Ratio and Proportion Questions for UP POLICE CONSTABLE

What is Ratio?

A ratio is a comparison of two quantities of the same type, expressed as a fraction or using a colon. For example, if there are 3 apples and 5 oranges, the ratio is 3:5. Ratios are always simplified to their lowest terms by dividing both parts by their GCD (Greatest Common Divisor). Think of ratio as 'how many parts of one thing per part of another.' In UP Police exams, you'll often see questions like 'divide Rs 1000 in ratio 2:3' or 'if A:B = 3:4, find their values.'

• Ratio is a comparison; always simplify to lowest terms
• If A:B = 3:4, then A = 3k and B = 4k for some constant k
• To find k, use the total or any given value
• Ratio has no units (it's dimensionless)
• Order matters: A:B is different from B:A

What is Proportion?

A proportion states that two ratios are equal. If a:b = c:d, we say 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: in a proportion, the product of extremes equals the product of means. That is, a × d = b × c. In UP Police exams, proportion questions often ask you to find a missing value when three values are given (Rule of Three or Direct Proportion).

• Proportion: a:b = c:d means a/b = c/d
• Extremes and Means: In a:b::c:d, a and d are extremes, b and c are means
• Key Property: a × d = b × c (product of extremes = product of means)
• Used to find missing values in equal ratios
• Common in word problems: 'If 5 workers build 10 walls, how many walls will 8 workers build?'

Compound Ratio and Continued Ratio

A compound ratio is formed by multiplying corresponding terms of two or more ratios. For example, if A:B = 2:3 and B:C = 4:5, the compound ratio A:B:C is found by making B equal in both ratios, then combining. Continued ratio (A:B:C) is useful when three or more quantities are compared. In UP Police exams, these appear less frequently but are important for partnership problems and alligation.

• Compound ratio: multiply ratios term by term after equalizing common terms
• Continued ratio A:B:C means A/B = (first ratio), B/C = (second ratio)
• To find A:B:C from A:B = 2:3 and B:C = 4:5: make B = 12 (LCM of 3 and 4), so A:B = 8:12 and B:C = 12:15, thus A:B:C = 8:12:15
• Used in profit-sharing, work distribution, and alligation problems

Inverse Ratio and Inverse Proportion

Inverse ratio is when one quantity increases and the other decreases proportionally. For example, if A:B = 3:4, then inverse ratio is B:A = 4:3. Inverse proportion (or indirect proportion) occurs when two quantities are related such that their product is constant. If x and y are inversely proportional, x × y = k (constant). Common in work problems: more workers = less time to complete a job.

• Inverse ratio of a:b is b:a
• Inverse proportion: x × y = constant, or x = k/y
• If 4 workers take 10 days, then 5 workers take (4×10)/5 = 8 days
• Recognize inverse proportion in problems mentioning 'more workers, less time' or 'faster speed, less time'

• Forgetting to simplify the ratio to lowest terms before solving. Always find GCD first.
• Confusing the order of ratio. A:B = 3:4 does NOT mean A = 3 and B = 4; it means A = 3k and B = 4k.
• In proportion problems, mixing up extremes and means. Remember: a:b::c:d ⟹ a×d = b×c (not a×c = b×d).
• Misidentifying whether a problem is direct or inverse proportion. Read carefully: 'more workers, less time' = inverse; 'more speed, more distance' = direct.
• Arithmetic errors when calculating k or the final value. Always double-check division and multiplication, especially under time pressure.

• RATIO = Reduce And Tally In Order. Always simplify first.
• PROPORTION = Product of Extremes = Product of Means. (a×d = b×c)
• DIRECT = Both increase together (more workers, more work done). INVERSE = One up, one down (more workers, less time).
• For continued ratio A:B:C, make the common term (B) equal in both ratios, then line them up.

• UP Police Constable typically asks 2-3 ratio/proportion questions. They are straightforward: divide a sum in a given ratio, or find a missing value using proportion. Expect 1-2 minutes per question.
• Recent papers show preference for practical word problems (money division, work-sharing, alligation) over pure ratio simplification. Read the problem carefully to extract the ratio.
• Inverse proportion questions are less common but appear in work/time or speed/time contexts. If you see 'workers and days' or 'speed and time,' think inverse.
• No calculator is allowed. Practice mental arithmetic for division and GCD. For example, GCD(48, 64) should be instant (= 16).
• Time-saving tip: If dividing a sum in ratio m:n, use the formula directly instead of finding k separately. This saves 10-15 seconds per question.