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Ratio proportion partnership

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Introduction to Ratio, Proportion, and Partnership

In everyday life and competitive exams, we often need to compare quantities, understand relationships between them, and share profits or workloads fairly. The concepts of ratio, proportion, and partnership form the foundation for solving such problems efficiently.

Ratio helps us compare two quantities directly, like the number of boys to girls in a class. Proportion tells us when two ratios are equal, allowing us to find unknown values. Partnership problems involve sharing profits or work based on contributions, which can depend on money invested, time, or both.

Mastering these topics is essential for success in aptitude tests and practical decision-making, such as dividing expenses, planning work schedules, or investing money.

Ratio

A ratio is a way to compare two quantities by showing how many times one quantity contains the other. It is written as A : B or as a fraction \(\frac{A}{B}\), where A and B are the two quantities.

For example, if there are 8 apples and 12 oranges, the ratio of apples to oranges is 8:12. This ratio can be simplified by dividing both numbers by their highest common factor (HCF), which is 4:

\[\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\]

So, the simplified ratio is 2:3.

8 units 12 units Ratio 8:12 simplified to 2:3

Key properties of ratios:

  • Ratios compare quantities of the same kind (e.g., length to length, money to money).
  • Ratios can be simplified like fractions.
  • Ratios can be scaled up or down by multiplying or dividing both terms by the same number.

Comparing Ratios

To compare two ratios, convert them to fractions and find a common denominator or cross multiply to see which is larger.

For example, compare 3:5 and 4:7:

Cross multiply:

\[3 \times 7 = 21, \quad 4 \times 5 = 20\]

Since 21 > 20, the ratio 3:5 is greater than 4:7.

Proportion

A proportion states that two ratios are equal. If \(\frac{A}{B} = \frac{C}{D}\), then A, B, C, and D are in proportion.

This equality allows us to find an unknown value when three values are known.

graph TD    A[Identify given ratios] --> B[Set up proportion \frac{A}{B} = \frac{C}{D}]    B --> C[Cross multiply: A x D = B x C]    C --> D[Solve for unknown variable]

Types of Proportion

  • Direct Proportion: When one quantity increases, the other increases at the same rate. Mathematically, \(A \propto B\) or \(\frac{A}{B} = k\), where \(k\) is a constant.
  • Inverse Proportion: When one quantity increases, the other decreases so that their product is constant. Mathematically, \(A \propto \frac{1}{B}\) or \(A \times B = k\).

Partnership

In partnership problems, two or more people invest money and/or time to earn a profit, which is then shared according to their contributions.

The share of profit for each partner depends on the effective capital, which is the product of the amount invested and the time for which it was invested.

Example of Profit Distribution Among Partners
Partner Investment (INR) Time (months) Effective Capital (Investment x Time) Profit Share (Ratio)
A 50,000 6 3,00,000 3,00,000 : 3,00,000 : 3,00,000 = 1 : 1 : 1
B 75,000 4 3,00,000
C 1,00,000 3 3,00,000

Here, all partners have equal effective capital, so the profit is shared equally.

Worked Examples

Example 1: Simplify the Ratio of 150 cm to 2.5 m Easy
Simplify the ratio of 150 cm to 2.5 m.

Step 1: Convert both quantities to the same unit. Since 1 m = 100 cm, 2.5 m = 2.5 x 100 = 250 cm.

Step 2: Write the ratio as 150 cm : 250 cm.

Step 3: Simplify by dividing both terms by their HCF, which is 50.

\[ \frac{150}{250} = \frac{150 \div 50}{250 \div 50} = \frac{3}{5} \]

Answer: The simplified ratio is 3:5.

Example 2: Find the Unknown in Proportion: 3:4 = x:16 Easy
Find the value of \(x\) if \(\frac{3}{4} = \frac{x}{16}\).

Step 1: Use cross multiplication:

\[ 3 \times 16 = 4 \times x \]

Step 2: Calculate the left side:

\[ 48 = 4x \]

Step 3: Solve for \(x\):

\[ x = \frac{48}{4} = 12 \]

Answer: \(x = 12\)

Example 3: Profit Sharing in Partnership Medium
Three partners invest INR 50,000, INR 75,000, and INR 1,00,000 for 6, 4, and 3 months respectively. If the total profit is INR 54,000, find each partner's share.

Step 1: Calculate effective capital for each partner:

  • Partner A: \(50,000 \times 6 = 3,00,000\)
  • Partner B: \(75,000 \times 4 = 3,00,000\)
  • Partner C: \(1,00,000 \times 3 = 3,00,000\)

Step 2: Total effective capital = \(3,00,000 + 3,00,000 + 3,00,000 = 9,00,000\)

Step 3: Each partner's share ratio = 3,00,000 : 3,00,000 : 3,00,000 = 1 : 1 : 1

Step 4: Divide total profit equally:

\[ \text{Each share} = \frac{54,000}{3} = 18,000 \]

Answer: Each partner receives INR 18,000.

Example 4: Inverse Proportion Application Medium
If 8 workers can complete a job in 15 days, how many days will 12 workers take to complete the same job?

Step 1: Recognize that the number of workers and days are inversely proportional.

Step 2: Use the formula for inverse proportion:

\[ \text{Workers} \times \text{Days} = k \]

Step 3: Calculate constant \(k\):

\[ 8 \times 15 = 120 \]

Step 4: Let the unknown number of days be \(x\) for 12 workers:

\[ 12 \times x = 120 \implies x = \frac{120}{12} = 10 \]

Answer: 12 workers will complete the job in 10 days.

Example 5: Complex Partnership with Changing Investments Hard
Two partners invest INR 60,000 and INR 90,000. The first partner invests for 8 months, the second for 6 months. If the total profit is INR 72,000, find their profit shares.

Step 1: Calculate effective capital for each partner:

  • Partner 1: \(60,000 \times 8 = 4,80,000\)
  • Partner 2: \(90,000 \times 6 = 5,40,000\)

Step 2: Total effective capital = \(4,80,000 + 5,40,000 = 10,20,000\)

Step 3: Profit sharing ratio = \(4,80,000 : 5,40,000\)

Simplify by dividing both by 60,000:

\[ \frac{4,80,000}{60,000} : \frac{5,40,000}{60,000} = 8 : 9 \]

Step 4: Total profit = INR 72,000

Sum of ratio parts = 8 + 9 = 17

Step 5: Partner 1's share:

\[ \frac{8}{17} \times 72,000 = 33,882.35 \approx 33,882 \]

Step 6: Partner 2's share:

\[ \frac{9}{17} \times 72,000 = 38,117.65 \approx 38,118 \]

Answer: Partner 1 gets INR 33,882 and Partner 2 gets INR 38,118.

Proportion Formula

\[\frac{A}{B} = \frac{C}{D} \implies A \times D = B \times C\]

Two ratios are equal if their cross products are equal

A, B, C, D = Quantities in proportion

Profit Sharing in Partnership

\[Share = \frac{Investment \times Time}{\sum (Investment \times Time)} \times Total\ Profit\]

Calculate each partner's share of profit based on investment and time

Investment = Amount invested by partner
Time = Duration of investment
Total Profit = Total profit to be shared

Tips & Tricks

Tip: Always convert all quantities to the same unit before forming ratios.

When to use: When given quantities in different metric units (e.g., cm and m).

Tip: Use cross multiplication to solve proportion problems quickly.

When to use: When two ratios are set equal and one term is unknown.

Tip: In partnership problems, multiply investment by time to get effective capital.

When to use: When partners invest different amounts for different time periods.

Tip: Check if the problem involves direct or inverse proportion before solving.

When to use: When quantities change in relation to each other.

Tip: Simplify ratios to lowest terms to avoid calculation errors.

When to use: Before comparing or using ratios in further calculations.

Common Mistakes to Avoid

❌ Not converting all quantities to the same unit before forming ratios.
✓ Always convert units to the same metric system (e.g., meters to centimeters) before calculating ratios.
Why: Leads to incorrect ratio values and wrong answers.
❌ Incorrectly applying direct proportion formula to inverse proportion problems.
✓ Identify the type of proportion first; use \(A \times B = k\) for inverse proportion.
Why: Misunderstanding the relationship causes wrong calculations.
❌ Ignoring the time factor in partnership problems.
✓ Multiply investment by the time period to find the effective capital for profit sharing.
Why: Profit depends on both amount invested and duration.
❌ Forgetting to simplify ratios before comparing or using them.
✓ Always reduce ratios to their simplest form to avoid confusion and errors.
Why: Simplification makes calculations easier and more accurate.
❌ Mixing up numerator and denominator while setting up proportions.
✓ Carefully write ratios in the correct order as given in the problem.
Why: Incorrect setup leads to wrong cross multiplication and answers.
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