⚖️ Ratios & Proportion

Day 1 | ⏱️ Part of 4-Hour Quant Block | TCS Interview Prep

📌 Subtopics

• Ratio basics and simplification
• Proportion (Direct & Inverse)
• Componendo and Dividendo
• Continued proportion
• Distributing amounts in a given ratio
• Partnership problems

📖 Comprehensive Notes

1. Ratio Basics and Simplification

A ratio is a comparison of two quantities of the same unit by division. The ratio of $a$ to $b$ is written as $a : b$ or $\frac{a}{b}$, where $a$ is the antecedent and $b$ is the consequent.

• Simplification: A ratio is simplified by dividing both terms by their HCF.
• Compounded Ratio: The compounded ratio of $(a : b)$ and $(c : d)$ is $ac : bd$.
• Duplicate Ratio: The duplicate ratio of $(a : b)$ is $a^2 : b^2$. Sub-duplicate ratio is $\sqrt{a} : \sqrt{b}$.
• Triplicate Ratio: The triplicate ratio of $(a : b)$ is $a^3 : b^3$. Sub-triplicate ratio is $\sqrt[3]{a} : \sqrt[3]{b}$.

Example: Find the compounded ratio of $2:3$, $6:5$, and $10:9$. $\text{Compounded Ratio} = \frac{2 \times 6 \times 10}{3 \times 5 \times 9} = \frac{120}{135} = \frac{8}{9}$ or $8:9$.

2. Proportion (Direct & Inverse)

An equality of two ratios is called a proportion. If $a : b = c : d$, then $a, b, c, d$ are in proportion, written as $a : b :: c : d$.

• $a$ and $d$ are called extremes, and $b$ and $c$ are called means.
• Product of extremes = Product of means: $a \times d = b \times c$
• Direct Proportion: If $x \propto y$, then $\frac{x_1}{y_1} = \frac{x_2}{y_2}$. (As one increases, the other increases).
• Inverse Proportion: If $x \propto \frac{1}{y}$, then $x_1 y_1 = x_2 y_2$. (As one increases, the other decreases).

Example: If $15$ men can build a wall in $12$ days, how many days will $20$ men take to build the same wall? This is inverse proportion: $\text{Men} \times \text{Days} = \text{Constant}$. $15 \times 12 = 20 \times D_2 \implies 180 = 20 \times D_2 \implies D_2 = 9$ days.

3. Componendo and Dividendo

If $\frac{a}{b} = \frac{c}{d}$, then:

• Componendo: $\frac{a+b}{b} = \frac{c+d}{d}$
• Dividendo: $\frac{a-b}{b} = \frac{c-d}{d}$
• Componendo & Dividendo (C&D): $\frac{a+b}{a-b} = \frac{c+d}{c-d}$

Example: If $\frac{x}{y} = \frac{5}{3}$, find $\frac{x+y}{x-y}$. Using C&D: $\frac{x+y}{x-y} = \frac{5+3}{5-3} = \frac{8}{2} = 4$.

4. Continued Proportion

Three numbers $a, b, c$ are in continued proportion if $a : b :: b : c$, which means $\frac{a}{b} = \frac{b}{c}$.

• Here, $b$ is called the mean proportional between $a$ and $c$. $b^2 = ac \implies b = \sqrt{ac}$
• $c$ is called the third proportional to $a$ and $b$. $c = \frac{b^2}{a}$
• If $a, b, c, d$ are in continued proportion, then $\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k$.

Example: Find the mean proportional between $4$ and $9$. Mean proportional $b = \sqrt{4 \times 9} = \sqrt{36} = 6$.

5. Distributing Amounts in a Ratio

To divide a quantity $T$ into parts in the ratio $a : b : c$:

• Sum of ratio terms: $S = a + b + c$
• First Part: $\frac{a}{S} \times T$, Second Part: $\frac{b}{S} \times T$, Third Part: $\frac{c}{S} \times T$.

Example: Divide $₹1200$ among A, B, and C in the ratio $2:3:5$. Sum of ratios: $2 + 3 + 5 = 10$. A’s share $= \frac{2}{10} \times 1200 = ₹240$. B’s share $= \frac{3}{10} \times 1200 = ₹360$. C’s share $= \frac{5}{10} \times 1200 = ₹600$.

6. Partnership Problems

When two or more people invest money in a business, the ratio of their profits depends on both the investment amount and the time duration of investment. $\text{Profit Ratio} = (I_1 \times T_1) : (I_2 \times T_2) : (I_3 \times T_3)$

• Simple Partnership: Capital is invested for the same duration. Profits are shared in the ratio of capital investments.
• Compound Partnership: Capitals are invested for different durations. Profits are shared in the ratio of $\text{Capital} \times \text{Time}$.

Example: A invests $₹10,000$ for $12$ months and B invests $₹15,000$ for $8$ months. Find their profit-sharing ratio. Ratio $= (10000 \times 12) : (15000 \times 8) = 120000 : 120000 = 1:1$.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

a : b = a/b
a : b :: c : d  →  a × d = b × c (cross multiply)
Componendo: (a+b)/b = (c+d)/d
Dividendo:  (a-b)/b = (c-d)/d

🔗 Related Topics

• Allegations and Mixtures — Alligation is a ratio-based technique
• Percentages — Fraction-to-percentage conversions
• Profit and Loss — Partnership and ratio-based profit sharing
• Time and Work — Work efficiency ratios
• Ages — Age ratio problems
• Number System HCF LCM — Simplifying ratios using HCF