Mensuration, mixtures and counting: the results worth memorising

Mensuration, mixtures and counting are the topics candidates most often leave to luck, yet each is small and rule-bound. Mensuration is recall, mixtures is one rule applied cleanly, and counting is a single question — does order matter — asked over and over. Here is the compact set that covers the exam versions.

Mensuration: memorise the short list, then scale

Bank and railway papers use a limited set of shapes, so hold these in recall rather than deriving them:

• Rectangle area l·b, triangle ½·base·height, circle area πr², circumference 2πr.
• Cuboid volume l·b·h, total surface area 2(lb + bh + hl).
• Cylinder volume πr²h, curved surface 2πrh; cone volume ⅓πr²h; sphere volume 4/3·πr³, surface 4πr²; hemisphere volume 2/3·πr³.

The habit that saves the most time is scaling by ratio: for similar figures, area scales as the square and volume as the cube of the linear ratio. Doubling a radius quadruples the area and multiplies the volume by eight — read straight off, without touching π.

Mixtures: alligation is the whole topic

Almost every mixture question is the rule of alligation — the ratio in which two ingredients at different values combine to a stated mean. The quantity of the cheaper over the quantity of the dearer equals (dearer − mean) : (mean − cheaper).

If milk at ₹40 and ₹60 a litre are mixed to sell at ₹45, the ratio is (60 − 45) : (45 − 40) = 15 : 5 = 3 : 1. The same rule finds a blend of two speeds to a known average, or two concentrations to a target — it is the weighted-average idea from averages read backwards.

The other standard type is repeated replacement: if x of a V-litre vessel is drawn off and refilled with water n times, the pure liquid left is V·(1 − x/V)ⁿ. Recognising this one formula answers the entire “how much wine remains” family in a single substitution.

Counting: order, or not

Permutations and combinations reduce to one decision.

• Order matters → permutation. Arranging r out of n is nPr = n!/(n−r)!. Ranks, seating, number-lock codes and “words that can be formed” are permutations.
• Order does not matter → combination. Choosing r out of n is nCr = n!/(r!(n−r)!). Teams, committees, handshakes and “ways to pick” are combinations.

Two facts remove most of the arithmetic: nCr = nC(n−r) (so nC8 out of 10 is just nC2 = 45), and probability is simply favourable ÷ total, with both counted the same way. When a probability question feels hard, it is almost always a counting question in disguise.

A worked reading

Two vessels hold spirit and water in the ratios 3 : 2 and 4 : 1. In what ratio should they be mixed to get a 5 : 3 spirit-to-water blend? Convert to spirit fractions: 3/5 and 4/5, target 5/8. By alligation the ratio is (4/5 − 5/8) : (5/8 − 3/5) = (7/40) : (1/40) = 7 : 1. No simultaneous equations — just fractions and one rule.

Where the marks leak

Surface area for volume, or the wrong surface. Read whether the question wants curved surface, total surface, or volume; they are different formulas for the same solid.

Alligation the wrong way round. The cheaper quantity pairs with the dearer-minus-mean distance; drawing the little cross the same way each time prevents the flip.

Permutation where a combination is meant. If swapping two chosen items gives the same outcome, it is a combination — dividing a permutation by r! is the correction.

Forgetting nCr = nC(n−r). Choosing 8 from 10 the long way wastes a minute that the symmetry saves.

Practice that transfers

These three reward recall and a couple of clean rules more than cleverness, so drill the formula list and the alligation cross until they are automatic. Work through the mensuration, mixtures and counting practice set, and pair it with percentage, profit and averages, since alligation is just the weighted average seen from the other end.