Ratio Proportion Partnership Questions for UP POLICE CONSTABLE

Ratio Basics

A ratio compares two quantities using division. If A:B = 3:5, it means for every 3 units of A, there are 5 units of B. Ratios are always simplified to lowest terms (like fractions). When you see 'ratio', think of it as a comparison tool—not an absolute value. For example, if profit is in ratio 2:3:5 among three partners, you don't know the actual profit amounts until you know the total or one part's value.

• Ratio a:b means a/b; always express in simplest form by dividing by GCD
• Equivalent ratios: 2:3 = 4:6 = 6:9 (multiply or divide both parts by same number)
• Compound ratio: (a:b) × (c:d) = (a×c):(b×d) — used when combining multiple ratios
• If a:b = 3:5 and total is 80, then a = (3/8)×80 = 30, b = (5/8)×80 = 50
• Inverse ratio: if a:b = 3:5, then b:a = 5:3

Proportion

Proportion states that two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion. This is written as a:b::c:d (read as 'a is to b as c is to d'). The key property: a×d = b×c (cross-multiplication). Proportions are used to find missing values when you know three of four quantities.

• If a:b::c:d, then a×d = b×c (product of extremes = product of means)
• Direct proportion: if x increases, y increases proportionally (y = kx, k is constant)
• Inverse proportion: if x increases, y decreases proportionally (y = k/x)
• Continued proportion: if a:b = b:c, then b is the mean proportional; b² = a×c
• Third proportional: if a:b = b:c, then c is third proportional to a and b

Partnership

Partnership is profit/loss sharing among business partners. Profit is divided in the ratio of capital invested and time period. If Partner A invests capital C_A for time T_A, and Partner B invests C_B for time T_B, their profit ratio is (C_A × T_A):(C_B × T_B). This combines ratio and proportion concepts. UP Police exams often test scenarios with unequal capital or unequal time periods.

• Profit ratio = (Capital × Time) ratio; if time is equal, profit ∝ capital only
• If capital is equal, profit ∝ time only
• Working partner may get salary/commission in addition to profit share
• Silent partner gets only profit share (no salary)
• Always identify: capital amounts, time periods, and any special allowances

Mixed Scenarios

UP Police exams frequently combine ratio, proportion, and partnership in one problem. You might need to find a ratio first, then use it to divide profit, or find a missing capital using proportion. The key is to break the problem into steps: identify what's given, what's asked, which concept applies, and solve methodically.

• Read carefully: identify capital, time, profit/loss, and any special conditions
• Convert all units to same basis (months to months, rupees to rupees)
• Use compound ratio when multiple ratios are involved
• Check your answer: does the ratio of profits match the ratio of (capital×time)?
• Watch for tricks: 'started after 3 months' means time is reduced; 'withdrew Rs 2000' means capital changes mid-year

• Forgetting to simplify ratios to lowest terms before using them in calculations — leads to wrong final answer
• Confusing ratio with actual value: a:b = 3:5 does NOT mean a=3 and b=5; you need total or one value to find actual amounts
• Ignoring time period in partnership: treating profit ratio as capital ratio only when time is unequal — major error
• Cross-multiplication errors in proportion: writing a×c = b×d instead of a×d = b×c
• Misreading 'started after 2 months' as 12 months instead of 10 months — affects partnership calculation significantly
• Not converting mixed time periods: forgetting to convert years to months or vice versa before multiplying

• RATIO = Reduce, Arrange, Total, Identify, Operate — steps to solve any ratio problem
• PARTNERSHIP = Capital × Time = Profit share — always multiply both factors
• PROPORTION = Cross-multiply (a×d = b×c) — the 'X' shape reminds you which to multiply
• COMPOUND = Multiply ratios term-by-term — (a:b) × (c:d) = (ac):(bd), like matrix multiplication
• TIME TRAP = Always check if time is equal; if unequal, MUST include in calculation

• UP Police Prelims: expect 1-2 direct ratio/proportion questions (easy, 30-45 seconds each); Mains: 2-3 questions including partnership scenarios (2-3 minutes each)
• Recent papers show preference for partnership with unequal time periods — practice 'A started, B joined after X months' type problems heavily
• Word problems are common: 'A and B started business with Rs X and Rs Y respectively' — always extract capital and time first
• Calculation-heavy: use approximation and simplification (e.g., 5000:3000 = 5:3) to save time; avoid long division
• Mixed topics: one question might ask ratio first, then use it for proportion or partnership — read the full question before solving
• Watch for 'profit in ratio' vs 'capital in ratio' — they're different unless time is equal