Ratio and Proportion Questions for SSC MTS

What is a Ratio

A ratio is just a comparison of two quantities of the same type. If Ravi has 30 rupees and Suresh has 50 rupees, the ratio of their money is 30:50, which simplifies to 3:5. Think of ratio as a fraction in disguise — 3:5 is the same as 3/5. Always simplify the ratio by dividing both parts by their HCF. The two numbers in a ratio are called antecedent (first) and consequent (second).

• Ratio a:b means a/b. Always reduce to lowest terms.
• Both quantities must be in the SAME unit before forming a ratio.
• Ratio has no unit — it is a pure number.
• If a:b = 3:5, then a = 3k and b = 5k for some common multiplier k.
• Multiplying or dividing both terms by the same number does NOT change the ratio.

Proportion and Its Types

Proportion means two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion. The key rule is: product of extremes = product of means, i.e., a*d = b*c. This is called cross multiplication and it is the fastest way to solve proportion problems. There are two types you must know: Direct Proportion (both increase or decrease together) and Inverse Proportion (one increases, other decreases). Imagine a tap filling water — more time means more water (direct). But more workers means less time to finish a job (inverse).

• a:b :: c:d means a*d = b*c (cross multiply).
• Direct proportion: x/y = constant. Example: more items, more cost.
• Inverse proportion: x*y = constant. Example: more speed, less time.
• Fourth proportional to a, b, c is x where a:b = c:x, so x = b*c/a.
• Mean proportional between a and b is square root of (a*b).
• Third proportional to a and b is b squared divided by a.

Ratio Operations — Compounding and Comparison

Sometimes you need to combine multiple ratios or compare them. Compounded ratio means multiplying two ratios together: (a:b) compounded with (c:d) = ac:bd. This comes up when two different ratios describe parts of the same thing. To compare two ratios, convert them to fractions and cross multiply — no need for decimals. SSC MTS loves questions where you are given A:B and B:C and asked to find A:B:C. The trick is to make the common term (B) equal in both ratios.

• Compounded ratio of a:b and c:d is ac:bd.
• Duplicate ratio of a:b is a^2:b^2.
• Sub-duplicate ratio of a:b is sqrt(a):sqrt(b).
• To find A:B:C from A:B and B:C, equalize B by taking LCM.
• If A:B = 2:3 and B:C = 4:5, make B same — A:B = 8:12 and B:C = 12:15. So A:B:C = 8:12:15.

Proportional Division and Mixtures

A very common SSC MTS question type: a total amount is split among 2 or 3 people in a given ratio. Just use the fraction method — each person's share = (their ratio part / total ratio parts) multiplied by total amount. For three people in ratio a:b:c, the shares are a/(a+b+c), b/(a+b+c), c/(a+b+c) of the total. This also appears in mixture problems where you find how much of each ingredient is present.

• For ratio a:b:c, individual shares out of total T are aT/(a+b+c), bT/(a+b+c), cT/(a+b+c).
• If ratio changes, find new total ratio and recalculate.
• Difference between two shares = (difference in ratio parts / total parts) * Total.
• Always add ratio parts first before dividing.

• Not converting units before forming a ratio — e.g., comparing 500 grams with 2 kg directly. Always bring both to same unit first.
• Confusing mean proportional (geometric mean = sqrt of product) with average. Mean proportional between 4 and 9 is 6, not 6.5.
• While combining A:B and B:C, forgetting to equalize B and directly writing the four numbers — gives a completely wrong answer.
• In proportional division, forgetting to add ratio parts and dividing by just one part — e.g., dividing total by 3 instead of (2+3+5)=10.
• Mixing up third proportional and fourth proportional formulas under exam pressure — remember: third proportional is b^2/a, fourth is bc/a.

• EXTREME MEANS rule: In a:b :: c:d, Extremes are a and d (outer), Means are b and c (inner). Extremes product = Means product. Think E=M (Extremes equal Means in product).
• For A:B:C combination, think of it as a relay race — B is the common runner who appears in both teams. Make B's value the same (LCM) before the final lineup.
• Mean proportional = Middle child of a and b = sqrt(a*b). It is the geometric middle.
• Direct proportion: SAME direction (both up or both down). Inverse proportion: OPPOSITE direction. Think of a seesaw for inverse.

• SSC MTS typically asks 2-3 questions from this topic per paper. They are usually easy to medium — no complex algebra. A student who knows the formulas should solve each in 45-60 seconds.
• Most common question types in recent MTS papers: (1) divide a sum in given ratio, (2) find fourth or mean proportional, (3) combine two ratios to get three-way ratio.
• If a question gives you two different ratios sharing a common term (like A:B and B:C), it is always asking for A:B:C. Go straight to LCM method — no thinking needed.
• Ratio questions in MTS do NOT require lengthy calculation. If your working is getting long, you are likely overcomplicating — check if a shortcut ratio comparison or fraction method is faster.
• Watch out for trick phrasing: 'ratio of A to B is 3:5' means A/B = 3/5 (A is smaller). Many students flip this and get the wrong answer.