

    Solution: P = ₹10000, A = ₹12000, R = 5%. SI = A - P = 12000 - 10000 = ₹2000. Using SI = (P * R * T) / 100: 2000 = (10000 * 5 * T) / 100 2000 = 100 * 5 * T 2000 = 500 * T T = 2000 / 500 = 4 years.

Key Point: For simple interest, the interest earned each year is the same. If you need to find interest for a period less than a year, convert it to a fraction of a year (e.g., 6 months = 6/12 = 0.5 years).

5. Compound Interest (CI)

Compound interest is the interest calculated on the principal amount and also on the accumulated interest of previous periods. It is also known as "interest on interest". This means the principal amount grows faster over time compared to simple interest.

Key Terms:

  • Principal (P): The initial amount of money.
  • Rate of Interest (R): The annual rate of interest.
  • Time (T): The duration for which the money is invested or borrowed, usually in years.
  • Amount (A): The total money after compounding the interest.
  • Compound Interest (CI): The difference between the Amount and the Principal (CI = A - P).

Formula for Amount (Compounded Annually):

A = P * (1 + R/100)T

Formula for Compound Interest:

CI = A - P CI = P * (1 + R/100)T - P CI = P * [(1 + R/100)T - 1]

Variations in Compounding Frequency:

If the interest is compounded semi-annually (half-yearly), quarterly, or monthly, the rate and time period need to be adjusted.

  • Compounded Semi-annually: Rate becomes R/2, Time becomes 2T. A = P * (1 + (R/2)/100)2T
  • Compounded Quarterly: Rate becomes R/4, Time becomes 4T. A = P * (1 + (R/4)/100)4T
  • Compounded Monthly: Rate becomes R/12, Time becomes 12T. A = P * (1 + (R/12)/100)12T

Difference between CI and SI for 2 years:

CI - SI = P * (R/100)2

Difference between CI and SI for 3 years:

CI - SI = P * (R/100)2 * (3 + R/100)

Example Problems:

  1. Calculate the compound interest on ₹10000 for 2 years at 5% per annum.

    Solution: P = ₹10000, R = 5%, T = 2 years. A = P * (1 + R/100)T A = 10000 * (1 + 5/100)2 A = 10000 * (1 + 1/20)2 A = 10000 * (21/20)2 A = 10000 * (441/400) A = 25 * 441 = ₹11025. CI = A - P = 11025 - 10000 = ₹1025.

  2. Find the amount and compound interest on ₹8000 for 1 year at 10% per annum, compounded semi-annually.

    Solution: P = ₹8000, R = 10%, T = 1 year. Compounded semi-annually means Rate = R/2 = 10%/2 = 5%, Time = 2T = 2 * 1 = 2 periods. A = P * (1 + (R/2)/100)2T A = 8000 * (1 + 5/100)2 A = 8000 * (1 + 1/20)2 A = 8000 * (21/20)2 A = 8000 * (441/400) A = 20 * 441 = ₹8820. CI = A - P = 8820 - 8000 = ₹820.

Quick Calculation for CI: For 2 years, CI = P * (R/100)2 + 2 * P * (R/100). It's the interest for the first year (P*R/100) plus the interest on that interest (P*R/100 * R/100) added to the interest for the second year (P*R/100).
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