Ratio and Proportion, Partnership, Simple and Compound Interest

Ratio and Proportion

Ratio is a comparison of two or more quantities by division. It tells us how many times one quantity contains another. For example, if there are 5 apples and 7 oranges, the ratio of apples to oranges is 5:7. Proportion is an equality of two ratios. If a:b = c:d, then a, b, c, and d are in proportion.

Types of Ratios

There are several types of ratios that are commonly used in quantitative aptitude problems:

  • Compound Ratio: The ratio of the products of the numerators to the products of the denominators of two or more ratios. For example, the compound ratio of a:b and c:d is ac:bd.
  • 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.
  • Continued Proportion: Three quantities a, b, and c are said to be in continued proportion if a:b = b:c. This implies b2 = ac, where b is the mean proportional.

Properties of Ratios and Proportions

Understanding these properties can simplify calculations:

  • Invertendo: If a:b = c:d, then b:a = d:c. (The ratios are inverted).
  • Alternendo: If a:b = c:d, then a:c = b:d. (The means and extremes are interchanged).
  • Componendo: If a:b = c:d, then (a+b):b = (c+d):d. (Unity is added to each antecedent).
  • Dividendo: If a:b = c:d, then (a-b):b = (c-d):d. (Unity is subtracted from each antecedent).
  • Componendo and Dividendo: If a:b = c:d, then (a+b):(a-b) = (c+d):(c-d). (This is derived by combining Componendo and Dividendo).
  • Addendo: If a:b = c:d = ... = k, then each of these ratios is equal to (a+c+...):(b+d+...).
  • Subtrahendo: If a:b = c:d = ... = k, then each of these ratios is equal to (a-c-...):(b-d-...).

Applications of Ratio and Proportion

Ratio and proportion are fundamental concepts used in various problems:

  • Sharing Quantities: Dividing a sum of money or any quantity among people according to a given ratio.
  • Ages: Problems involving the ratio of ages of people now, in the past, or in the future.
  • Mixtures and Alligations: Determining the ratio in which different ingredients should be mixed or how a mixture should be altered.
  • Partnership: Calculating profit sharing based on investment and duration.
  • Speed, Distance, and Time: When speeds are in a certain ratio, the time taken to cover the same distance will be in the inverse ratio.

Example Problem (Ratio and Proportion)

Two numbers are in the ratio 3:5. If 5 is added to each number, the new ratio becomes 2:3. Find the numbers.

Solution:

Let the numbers be 3x and 5x. According to the problem, (3x + 5) / (5x + 5) = 2 / 3. Cross-multiplying, we get: 3(3x + 5) = 2(5x + 5) 9x + 15 = 10x + 10 15 - 10 = 10x - 9x 5 = x So, the numbers are 3x = 3 * 5 = 15 and 5x = 5 * 5 = 25. The numbers are 15 and 25.

Shortcut: When dealing with ratios where a constant is added or subtracted, if the difference between the original ratio terms and the new ratio terms, after cross-multiplication, is constant, you can use that difference to find the value of 'x' more quickly.

Partnership

A partnership is a business arrangement where two or more individuals agree to share in the profits or losses of a business. The individuals who form a partnership are called partners. Partnership problems typically involve the distribution of profits based on the capital invested and the duration of the investment.

Types of Partnerships

Partnerships can be broadly classified into two types:

  • Simple Partnership: This occurs when all partners invest their capital for the same period. In this case, the profits are shared in the ratio of their capitals.
  • Compound Partnership: This occurs when partners invest their capital for different periods. In this case, the profit is shared in the ratio of the product of their capitals and the time periods for which they invested.

Key Concepts and Formulas

The fundamental principle in partnership problems is that the profit share is directly proportional to the product of capital invested and the time period.

Let's say two partners, A and B, invest capitals CA and CB for time periods TA and TB respectively. The ratio of their profits (PA : PB) is given by: PA : PB = (CA × TA) : (CB × TB)

If there are multiple partners, say A, B, and C, with capitals CA, CB, CC and time periods TA, TB, TC, then the ratio of their profits is: PA : PB : PC = (CA × TA) : (CB × TB) : (CC × TC)

Working Partner: A partner who actively participates in the business management is called a working partner. Often, a working partner receives a fixed salary or a commission in addition to their share of profit. This share is usually calculated after deducting the working partner's salary from the total profit.

Sleeping Partner: A partner who only contributes capital but does not participate in the business management is called a sleeping partner.

Steps to Solve Partnership Problems

  1. Identify the partners involved.
  2. Note down the capital invested by each partner.
  3. Note down the duration for which each partner invested their capital. Ensure the time units are consistent (e.g., all in months or all in years).
  4. Calculate the "effective investment" for each partner by multiplying their capital with the duration of their investment. This gives you the ratio of their shares.
  5. If there is a working partner who receives a salary or commission, calculate their share first.
  6. The remaining profit is distributed among all partners (including the working partner) in the ratio of their effective investments.
  7. Calculate the total profit share for each partner.

Example Problem (Partnership)

A, B, and C started a business by investing Rs. 50,000, Rs. 75,000, and Rs. 1,00,000 respectively. After one year, they made a profit of Rs. 1,50,000. If A is a working partner and receives 20% of the profit for his services, find the share of each partner.

Solution:

Step 1: Calculate A's share as a working partner. A's service charge = 20% of Rs. 1,50,000 = 0.20 × 1,50,000 = Rs. 30,000.

Step 2: Calculate the remaining profit to be distributed. Remaining profit = Total profit - A's service charge Remaining profit = 1,50,000 - 30,000 = Rs. 1,20,000.

Step 3: Determine the ratio of investments. Investments are A: Rs. 50,000, B: Rs. 75,000, C: Rs. 1,00,000. The ratio of their investments is 50,000 : 75,000 : 1,00,000. Simplifying this ratio by dividing by 25,000: Ratio = 2 : 3 : 4.

Step 4: Distribute the remaining profit according to the investment ratio. The sum of the ratio parts = 2 + 3 + 4 = 9. A's share from remaining profit = (2/9) × 1,20,000 = Rs. 26,666.67 (approx.) B's share from remaining profit = (3/9) × 1,20,000 = Rs. 40,000. C's share from remaining profit = (4/9) × 1,20,000 = Rs. 53,333.33 (approx.)

Step 5: Calculate the total share of each partner. A's total share = A's service charge + A's share from remaining profit A's total share = 30,000 + 26,666.67 = Rs. 56,666.67. B's total share = Rs. 40,000. C's total share = Rs. 53,333.33.

Check: 56,666.67 + 40,000 + 53,333.33 = 1,50,000.

Memory Trick: Think of partnership profit sharing as "Capital × Time = Profit Share Ratio". For working partners, always settle their salary/commission first from the total profit.

Simple and Compound Interest

Interest is the additional amount paid by a borrower to a lender for the use of money. It is usually expressed as a percentage of the principal amount.

Simple Interest (SI)

Simple Interest is calculated only on the initial principal amount. The interest earned each year is the same.

Formula: Simple Interest (SI) = (Principal × Rate × Time) / 100 SI = (P × R × T) / 100

Where:

  • P = Principal (the initial sum of money borrowed or invested)
  • R = Rate of Interest (per annum, usually)
  • T = Time (in years, usually)

Amount (A): The total amount to be repaid or received at the end of the term is the sum of the principal and the simple interest. Amount (A) = Principal (P) + Simple Interest (SI) A = P + (P × R × T) / 100

Compound Interest (CI)

Compound Interest is calculated on the principal amount as well as on the accumulated interest of previous periods. It is interest on interest. This means the interest earned in each period is added to the principal for the next period's calculation.

Formula for Amount (A) after n years: Amount (A) = P (1 + R/100)n

Where:

  • P = Principal
  • R = Rate of Interest (per annum)
  • n = Number of years

Compound Interest (CI): Compound Interest (CI) = Amount (A) - Principal (P) CI = P (1 + R/100)n - P CI = P [ (1 + R/100)n - 1 ]

Interest Compounded Annually, Half-Yearly, Quarterly

The formulas need adjustment if the interest is compounded more than once a year.

  • Compounded Half-Yearly: The rate of interest becomes R/2, and the number of periods becomes 2n. Amount (A) = P (1 + (R/2)/100)2n CI = P [ (1 + R/200)2n - 1 ]
  • Compounded Quarterly: The rate of interest becomes R/4, and the number of periods becomes 4n. Amount (A) = P (1 + (R/4)/100)4n CI = P [ (1 + R/400)4n - 1 ]
  • Compounded at Different Rates for Different Periods: If the rate is R1% for the first year, R2% for the second year, and so on, then: Amount (A) = P (1 + R1/100) (1 + R2/100) ...

Difference between CI and SI

The difference between compound interest and simple interest for a given period is a common topic.

  • For 2 years: Difference (CI - SI) = P (R/100)2
  • For 3 years: Difference (CI - SI) = P (R/100)2 (3 + R/100)

These formulas are very useful for quickly finding the principal or rate when the difference is given.

Example Problem (Simple Interest)

Find the Simple Interest on Rs. 5,000 at 8% per annum for 3 years.

Solution:

P = 5,000 R = 8% T = 3 years SI = (P × R × T) / 100 SI = (5000 × 8 × 3) / 100 SI = 50 × 8 × 3 SI = 400 × 3 SI = Rs. 1,200.

Amount = P + SI = 5000 + 1200 = Rs. 6,200.

Example Problem (Compound Interest)

Calculate the Compound Interest on Rs. 10,000 for 2 years at 5% per annum, compounded annually.

Solution:

P = 10,000 R = 5% n = 2 years Amount (A) = P (1 + R/100)n A = 10000 (1 + 5/100)2 A = 10000 (1 + 1/20)2 A = 10000 (21/20)2 A = 10000 × (441 / 400) A = 25 × 441 A = Rs. 11,025.

Compound Interest (CI) = A - P CI = 11025 - 10000 CI = Rs. 1,025.

Example Problem (Difference between CI and SI)

The difference between the compound interest and simple interest on a certain sum for 2 years at 10% per annum is Rs. 40. Find the sum.

Solution:

Difference (CI - SI) = Rs. 40 R = 10% n = 2 years We know that for 2 years, Difference = P (R/100)2 40 = P (10/100)2 40 = P (1/10)2 40 = P (1/100) P = 40 × 100 P = Rs. 4,000.

The sum is Rs. 4,000.

Interest Shortcut: For SI, think of it as adding R% of P every year. For CI, the interest of the first year becomes part of the principal for the second year. For 2-year CI-SI difference, it's just the interest on the first year's SI. For 3 years, it's the interest on the 2-year CI-SI difference plus the interest on the first year's SI.
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