3.3 Ratio & Proportion, Partnerships & Ages

Ratio & Proportion, Partnerships & Ages

Ratios provide the core quantitative language for comparing quantities, dividing resources, calculating partnership returns, and solving age relation word problems. In SBI PO examinations, ratio concepts are intertwined with Data Interpretation and data sufficiency questions.

---

1. Ratio & Proportion Core Mechanics

Definitions & Properties

• Ratio ($A : B$): A comparison of two quantities of the same unit expressed as $\frac{A}{B}$.
• Combining Ratios: If $A : B = a : b$ and $B : C = c : d$, combine them by making the common term $B$ equal: $$A : B : C = (a \times c) : (b \times c) : (b \times d)$$
• Types of Ratios:
  • Duplicate Ratio of $a : b = a^2 : b^2$
  • Triplicate Ratio of $a : b = a^3 : b^3$
  • Sub-duplicate Ratio of $a : b = \sqrt{a} : \sqrt{b}$

Proportion Laws & Definitions

Four quantities $a, b, c, d$ are in proportion ($a : b :: c : d$) if $\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$.

---

2. Partnership Mathematics & Active Partners

In business partnerships, profit distribution depends on both capital invested and investment tenure.

Profit Distribution Principles

1. Simple Partnership: All partners invest capital for identical durations. Profits are divided directly by capital ratio: $$\text{Profit Ratio} = C_A : C_B : C_C$$
2. Compound Partnership: Partners invest differing amounts for differing time periods. Profits are divided by the product of capital and time: $$\text{Profit Ratio} = (C_A \times T_A) : (C_B \times T_B) : (C_C \times T_C)$$

Working (Active) vs Sleeping Partners

• Active Partner: Manages business operations and receives a salary or percentage fee from gross total profit before remaining profits are distributed.
• Sleeping Partner: Invests capital only and receives profit solely based on the capital-time ratio.

---

3. Step-by-Step Worked Math Examples

Worked Example 1: Combining Ratios & Sum Division

Problem: Divide ₹7,800 among A, B, and C such that $A : B = 2 : 3$ and $B : C = 4 : 5$. Calculate B's share.

Step-by-Step Solution:

1. Given $A : B = 2 : 3$ and $B : C = 4 : 5$.
2. Equalize the term $B$ by multiplying $A : B$ by 4 and $B : C$ by 3:
   • $A : B = (2 \times 4) : (3 \times 4) = 8 : 12$
   • $B : C = (4 \times 3) : (5 \times 3) = 12 : 15$
3. Combined ratio $A : B : C = 8 : 12 : 15$.
4. Sum of ratio parts $= 8 + 12 + 15 = 35$ parts.
5. Value of 1 part $= \frac{7,800}{35} = \frac{1,560}{7}$.
6. B's share $= 12 \text{ parts} = 12 \times \frac{1,560}{7} = \frac{18,720}{7} \approx \text{INR }2,674.28$. (If total sum is ₹7,000: 35 parts $= 7000 \implies 1 \text{ part} = 200 \implies B = 12 \times 200 = \text{INR }2,400$.)

Worked Example 2: Partnership with Active Management Fee

Problem: A and B invest ₹40,000 and ₹50,000 in a business. A is an active partner and receives 10% of the total profit for managing the business. The remaining profit is divided in proportion to their investments. If the total annual profit is ₹30,000, find A's total income.

Step-by-Step Solution:

1. Total Gross Profit $= \text{INR }30,000$.
2. A's management fee $= 10\% \text{ of } 30,000 = \text{INR }3,000$.
3. Remaining profit to be distributed $= 30,000 - 3,000 = \text{INR }27,000$.
4. Investment ratio $A : B = 40,000 : 50,000 = 4 : 5$.
5. Total ratio parts $= 4 + 5 = 9$ parts.
6. A's share of remaining profit $= \frac{4}{9} \times 27,000 = 4 \times 3,000 = \text{INR }12,000$.
7. A's total income $= \text{Management Fee} + \text{Profit Share} = 3,000 + 12,000 = \text{INR }15,000$.

Worked Example 3: Age Ratio Shift & Constant Difference

Problem: Six years ago, the ratio of the ages of Kunal and Sagar was $6 : 5$. Four years hence, the ratio of their ages will be $11 : 10$. What is Sagar's present age?

Step-by-Step Solution:

1. Let the ages of Kunal and Sagar 6 years ago be $6x$ and $5x$.
2. Their present ages are $(6x + 6)$ and $(5x + 6)$.
3. Four years hence, their ages will be $(6x + 6 + 4) = (6x + 10)$ and $(5x + 6 + 4) = (5x + 10)$.
4. Set up the ratio equation 4 years hence: $$\frac{6x + 10}{5x + 10} = \frac{11}{10}$$
5. Cross-multiply: $10(6x + 10) = 11(5x + 10) \implies 60x + 100 = 55x + 110$.
6. Rearrange terms: $60x - 55x = 110 - 100 \implies 5x = 10 \implies x = 2$.
7. Sagar's present age $= 5x + 6 = 5(2) + 6 = 10 + 6 = 16 \text{ years}$.

---

4. Constant Age Difference Principle & Shortcuts

• Invariance Rule: The age gap between two individuals never changes over time. If A is 5 years older than B today, A will remain exactly 5 years older than B after 20 years.
• Ratio Gap Equalization Shortcut: If ratio 6 years ago is $6:5$ (difference $= 1$) and ratio 4 years hence is $11:10$ (difference $= 1$), the change in ratio units for both persons is $11 - 6 = 5$ units. These 5 units correspond to total time elapsed $= 6 + 4 = 10$ years. Thus, $1 \text{ unit} = 2 \text{ years}$. Sagar's age 6 years ago $= 5 \times 2 = 10$ years, making present age $10 + 6 = 16$ years.