Aptitude · Mixtures and Alligations · Model 4

Where mixtures meet profit, and the selling price ruins the cross

The examiner blends two goods, sells the blend, and quotes a selling price. The cross needs cost prices. Almost every mark in this model turns on converting the selling price back to a cost price before you draw anything.

Convert the selling price, then run the cross →

The middle of the cross is always a cost price. If the question gives a selling price, convert it first — the cross will not warn you.

01 The idea

Two chapters, one deliberate collision

This model exists because papers like combining chapters. You are given two goods and their cost prices, told the blend was sold at some price for some profit, and asked for the mixing ratio. Every ingredient of the question is familiar; the join is what is being tested.

The trap is structural rather than arithmetic. The alligation cross demands that all three values be the same kind of quantity, and a selling price is not a cost price. Drop ₹99 into the middle of a cross between ₹80 and ₹110 and you will get 11 : 19 — a clean, wrong ratio, with nothing to indicate the error.

So the model has a fixed opening move: use the profit to convert the selling price into the blend's cost price. If the blend sold at ₹99 with a 10% profit, then ₹99 is 110% of its cost, so the cost is ₹90. Only ₹90 may go in the middle.

The reverse direction is set just as often. Given the ratio, you compute the blend's cost, apply the required profit, and quote a selling price. Same two ideas in the other order, and the same discipline about which number is which.

Selling price into cost price first, using the profit. Then the cross is ordinary — and only then.

Cost price of the blendWhat the mixture cost the trader per unit, which is the weighted average of the ingredient costs. This is the only value that belongs in the middle of the cross.

Selling priceWhat the customer pays. Related to the blend's cost by the profit percentage, and never usable directly on a cross whose other values are costs.

Profit percentageProfit as a percentage of the cost price, not of the selling price. So SP = CP × (100 + p)/100.

02 Worked example

₹80 and ₹110 tea, sold at ₹99 for 10%

One question with the trap fully armed. A trader mixes tea costing ₹80/kg with tea costing ₹110/kg and sells the blend at ₹99/kg, making a profit of 10%. In what ratio were the two teas mixed?

Notice what kind of number ₹99 isIt is a selling price. The other two figures are costs. Three values, and one of them does not belong — yet.costs: 80 and 110 selling price: 99 profit: 10%

Convert the selling price to a costA 10% profit means the selling price is 110% of the blend's cost price. So divide rather than subtract 10%.99 = 110% of CP ⇒ CP = 99 × 100/110 = ₹90

Now the three values matchCheaper ₹80, dearer ₹110, mean ₹90 — all cost prices per kilo.cheaper = 80 mean = 90 dearer = 110

Cross as usualDearer minus mean gives the cheaper tea's parts; mean minus cheaper gives the dearer's.110 − 90 = 20 90 − 80 = 10

Simplify and checkTwo to one, cheap to expensive. Verify the blend really costs ₹90.20 : 10 = 2 : 1, and (2×80 + 1×110)/3 = 270/3 = ₹90 ✓

Now see what the trap would have cost. Putting ₹99 in the middle gives (110 − 99) : (99 − 80) = 11 : 19 — not obviously absurd, and a blend in that ratio actually costs (11×80 + 19×110)/30 = ₹99, which is the trader's selling price and therefore a blend on which he makes no profit at all. The wrong answer is internally consistent, which is precisely why it is dangerous.

03 The method

The two directions, and the percentage that connects them

Every question in this model is one of two journeys between a cost and a selling price. The cross sits at the cost end of both.

SP = CP × (100 + p)/100, so going backwards CP = SP × 100/(100 + p). On a loss of l%, replace (100 + p) with (100 − l).

Divide, do not subtract. A 10% profit on a selling price of ₹99 does not make the cost ₹99 − ₹9.90 = ₹89.10. The profit is a percentage of the cost, so ₹99 is 110% of the cost and the cost is ₹99 × 100/110 = ₹90. Subtracting the percentage from the selling price is the second trap in this model, sitting right behind the first.

The question gives you	First move	Then
Two costs, SP and profit%	SP → CP by dividing	Cross for the ratio
Two costs and the ratio	Weighted average for the CP	Apply profit for the SP
Two costs, CP and profit% wanted	Cross or average as needed	Compare SP with CP
A loss instead of a profit	`CP = SP × 100/(100−l)`	Cross unchanged
SP put straight in the middle	`invalid`	Gives a zero-profit blend
Profit subtracted from SP	`invalid`	Profit is a % of cost, not of SP

05 Cheat sheet

Model 4 on one page

Two conversions and one cross. The last three rows are the ways this model is failed.

Case	Route	On the tea question
SP to CP	`CP = SP × 100/(100+p)`	99 × 100/110 = 90
CP to SP	`SP = CP × (100+p)/100`	90 × 1.1 = 99
Ratio from costs	`(d−m) : (m−c)`	20 : 10 = 2 : 1
CP from a known ratio	`weighted average`	(2×80+110)/3 = 90
Loss instead of profit	`CP = SP × 100/(100−l)`	—
SP used as the mean	`wrong`	gives 11 : 19
Profit subtracted from SP	`wrong`	gives 89.10, not 90

The mean is always a cost priceThe two outer values are costs, so the middle must be too. A selling price in the middle returns a ratio for a blend sold at cost — zero profit, which is never what was asked.

Divide by (100 + p), do not subtract p%Profit is a percentage of the cost, not of the selling price. Subtracting 10% from ₹99 gives ₹89.10; the correct cost is ₹90.

Check the blend's cost at the endTake your ratio, compute the weighted average of the two costs, and confirm it equals the cost you put in the middle. This catches both traps at once.

06 Where & why

Where this model shows up

Cross-chapter questions are how papers separate students who learned methods from students who learned when methods apply.

Bank PO Mains · SSC CGL

Blend, sell, find the ratio

The standard form, and the selling-price conversion is the whole difficulty. Both the right answer and the trap answer are normally among the options.

TCS Digital · Infosys

Ratio given, selling price wanted

The reverse journey: average the costs, then mark up. Easier, because there is no temptation to put the wrong number on a cross.

Dishonest-dealer overlap

Blending as a way to profit

Adding water to milk, or cheap grain to expensive, is profit through mixing. The cost of the added component is zero or low, which is just a cross with one value at 0.

Any 'worth' phrasing

“A mixture worth ₹x”

The word worth almost always means cost price, but read the sentence — if the trader sells at that figure, it is a selling price and needs converting.

The one habit that carries this model: before any number enters the middle of a cross, say out loud whether it is a cost or a selling price. Everything else here you already know from the two chapters separately.

07 Interview questions

What gets asked

Ten, and the first three are the model in its entirety.

Tea at ₹80 and ₹110 is blended and sold at ₹99 for a 10% profit. Find the mixing ratio.

2 : 1, cheap to expensive. First convert: ₹99 is 110% of the blend's cost, so the cost is ₹90. Then cross 80, 90 and 110 to get 20 : 10 = 2 : 1. Check: (2×80 + 110)/3 = ₹90.

Why can't ₹99 go straight into the middle of the cross?

Because the other two values are cost prices and ₹99 is a selling price — the cross requires all three to be the same kind of quantity. Using it anyway gives 11 : 19, which is the ratio of a blend costing ₹99 to make, meaning the trader sells at cost and makes nothing.

How do you convert a selling price to a cost price?

Divide by (100 + profit%) and multiply by 100. So ₹99 at 10% profit is 99 × 100/110 = ₹90. On a loss of l%, use (100 − l) instead.

Why not just subtract 10% from ₹99?

Because the profit is a percentage of the cost, not of the selling price. Subtracting gives ₹89.10, whereas the true cost is ₹90. The two differ because 10% of the cost and 10% of the selling price are different amounts — and this is a distinct error from the cross-position one, so it is possible to make both.

Given the ratio instead, how do you find the selling price?

Weighted-average the two costs to get the blend's cost, then mark it up. For 2 : 1 on ₹80 and ₹110, the cost is ₹90, and a 10% profit makes the selling price ₹99. This direction is the same two ideas in reverse order.

What if the trader makes a loss?

Only the conversion changes: CP = SP × 100/(100 − l). A blend sold at ₹90 at a 10% loss cost ₹100 to make. The cross afterwards is untouched, since it only ever deals in costs.

Does the word 'worth' mean cost price or selling price?

Usually cost price — “a mixture worth ₹90 a kilo” normally means it costs that to produce. But read the sentence rather than the word: if the trader sells at that figure, it is a selling price and must be converted first.

How would you check a Model 4 answer quickly?

Take your ratio, compute the weighted average of the two ingredient costs, and confirm it matches the cost you used as the mean. Then apply the profit and confirm you land on the stated selling price. Two multiplications, and it catches both of the model's traps.

Where does adding water to milk fit in?

Hard

Two grades of coffee costing ₹150/kg and ₹250/kg are blended and the blend is sold at ₹209/kg at a profit of 10%. (a) Find the mixing ratio. (b) A student answers by subtracting 10% from ₹209 to get a cost of ₹188.10 and crossing with that. Without simplifying it fully, show that their ratio cannot reduce to anything tidy, and say why an ugly ratio is itself a warning sign in a question whose numbers were chosen by an examiner. (c) A second student puts ₹209 straight into the middle. Give their ratio, and show that their blend would be sold at cost. (d) Describe a single check that would have caught both students' errors.

Follow-up

Three answers to the same question, two of them wrong in different ways — one from the arithmetic of percentages and one from the structure of the cross. Part (d) is the payoff: one verification step catches both, which is why it is worth building into your working rather than remembering two separate warnings.

Show the hint

For (d), think about what you can compute from a candidate ratio and compare against something the question already told you.