A provisions shop-owner is found to mix 25 kg of rice worth ₹ 32/kg and 20 kg of rice worth ₹ 35/kg and the mixed rice is sold at 15% profit. What is the selling price of the mixed rice ?

Correct — B, (b) ₹ 38·33/kg.

THE STEP THAT DECIDES THIS QUESTION is that the cost of the mixture is a WEIGHTED average, not a simple one. The two grades are not mixed in equal quantities, so the cheaper rice pulls the average price down towards itself in proportion to how much of it there is.

WORK IN TOTALS, WHICH IS THE SAFEST ROUTE.

25 kg at ₹ 32/kg costs 25 × 32 = ₹ 800 20 kg at ₹ 35/kg costs 20 × 35 = ₹ 700 Total: 45 kg costing ₹ 1500

Cost price of the mixture = 1500 ÷ 45 = ₹ 33·33/kg, which is exactly 100/3.

NOW APPLY THE MARK-UP. A profit of 15 per cent is reckoned on the cost price, so the selling price is 1·15 times it:

(100/3) × 1·15 = 115/3 = ₹ 38·33/kg

and that is option (b). Keeping the cost as the fraction 100/3 rather than as the decimal 33·33 avoids carrying a rounding error into the multiplication; 115 divided by 3 is 38·333…, which the paper prints as 38·33.

A BOUNDING ARGUMENT THAT SETTLES THE OPTION LIST ALMOST WITHOUT ARITHMETIC, and which is the most useful part of this card. Any weighted average of ₹ 32 and ₹ 35 must lie strictly BETWEEN those two numbers. After a mark-up of 15 per cent, the selling price must therefore lie strictly between

32 × 1·15 = ₹ 36·80 and 35 × 1·15 = ₹ 40·25

Two of the four options fall outside that band altogether — 35·40 and 36·50 are both below ₹ 36·80 — and can be struck out on sight, without knowing anything about the quantities involved. The band can then be narrowed. Since there is MORE of the cheaper rice, 25 kg against 20, the average must lie nearer to 32 than to 35, and so below the midpoint of ₹ 33·50; the selling price must therefore be below 33·50 × 1·15 = ₹ 38·525. That leaves the interval from ₹ 36·80 to ₹ 38·525, and only two options survive it. The calculation then picks between them.

CHECKING EACH OPTION BACKWARDS is the other quick discipline, and it makes the point sharply. Divide each printed price by 1·15 to see what cost per kilogram it implies:

₹ 35·40 → a cost of ₹ 30·78/kg — cheaper than either ingredient. Impossible. ₹ 38·33 → a cost of ₹ 33·33/kg — exactly the weighted average of 32 and 35 in the ratio 25 : 20. ₹ 36·50 → a cost of ₹ 31·74/kg — again cheaper than either ingredient. Impossible. ₹ 37·42 → a cost of ₹ 32·54/kg — inside the range, but it corresponds to the dearer rice being only about an eighth of the mixture instead of nearly half.

THE SIMPLE-AVERAGE TRAP is worth naming even though it is not on the list. Averaging the two prices as (32 + 35)/2 = ₹ 33·50 and marking that up gives ₹ 38·53, which is close to the answer but not equal to it, and the closeness is only an accident of these particular quantities being nearly equal. With 40 kg and 5 kg the two methods would diverge sharply.

The four options are printed OUT OF ASCENDING ORDER, their decimals use a raised middle dot, and the rates are set as ₹ 32/kg with a space after the currency sign and none around the solidus — all as the booklet has them.

Why the others are wrong

• (a)₹ 35·40/kg — This option is impossible before any calculation is done. Dividing ₹ 35·40 by 1·15 gives an implied cost of about ₹ 30·78 per kilogram, which is cheaper than either of the two grades that went into the mixture. No blend of rice at ₹ 32 and rice at ₹ 35 can cost less than ₹ 32, whatever the proportions — a weighted average always lies between the values being averaged, and it equals one of them only when the other has zero weight. So a selling price below 32 × 1·15 = ₹ 36·80 cannot arise from this stem at all, and this option is nearly a rupee and a half below that floor. It is the lowest figure on the list and it is offered first, which is worth noticing: the options here are printed out of ascending order, so a candidate who assumes the list runs from smallest to largest and reads only the first two will be working from a false picture of the range on offer.
• (c)₹ 36·50/kg — This option fails the same test as the lowest one, though less obviously. Dividing ₹ 36·50 by 1·15 gives an implied cost of about ₹ 31·74 per kilogram, again below the cheaper of the two ingredients, so no mixture of these two grades sold at this profit could fetch it. The floor is ₹ 36·80 and this option sits thirty paise underneath it. What makes it more dangerous than the lowest option is that it LOOKS as though it could be a mixture price — it lies between 32 and 40, it has a plausible shape, and a candidate who has computed the cost of the mixture correctly at ₹ 33·33 but then forgotten to apply the mark-up at all will be looking for something in the middle thirties and may settle here. Applying the profit is not an optional flourish: the question asks for the SELLING price, and the cost price is only an intermediate step.
• (d)₹ 37·42/kg — This is the one wrong option that survives the bounding test, and so the only one that needs to be settled by arithmetic. Dividing ₹ 37·42 by 1·15 gives an implied cost of about ₹ 32·54 per kilogram, which does lie between the two ingredient prices — but it corresponds to the wrong proportions. A cost of ₹ 32·54 would mean the dearer rice made up (32·54 − 32)/(35 − 32), that is about eighteen per cent of the mixture, whereas the stem gives 20 kg out of 45, which is a little over forty-four per cent. The narrower bound disposes of it: because there is MORE of the cheaper grade, the blended cost must be below the midpoint of ₹ 33·50 but it must still be a genuine average of the two, and 20 kg of the dearer grade lifts it to ₹ 33·33 — only seventeen paise short of the midpoint, not the eighty paise this option would require. The correct figure is 1500 ÷ 45, and there is no substitute for computing it.

MIXTURE PROBLEMS ARE WEIGHTED-AVERAGE PROBLEMS, and the whole topic follows from that one sentence.

When quantities q1 and q2 of two ingredients costing p1 and p2 are blended, the cost of the mixture per unit is

(q1·p1 + q2·p2) / (q1 + q2)

which is the average of p1 and p2 weighted by the quantities. Three properties follow immediately and are worth holding as facts rather than as formulae.

The result always lies strictly BETWEEN p1 and p2, and never outside them. It lies NEARER to the price of whichever ingredient there is more of. It equals the simple average (p1 + p2)/2 only when the two quantities are equal.

THE RULE OF ALLIGATION is the same relation solved for the quantities instead of for the price, and it is the form in which these problems are usually set when the mixture price is given and the ratio is wanted:

q1 : q2 = (p2 − m) : (m − p1)

where m is the mean price of the mixture. In words, the quantities are in the inverse ratio of their distances from the mean price. It is worth checking that against this problem the other way round: with m = 100/3, the distances are 35 − 100/3 = 5/3 and 100/3 − 32 = 4/3, so the ratio of the cheaper to the dearer is 5/3 : 4/3 = 5 : 4 — which is 25 kg to 20 kg exactly, as the stem says.

THE SECOND HALF OF THE QUESTION IS PLAIN PROFIT AND LOSS. Profit is reckoned on the COST price unless a question says otherwise, so a 15 per cent profit means a selling price of 1·15 times cost. The multiplier form is safer than computing the profit separately and adding it, because it is one operation rather than two. Related conversions worth keeping: a loss of x per cent means a multiplier of (1 − x/100); a margin of x per cent ON THE SELLING PRICE means dividing the cost by (1 − x/100), which is a different and larger number.

THE STRUCTURE OF THE WHOLE PROBLEM is therefore two steps that must be kept apart: BLEND, then MARK UP. Most errors in this area come from merging them — marking up each ingredient separately and then averaging, which happens to give the same answer here because the mark-up is uniform, or averaging the prices without weighting, which does not. Keeping the two steps distinct also makes the answer checkable, since each step can be verified against a bound: the blended cost must lie between the ingredient costs, and the selling price must be 1·15 times the blended cost.

Key facts

• The mixture costs 25 × 32 + 20 × 35 = ₹ 1500 for 45 kg, so its cost is 1500 ÷ 45 = ₹ 33·33 per kilogram, exactly 100/3.
• A profit of 15 per cent is reckoned on the cost price, so the selling price is (100/3) × 1·15 = 115/3 = ₹ 38·33 per kilogram.
• A weighted average always lies strictly between the two values being averaged, so the mixture must cost between ₹ 32 and ₹ 35 per kilogram.
• The selling price must therefore lie between 32 × 1·15 = ₹ 36·80 and 35 × 1·15 = ₹ 40·25, which rules out two of the four options at once.
• Because there is more of the cheaper grade — 25 kg against 20 — the blended cost lies nearer ₹ 32 and below the midpoint of ₹ 33·50.
• Back-dividing each option by 1·15 shows what cost it implies: ₹ 30·78 and ₹ 31·74 for two of them, both cheaper than either ingredient and therefore impossible.
• The rule of alligation states that the quantities are in the inverse ratio of their distances from the mean price, which here gives 5 : 4, matching 25 kg to 20 kg.
• The simple average of ₹ 32 and ₹ 35 is ₹ 33·50 and marking it up gives ₹ 38·53 — close here only because the two quantities happen to be nearly equal.
• The four options are printed out of ascending order on this item, which is uncommon on this paper and worth noticing before comparing a computed figure against them.

Common traps

• Taking the simple average of ₹ 32 and ₹ 35 instead of weighting by the 25 kg and 20 kg actually mixed.
• Stopping at the cost of the mixture and forgetting the 15 per cent mark-up. The question asks for the selling price.
• Assuming the option list is in ascending order. On this item it is not, and a computed figure has to be matched against all four.
• Reckoning the profit on the selling price rather than on the cost, which would give about ₹ 39·22 and match no option.
• Rounding ₹ 33·33 before multiplying. Keeping the cost as 100/3 gives 115/3 exactly and avoids carrying an error into the second step.

• practice — not a real PYQ

  In what ratio must rice costing ₹ 30 per kg be mixed with rice costing ₹ 40 per kg so that the mixture costs ₹ 34 per kg ?

  • (a)2 : 3
  • (b)3 : 2
  • (c)4 : 3
  • (d)5 : 4

  Answer(b) 3 : 2 — by the rule of alligation the quantities are in the inverse ratio of their distances from the mean price, that is (40 − 34) : (34 − 30) = 6 : 4 = 3 : 2, so three parts of the cheaper grade go with two of the dearer.

• practice — not a real PYQ

  A dealer mixes 30 kg of tea worth ₹ 200 per kg with 20 kg of tea worth ₹ 250 per kg and sells the whole of the mixture at a profit of 10 per cent. What is the selling price of the mixture per kg ?

  • (a)₹ 231
  • (b)₹ 242
  • (c)₹ 247
  • (d)₹ 253

  Answer(b) ₹ 242 — the blend costs 30 × 200 + 20 × 250 = ₹ 11,000 for 50 kg, that is ₹ 220 per kg, and a mark-up of 10 per cent gives 220 × 1·1 = ₹ 242. The simple average of ₹ 225 would have given ₹ 247·50, which is why the weighting matters.