Aptitude · Mixtures and Alligations · Model 1

Every disguise the basic cross wears

The cross itself is one subtraction each way. What makes Model 1 questions hard is that the three values are often not handed to you — one of them is hidden behind a total, a count, or a purity, and finding it is the actual question.

Work the levelled source questions with the cross →

When a mixture question looks unfamiliar, the missing piece is almost always the mean. Build it first, then the cross is routine.

01 The idea

The cross is easy; identifying the three values is not

You already have the rule: dearer minus mean over mean minus cheaper. Applying it to ₹40 rice, ₹60 rice and a ₹45 target takes seconds. Exam questions rarely look like that.

Instead they give you a total and a count and expect you to divide. “₹12.40 in 80 coins” contains a mean of 15.5 paise that appears nowhere in the sentence. Or they give you gold in carats, where the numbers are purities rather than prices. Or two group averages and a combined one, where the answer is a ratio of headcounts.

So the working habit for Model 1 is to name the three values out loud before drawing anything. Which number is the cheaper, which the dearer, which the mean — and if the mean is absent, what has to be divided by what to produce it.

The second habit is checking that the ratio you get is the ratio the question asked for. The cross always returns cheaper-to-dearer. If the question wanted dearer-to-cheaper, or wanted an actual weight rather than a ratio, there is one more step and it is easy to skip.

Name the cheaper, the dearer and the mean before you draw the cross. If the mean is missing, it is a total divided by a count.

Hidden meanA mean the question does not state. Usually total value divided by total quantity — ₹12.40 across 80 coins is a mean of 15.5 paise per coin.

Parts and quantitiesThe cross gives parts. To get an actual weight or count, add the parts, divide the real total by that, and multiply back up.

PurityCarat or percentage-fineness, treated exactly like a price. 18-carat and 12-carat mixed to 15-carat behaves identically to ₹18 and ₹12 mixed to ₹15.

02 Worked example

100 kg of tea, and a mean you are given

Start with the form where the total matters. A merchant has 100 kg of tea. Part of it he sells at ₹110/kg and the rest at ₹130/kg. If the average price of the whole stock is ₹118/kg, find the quantity of the ₹110 tea.

Name the three valuesCheaper ₹110, dearer ₹130, mean ₹118. All three are prices per kilo, so the cross is legal.cheaper = 110 mean = 118 dearer = 130

Cross both waysDearer minus mean gives the cheaper tea's parts; mean minus cheaper gives the dearer's.130 − 118 = 12 118 − 110 = 8

Simplify the ratioTwelve to eight, cheaper to dearer.12 : 8 = 3 : 2

Turn parts into kilosThe ratio has 5 parts in total and they must cover the whole 100 kg.5 parts = 100 kg ⇒ 1 part = 20 kg

Answer what was askedThe ₹110 tea is the cheaper one, which is 3 parts.3 × 20 = 60 kg of the ₹110 tea (and 40 kg of the ₹130)

Two places to go wrong here, neither of them the cross. The first is stopping at 3 : 2, which is not a quantity. The second is answering 40 kg — the dearer tea — because the ratio comes out cheaper-first and the question asked for the cheaper. Read the last line again before you commit: 60 + 40 = 100, so both numbers are available and only one is right.

03 The method

Four disguises, and how to undress each

Every row below is the same cross. The column that matters is the middle one — what you have to do before the cross can start.

Cheaper : dearer = (dearer − mean) : (mean − cheaper), then one part = total / (sum of parts) whenever an actual quantity is wanted.

When the mean is missing, divide. Total value over total quantity gives it every time — rupees over kilos, paise over coins, marks over students. Work in one unit throughout: convert ₹12.40 to 1240 paise before dividing by 80 coins, never after.

Disguise	Get the mean by	Then the cross gives
Two prices, target given	Reading it off	Ratio of the two goods
Total value and count	Total / count	Ratio of the two counts
Coins of two denominations	Paise total / number of coins	Ratio of the coin counts
Two group averages	Reading the combined average	Ratio of the group sizes
Gold purity	Reading the target carat	Ratio of the two alloys
Decimal values	Reading it off	Multiply out the decimals at the end

05 Cheat sheet

Model 1 on one page

The first row is the rule. The rest is what to do before and after it.

Step	Do this	On the tea question
1. Name the values	`cheaper, dearer, mean`	110, 130, 118
2. Build a missing mean	`total / count`	Given, so nothing to do
3. Cross	`(d−m) : (m−c)`	12 : 8
4. Simplify	`divide by the hcf`	3 : 2
5. Clear decimals	`multiply both sides`	Not needed here
6. Parts to quantity	`total / sum of parts`	100/5 = 20 kg per part
7. Answer the question asked	`cheaper or dearer?`	Cheaper = 3 parts = 60 kg

The ratio comes out cheaper-firstAlways. If the question wanted the dearer quantity, the answer is the second number, not the first — and both will be among the options.

A ratio is not a quantity3 : 2 is not an answer to “how many kilos”. Add the parts, divide the total, multiply back.

One unit throughoutConvert ₹12.40 to 1240 paise before dividing by 80 coins. Mixing rupees and paise mid-question produces a mean that is out by a factor of 100.

06 Where & why

Where Model 1 shows up

This is the form the basic cross is actually set in, and the difficulty is always in the reading rather than the arithmetic.

TCS · Wipro · Accenture

Two prices, ratio wanted

The easiest version, set as a speed question. The cross plus a simplification and you are done inside twenty seconds.

SSC CGL · Bank Prelims

Decimal prices and hidden means

₹64.50 against ₹84.00, or a coin total that has to be divided down. The maths is identical; the setup is the test.

Bank PO Mains

Purity and alloy questions

Carat values behave exactly like prices. The 12/18/15 case coming out 1 : 1 is a favourite because the symmetry looks like a mistake.

Averages and data interpretation

Group sizes from group averages

Given two averages and a combined one, the cross returns the ratio of the group sizes. Worth recognising well outside this chapter.

The whole of Model 1 reduces to two questions asked before any arithmetic: what are my three values, and does the question want a ratio or a quantity? Get both right and there is nothing left to get wrong.

07 Interview questions

What gets asked

Ten, weighted towards the setup rather than the subtraction — which is where the marks actually move.

Walk me through the tea question.

Tea at ₹110 and ₹130 averaging ₹118 over 100 kg. The cross gives (130 − 118) : (118 − 110) = 12 : 8 = 3 : 2. That is 5 parts over 100 kg, so 20 kg a part, making 60 kg of the ₹110 tea and 40 kg of the ₹130.

The cross gave you 3 : 2. Which is which?

Cheaper first. The 3 belongs to the ₹110 tea, because the difference taken from the dearer value crosses over to the cheaper item. Getting this backwards gives 40 kg, which will be one of the options.

How do you handle ₹12.40 in 80 coins of 10 and 20 paise?

Build the mean first: 1240 paise over 80 coins is 15.5 paise per coin. Then cross 10, 15.5 and 20 to get 4.5 : 5.5 = 45 : 55 = 9 : 11. Converting to paise before dividing is essential — dividing ₹12.40 by 80 gives 0.155 rupees, which is the same thing but easy to misread.

What if the cross gives you a decimal ratio?

Multiply both sides by whatever clears it. Pulses at ₹64.50 and ₹84.00 blended to ₹72.50 give 11.5 : 8; doubling gives 23 : 16. A ratio is unchanged by scaling both sides, so this is always legal.

In what ratio do you mix 12-carat and 18-carat gold to get 15-carat?

1 : 1. The differences are 18 − 15 = 3 and 15 − 12 = 3, so 3 : 3. The target sits exactly halfway between the two, and an equal blend is the correct and slightly counter-intuitive answer.

When is the answer always 1 : 1?

Whenever the mean is exactly midway between the two values, since both differences are then equal. It is worth spotting on sight, because it lets you answer a purity or percentage question without doing the subtraction at all.

Does alligation work for percentages as well as prices?

Yes, and for purities, speeds, marks and averages. The cross is a weighted-average tool and does not care what is being averaged, provided all three values are the same kind of quantity.

How do you convert the ratio into an actual quantity?

Add the parts, divide the real total by that sum to get one part, then multiply by the parts you want. For 3 : 2 over 100 kg, that is 5 parts, 20 kg a part, so 60 kg and 40 kg — and they should add back to the total, which is your check.

A question gives the selling price of the mixture. What do you do?

Hard

A trader blends tea costing ₹80/kg with tea costing ₹110/kg and sells the blend at ₹99/kg, making a profit of 10%. (a) Find the cost price of the blend. (b) Find the ratio in which the two teas were mixed. (c) A student puts ₹99 straight into the middle of the cross and reports the ratio they get. State what ratio they would report, and explain in one sentence why the method is invalid even though it produces a plausible-looking answer.

Follow-up

Part (c) is the unit condition set as a trap. The student's cross averages a selling price against two cost prices, which is meaningless — but it returns a tidy ratio, and a tidy wrong answer is far more dangerous than an obviously broken one.

Show the hint

For (a), ₹99 is 110% of the blend's cost. Only once you have that cost may it go in the middle.