Ratio, Proportion & Partnership
🟢 Lite — Quick Review
Ratio, Proportion and Partnership form one of the most versatile arithmetic topics in the MAT Quantitative section. The underlying idea is straightforward — a ratio $a:b$ expresses how many times the first quantity contains the second — but the topic ramifies into mixture problems, partnership profit-sharing, direct and inverse variation, and the alligation method for finding mean prices. Typically 3–5 MAT questions draw from this cluster, often disguised in word problems about dividing inheritance, mixing alloys, adjusting concentrations, or sharing business profits.
The core ratio identity: if a quantity is divided in the ratio $a:b$, the two parts are $\frac{a}{a+b} \times \text{total}$ and $\frac{b}{a+b} \times \text{total}$. The common factor $k$ (where parts are $ak$ and $bk$) is found from total or from any known part.
For proportions, the fundamental property is that in $a:b = c:d$, the product of extremes equals the product of means: $a \times d = b \times c$. Three numbers $a, b, c$ are in continued proportion when $a:b = b:c$, which means $b^2 = ac$.
Partnership profit-sharing follows the rule: profit is divided in the ratio of $\text{capital} \times \text{time}$ for each partner. Simple partnership (same time period) uses just capital ratios; compound partnership accounts for different investment durations.
⚡ MAT exam tip: When a ratio changes after a transfer (e.g., B gives some money to A), set up equations using the unchanged total. The ratio $2:3$ becoming $5:4$ after B gives ₹100 to A means: $2x + 100 = 5y$ and $3x - 100 = 4y$ for some common factor — solve simultaneously. For mixture/alligation problems, draw the alligation chart visually: it eliminates algebraic confusion.
🟡 Standard — Regular Study
Ratio — Definitions and Properties
A ratio $a:b$ compares two quantities of the same kind. Both $a$ and $b$ must be non-zero. The ratio is said to be in its simplest form when $\text{HCF}(a,b) = 1$.
Key properties:
- $\frac{a}{b} = \frac{ka}{kb}$ for any non-zero $k$ (fundamental property of ratios)
- If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a}{c} = \frac{b}{d}$ (invertendo)
- $\frac{a+b}{b} = \frac{c+d}{d}$ (addendo)
- $\frac{a-b}{b} = \frac{c-d}{d}$ (subtrahendo)
Dividing a quantity in ratio $a:b$:
Example: Divide ₹7,800 among three persons X, Y, Z in the ratio 3:5:8. Total parts = $3 + 5 + 8 = 16$ Value per part = $\frac{7800}{16} = 487.50$ X gets $3 \times 487.50 = ₹1{,}462.50$ Y gets $5 \times 487.50 = ₹2{,}437.50$ Z gets $8 \times 487.50 = ₹3{,}900.00$
Types of ratios:
| Type | Definition | Example |
|---|---|---|
| Duplicate | $a^2 : b^2$ | $3:5 \rightarrow 9:25$ |
| Sub-duplicate | $\sqrt{a}:\sqrt{b}$ | $9:16 \rightarrow 3:4$ |
| Triplicate | $a^3 : b^3$ | $2:3 \rightarrow 8:27$ |
| Compounded | $(a:c) : (b:d)$ from $(a:b)$ and $(c:d)$ | $(2:3)$ and $(4:5) \rightarrow 8:15$ |
Ratio in practical mixtures:
Example: A solution contains milk and water in the ratio 5:3. If there are 25 litres of milk, how much total solution? $5$ parts $= 25 \implies 1$ part $= 5$ litres Water $= 3 \times 5 = 15$ litres Total $= 40$ litres
Proportion
When two ratios are equal, four numbers are in proportion: $a:b = c:d$, written $a:b :: c:d$.
Key property: In $a:b :: c:d$, the product of extremes ($a \times d$) equals the product of means ($b \times c$).
Mean proportional: If $a, b, c$ are in continued proportion ($a:b = b:c$), then $b^2 = ac$ and $b$ is the mean proportional between $a$ and $c$.
Example: Find the mean proportional between 9 and 64. $$b^2 = 9 \times 64 = 576 \implies b = \sqrt{576} = 24$$
Direct proportion: $y \propto x \implies y = kx$ (as one increases, the other increases at constant rate)
Inverse proportion: $y \propto \frac{1}{x} \implies y = \frac{k}{x}$ (as one increases, the other decreases)
Joint variation: $z \propto xy \implies z = kxy$
Example of joint variation: The time taken to complete a job varies directly with the work load and inversely with the number of workers. If 6 workers complete 240 units of work in 8 hours, how long for 8 workers to complete 320 units? $$T \propto \frac{W}{n} \implies T = k \cdot \frac{W}{n}$$ $$k = \frac{8 \times 6}{240} = \frac{1}{5}$$ $$T = \frac{1}{5} \times \frac{320}{8} = \frac{1}{5} \times 40 = 8 \text{ hours}$$
Partnership
Simple Partnership: All partners invest for the same duration. Profit/loss is divided in the ratio of investments.
Example: A invests ₹60,000 and B invests ₹40,000 in a business. They earn a profit of ₹25,000. Divide. $$\text{Investment ratio} = 60000 : 40000 = 3 : 2$$ $$\text{Total parts} = 5$$ $$A = \frac{3}{5} \times 25000 = ₹15{,}000, \quad B = \frac{2}{5} \times 25000 = ₹10{,}000$$
Compound Partnership: Partners invest for different time periods. Profit is divided in the ratio of capital × time for each partner.
Example: A invests ₹50,000 for 9 months, B invests ₹30,000 for the full year. Profit is ₹27,000. $$A = 50000 \times 9 = 4{,}50{,}000$$ $$B = 30000 \times 12 = 3{,}60{,}000$$ $$\text{Ratio} = 450000 : 360000 = 5 : 4$$ $$A = \frac{5}{9} \times 27000 = ₹15{,}000, \quad B = \frac{4}{9} \times 27000 = ₹12{,}000$$
Alligation — Mixing Ingredients
The alligation method finds the ratio in which two ingredients at different prices must be mixed to produce a mixture at a given mean price.
Alligation formula: $$\text{Ratio} = \frac{\text{Mean price} - \text{Cheaper price}}{\text{Dearer price} - \text{Mean price}}$$
Example: Mix sugar at ₹38/kg and ₹52/kg to get a mixture worth ₹44/kg. $$\text{Ratio} = \frac{52 - 44}{44 - 38} = \frac{8}{6} = \frac{4}{3} = 4:3$$ For every 4 kg of the ₹38 variety, use 3 kg of the ₹52 variety.
Alligation extended to three or more ingredients:
Example: Tea priced at ₹80/kg, ₹120/kg, and ₹160/kg mixed to get ₹110/kg. Start with the mean price of 110:
- Difference from ₹80: $110 - 80 = 30$
- Difference from ₹160: $160 - 110 = 50$
- For ₹120: It’s already at 120, between 80 and 160
The ratio can be found by considering all three prices: $$(120-80) : (160-120) = 40 : 40 = 1 : 1$$ Then check the ₹110 mixture: mixing ₹80 and ₹160 in some ratio… Using all three: $$\text{Ratio} = (120-110) : (110-80) : (160-110) = 10 : 30 : 50 = 1 : 3 : 5$$
Actually for ₹110 (between ₹80 and ₹160): $$\text{Ratio} = (160-110) : (110-80) = 50 : 30 = 5 : 3$$ So mix ₹160 and ₹80 in ratio 5:3. The ₹120/kg tea is not needed to achieve the mean price — it sits on the other side of the mean.
Common mistakes: Using the arithmetic mean $(\frac{38+52}{2} = 45)$ as the mean price instead of the stated mean price in the problem. Confusing ratio of ingredients with ratio of prices. Forgetting that parts must be integers in simplest form.
🔴 Extended — Deep Study
Complex Ratio Division — Multiple Changes
Example: ₹13,600 is divided among A, B, C such that after B gives ₹200 to A and ₹300 to C, the ratio becomes 1:1:2. Find original shares.
Let original shares be $a, b, c$. After transfers: $A = a + 200$, $B = b - 500$, $C = c + 300$. Given: $(a+200) : (b-500) : (c+300) = 1 : 1 : 2$. Total unchanged: $a + b + c = 13600$. Let common factor = $k$: $a + 200 = k$, $b - 500 = k$, $c + 300 = 2k$ So $a = k - 200$, $b = k + 500$, $c = 2k - 300$. $$(k-200) + (k+500) + (2k-300) = 13600 \implies 4k = 13600 \implies k = 3400$$ $$a = 3200, \quad b = 3900, \quad c = 6500$$
Partnership with Changing Capital
When partners add or withdraw capital mid-business, calculate capital-time for each distinct period.
Example: A and B start a business with ₹20,000 and ₹30,000 respectively. After 5 months, A adds ₹10,000 and B withdraws ₹5,000. Total profit after 11 months is ₹78,000. Find each share.
A’s capital-time:
- ₹20,000 for 5 months = $20{,}000 \times 5 = 1{,}00{,}000$
- ₹30,000 for remaining 6 months = $30{,}000 \times 6 = 1{,}80{,}000$
- A’s total = $2{,}80{,}000$
B’s capital-time:
- ₹30,000 for 5 months = $1{,}50{,}000$
- ₹25,000 for remaining 6 months = $1{,}50{,}000$
- B’s total = $3{,}00{,}000$
$$\text{Ratio} = 280000 : 300000 = 14 : 15$$ $$A = \frac{14}{29} \times 78000 = ₹37{,}655.17$$ $$B = \frac{15}{29} \times 78000 = ₹40{,}344.83$$
Working Partner vs Sleeping Partner
A working partner may receive a commission from profits before the remainder is divided in the capital ratio.
Example: A (working) and B (sleeping) invest ₹60,000 and ₹90,000 respectively. A is entitled to 15% of profit as management commission before dividing the rest. Total profit = ₹50,000.
- A’s commission: $0.15 \times 50000 = ₹7{,}500$
- Remaining profit: $50000 - 7500 = ₹42{,}500$
- Divide in capital ratio $60{,}000 : 90{,}000 = 2 : 3$: $$A = \frac{2}{5} \times 42500 = ₹17{,}000$$ $$B = \frac{3}{5} \times 42500 = ₹25{,}500$$
- A’s total: $7500 + 17000 = ₹24{,}500$; B’s total: ₹25,500
Consecutive Ratio Change Problems
Example: The incomes of two persons P and Q are in the ratio 7:5 and their expenditures are in the ratio 3:2. Each saves ₹4,000. Find their incomes.
Let incomes be $7x$ and $5x$, expenditures be $3y$ and $2y$. $$7x - 3y = 4000 \quad \text{(1)}$$ $$5x - 2y = 4000 \quad \text{(2)}$$
Multiply (1) by 2 and (2) by 3: $$14x - 6y = 8000$$ $$15x - 6y = 12000$$ Subtract first from second: $x = 4000$ From (1): $7(4000) - 3y = 4000 \implies y = \frac{24000}{3} = 8000$ $$P = 7 \times 4000 = ₹28{,}000, \quad Q = 5 \times 4000 = ₹20{,}000$$ P saves: $28000 - 24000 = ₹4{,}000$ ✓; Q saves: $20000 - 16000 = ₹4{,}000$ ✓
Time and Work with Ratios
Example: If A and B together can complete a task in 18 days, and A alone can do it in 30 days, how long does B alone take? $$\frac{1}{T} = \frac{1}{30} + \frac{1}{T_B} \implies \frac{1}{18} - \frac{1}{30} = \frac{1}{T_B}$$ $$\frac{5-3}{90} = \frac{2}{90} = \frac{1}{45} \implies T_B = 45 \text{ days}$$
Efficiency ratio: If A is twice as efficient as B, A takes half the time B takes for the same job. Efficiency ratio $3:5$ means time ratio $5:3$.
Applications to Business and Commerce
Example: Three partners A, B, C invest ₹25,000 (for 8 months), ₹35,000 (for 6 months), and ₹40,000 (for the full year) respectively. Profit = ₹22,800. Find each share. $$A = 25000 \times 8 = 2{,}00{,}000$$ $$B = 35000 \times 6 = 2{,}10{,}000$$ $$C = 40000 \times 12 = 4{,}80{,}000$$ $$\text{Total} = 8{,}90{,}000$$ $$A = \frac{200}{890} \times 22800 = ₹5{,}123.60$$ $$B = \frac{210}{890} \times 22800 = ₹5{,}382.02$$ $$C = \frac{480}{890} \times 22800 = ₹12{,}294.38$$
Practice Problems with Full Solutions
Q1: Divide ₹12,600 into three parts in the ratio 2:3:4 such that each yields the same interest at 8% p.a. in 1 year. Each part gets interest of $P \times \frac{8}{100}$. For interest to be equal, principals must be equal. But the ratio 2:3:4 means unequal principals… This is a trick question. The parts are $2x, 3x, 4x$ and each interest is $0.08 \times 2x, 0.08 \times 3x, 0.08 \times 4x$. Since $x$ is common, interest amounts are in the same ratio 2:3:4 — they cannot be equal unless $x = 0$. The question may be misstated. The intended interpretation is probably that each gets the same amount of interest at the same rate, which requires equal principals, contradicting the ratio. Resolution: the problem as stated has no solution with non-zero $x$.
Q2: A 60-litre mixture contains milk and water in ratio 7:3. How much water must be added to make the ratio 3:2? Milk = $60 \times \frac{7}{10} = 42$ litres Water = $60 \times \frac{3}{10} = 18$ litres Add $x$ litres water: $\frac{42}{18+x} = \frac{3}{2}$ $$84 = 3(18+x) \implies 84 = 54 + 3x \implies x = 10 \text{ litres}$$
Q3: The incomes of P and Q are in ratio 4:3 and expenditures in ratio 12:7. If each saves ₹6,000, find their incomes. Let incomes be $4x, 3x$ and expenditures be $12y, 7y$. $$4x - 12y = 6000 \quad \text{(1)}$$ $$3x - 7y = 6000 \quad \text{(2)}$$ Multiply (1) by 3 and (2) by 4: $$12x - 36y = 18000$$ $$12x - 28y = 24000$$ Subtract: $-8y = -6000 \implies y = 750$ From (1): $4x - 12(750) = 6000 \implies 4x = 6000 + 9000 = 15000 \implies x = 3750$ $$P = 4 \times 3750 = ₹15{,}000, \quad Q = 3 \times 3750 = ₹11{,}250$$ Check: P saves $15000 - 9000 = 6000$ ✓; Q saves $11250 - 5250 = 6000$ ✓
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