Aptitude · Mixtures and Alligations · Model 3

Two jars that are already mixtures, poured into one

When the things you combine are themselves blends, the cross still works — but only after you convert each jar into a single number. Getting that conversion right, and consistent, is the whole model.

Set two jar concentrations and a target, and pour →

Turn each mixture into one concentration first. Then it is an ordinary alligation between two numbers.

01 The idea

Reduce each jar to a single fraction

Jar A holds milk and water in the ratio 4 : 1. Jar B holds them 2 : 3. Pour them together in some ratio and you get a third mixture. The question is usually what ratio of jars produces a target blend — and the cross answers it, once the jars are numbers rather than ratios.

A ratio of 4 : 1 is not a number you can put on a cross. Convert it to the fraction of the jar that is milk: four parts milk out of five parts total, so 4/5, or 80%. Jar B is two parts out of five, so 2/5, or 40%. Now you have two percentages and the cross applies exactly as it did to two prices.

The one discipline that matters is consistency. If you measure jar A by its milk, measure jar B by its milk and the target by its milk. Mixing a milk fraction against a water fraction produces a wrong answer that looks entirely reasonable, and it is the error this model is set to catch.

Everything else is the familiar cross. The target concentration goes in the middle, the two jar concentrations go either side, and the differences cross over to give the ratio in which the jars must be poured — not the ratio of milk to water in the result, which is a different question the examiner may also ask.

Convert each mixture to the fraction of one component, keep the same component throughout, then run the ordinary cross.

ConcentrationThe fraction of a mixture that is the component you care about. A 4 : 1 milk-to-water jar has milk concentration 4/5, because the ratio's parts add to 5.

Consistency ruleAll three values must measure the same component. Milk fraction against milk fraction against milk fraction — never one of them as water.

Ratio of jarsWhat the cross returns here: how much of jar A to how much of jar B. Distinct from the milk-to-water ratio of the finished blend.

02 Worked example

4 : 1 and 2 : 3, blended to 1 : 1

Two jars, both already mixtures. Jar A contains milk and water in the ratio 4 : 1 and jar B in the ratio 2 : 3. In what ratio must A and B be mixed so that the result has milk and water in equal measure?

Turn jar A into one numberFour parts milk and one part water means five parts in all, of which four are milk.jar A milk = 4/(4+1) = 4/5 = 0.8

Turn jar B into one numberTwo parts milk and three water, so five parts again, of which two are milk. Same component measured, as required.jar B milk = 2/(2+3) = 2/5 = 0.4

Turn the target into one numberEqual milk and water is 1 : 1, which is one part in two — a half. Still measuring milk.target milk = 1/(1+1) = 1/2 = 0.5

Cross the three concentrationsLower value 0.4 on the left, higher 0.8 on the right, target 0.5 in the middle. The differences cross over as always.0.8 − 0.5 = 0.3 (parts of B) 0.5 − 0.4 = 0.1 (parts of A)

Read the ratio in the order askedThe question wants A to B. A is the richer jar and gets the smaller share, which fits — the target is much nearer the weaker jar.A : B = 0.1 : 0.3 = 1 : 3

Check it by counting actual milk. Take 1 litre of A and 3 of B, so 4 litres in total. The milk is 1 × 0.8 + 3 × 0.4 = 0.8 + 1.2 = 2 litres, which is exactly half of 4. Do this check whenever a blend-of-blends answer feels uncertain: multiply each jar's volume by its concentration, add, and compare with the target.

03 The method

The conversion, and the error it exists to prevent

The cross is unchanged from Model 1. Everything specific to this model happens before the cross is drawn.

A mixture in the ratio a : b has first-component concentration a/(a+b). Convert all three — two jars and the target — then A : B = (conc_B − target) : (target − conc_A) with the differences taken larger minus smaller.

Measure the same component three times. If jar A is 80% milk and jar B is 40% milk, the target must also be expressed as milk. Writing jar B as “60% water” and crossing it against milk percentages gives a confident wrong ratio. A second habit worth having: the answer is the ratio of jars, so if the question asks for the final milk-to-water ratio instead, there is another step.

Mixture given as	Milk concentration	Water concentration
4 : 1	`4/5 = 80%`	`1/5 = 20%`
2 : 3	`2/5 = 40%`	`3/5 = 60%`
1 : 1	`1/2 = 50%`	`1/2 = 50%`
3 : 2	`3/5 = 60%`	`2/5 = 40%`
7 : 3	`7/10 = 70%`	`3/10 = 30%`
Pure milk	`1 = 100%`	`0`
Pure water	`0`	`1 = 100%`

05 Cheat sheet

Model 3 on one page

Rows one and two are the conversion. Rows three onward are the ordinary cross, plus the two special cases worth recognising instantly.

Step or case	Do this	On 4:1 and 2:3 → 1:1
Convert a jar	`a/(a+b)`	4/5 = 0.8 and 2/5 = 0.4
Convert the target	`same component`	1/2 = 0.5
Cross	`(hi−m) : (m−lo)`	0.3 : 0.1
Order the answer	`as the question asks`	A : B = 1 : 3
Adding pure milk	`stronger = 100%`	—
Adding pure water	`weaker = 0%`	—
Verify	`Σ(volume × conc) = target × total`	0.8 + 1.2 = 2 of 4 ✓

Same component, all three valuesMilk against milk against milk. Slipping one of them in as a water percentage is the error this model is built to punish, and the wrong answer looks perfectly plausible.

Pure means 100% or 0%Adding pure milk is a jar at 100%; diluting with water is a jar at 0%. Both are ordinary crosses once you see that.

The cross gives the ratio of jarsNot the milk-to-water ratio of the result. If the question wants that instead, convert back after you have the jar ratio.

06 Where & why

Where this model shows up

Blend-of-blends questions are the standard way papers make alligation harder without making it longer.

Bank PO Mains · SSC CGL

Two jars given as ratios

The classic form: two milk-water ratios and a target ratio, jar ratio wanted. The conversion step is where the marks are lost, not the cross.

TCS Digital · Infosys

Dilution and fortification

“How much water must be added…” and “how much pure milk…” — both are this model with one jar at 0% or 100%.

Chemistry-flavoured questions

Acid and alcohol solutions

Already stated as percentages, so no conversion is needed and the question is a plain cross. These are the easy version of the model.

Combined with Model 2

Replace, then blend

Papers chain them: draw off and replace to get a concentration, then mix that with another jar. The output of Model 2 is the input to this one.

Whenever two things being combined are described by ratios rather than by single numbers, your first line should be a conversion — and your last should be a check that the component you measured never changed.

07 Interview questions

What gets asked

Ten, concentrated on the conversion and the two pure-liquid special cases.

Jar A is milk and water 4 : 1. What is its milk concentration?

Four fifths, or 80%. The ratio's parts add to five and four of them are milk, so the fraction is 4/(4+1). A ratio cannot go on the cross directly; this conversion is what makes it usable.

Mix 4 : 1 with 2 : 3 to get equal milk and water. In what ratio?

One part of the 4 : 1 jar to three of the 2 : 3 jar. The concentrations are 0.8, 0.4 and a target of 0.5, so the cross gives 0.3 : 0.1 and, ordered as A to B, that is 1 : 3. Check: one litre at 80% plus three at 40% gives 2 litres of milk in 4, which is half.

What is the single most common error in this model?

Measuring different components. Expressing one jar by its milk and another by its water, then crossing them. The cross returns a tidy ratio either way, so nothing signals the mistake — you have to prevent it rather than detect it.

How do you handle “how much water must be added”?

Treat the water as a jar at 0% milk. So diluting an 80% mixture down to 60% is a cross on 0, 60 and 80, giving 20 : 60 = 1 : 3 — one part water to three parts of the original mixture.

And “how much pure milk must be added”?

The mirror image: pure milk is a jar at 100%. Strengthening a 50% mixture to 75% crosses 50, 75 and 100, giving 25 : 25 = 1 : 1 — equal volumes of the mixture and of pure milk.

Does the cross give the ratio of the jars or the ratio of milk to water in the result?

The ratio of the jars. The milk-to-water ratio of the result is whatever the target was, which you already knew. Questions sometimes ask for the final ratio after adding a stated volume, and that needs a further step rather than a reading of the cross.

Verify a blend-of-blends answer for me.

Multiply each jar's volume by its concentration and add, then compare with the target times the total volume. For 1 litre at 0.8 and 3 at 0.4: 0.8 + 1.2 = 2, and the target 0.5 times 4 litres is also 2. Agreement means the ratio is right.

A 45-litre jar is milk and water 4 : 1. How much milk does it hold?

Thirty-six litres. The concentration is 4/5, and 4/5 of 45 is 36, leaving 9 of water. Converting a ratio to an actual quantity like this is usually the first line of a longer question.

Can this handle three jars?

Hard

A 50-litre vessel holds pure milk. 10 litres are drawn off and replaced with water, and this is done twice. (a) Find the concentration of milk in the vessel now. (b) The resulting mixture is then blended with a second mixture that is 30% milk, to produce a blend that is 50% milk. Find the ratio in which the two are mixed. (c) State which of the two models you used at each stage and why the output of the first is a legitimate input to the second.

Follow-up

This is the join that mains papers set. Each half is routine; the difficulty is realising that repeated replacement produces exactly the one number — a concentration — that alligation needs as an input, so the two models compose cleanly.

Show the hint

For (a) use the surviving-fraction power; the answer is a percentage, which is precisely what part (b)'s cross wants on one side.