SSC CGL Maths · Ratio, Proportion and Variation

Mixtures and Alligation

The price of a mixture is the weighted average of the prices that go into it, and alligation runs that average backwards to give the ratio in which two kinds were mixed.

All Ratio, Proportion and Variation notes

Why this matters

Thirty questions in the bank, nineteen EASY and eleven MODERATE. Fifteen find a mean price or a missing price, four ask straight for a mixing ratio, and eleven are sold at a profit or a loss, which is where most wrong answers come from: the cross must use the cost price, not the selling price.

Concept 1 of 3: Mean price of a mixture

If you buy 6 kg of one tea and 4 kg of another, the mixture costs exactly what you paid in total, spread over 10 kg. So its price per kg is a weighted average, and it always lies between the cheapest and the dearest price, nearer to the kind you used more of.

Definition

Mean price $= \frac{q_1p_1 + q_2p_2 + \cdots}{q_1 + q_2 + \cdots}$ : total cost over total quantity.
A missing price: total value of the mixture (its quantity times its mean price) minus the known values is the value of the missing kind. With a ratio 2 : 1 : 3, take 2 kg, 1 kg and 3 kg.
Fast route (balance): the amounts above and below the mean cancel: $q_1(p_1 - m) + q_2(p_2 - m) = 0$ . With a convenient mean this is quicker than the totals.
Change every price to the same unit first: ₹250 for 5 kg is ₹50 per kg.
Option check: the mean must lie between the lowest and highest price.

Mean price

m = \frac{q_1p_1 + q_2p_2}{q_1 + q_2} \qquad q_1(p_1 - m) + q_2(p_2 - m) = 0

qthe quantity of each kind
pits price per unit
mthe mean price of the mixture

Worked example

6 kg of tea at ₹180 per kg is mixed with 4 kg of tea at ₹230 per kg. What is the price per kg of the mixture?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 3 · Q65Easy

Example 1 · Ratio, Proportion and Variation · Mixtures and Alligation

A shopkeeper mixes two qualities of tea. The first quality costs ₹150 per kg and is mixed with the second quality in the ratio

4 : 5

. If the mixture is sold at ₹130 per kg (at no profit, no loss), what is the cost per kg of the second quality of tea?

Averaging the prices without the quantities

6 kg at ₹180 and 4 kg at ₹230 do not average ₹205. The mixture has more of the cheaper tea, so its price is ₹200, nearer ₹180.

Mixing price units

A price given per 5 kg must be turned into a price per kg before it meets prices per kg. ₹360 for 5 kg is ₹72 per kg.

Drill 14 more on mean price of a mixture

Concept 2 of 3: The alligation cross for a mixing ratio

The mean price is a balance point between the two prices. The kind whose price is closer to the mean must be present in a larger amount to hold the balance there, so each kind's share is the OTHER kind's distance from the mean.

Definition

For a cheaper kind at $p_1$ , a dearer kind at $p_2$ and a mean m between them: cheaper : dearer $= (p_2 - m) : (m - p_1)$ .
Write the two prices at the top, the mean in the middle, and subtract across the diagonals.
It works for any weighted average of two kinds: prices, strengths of solutions, rates of profit. Water counts as price 0.
Once the ratio is known, scale it to the quantity given in the stem.
It is a two-kind rule. With three kinds, go back to totals.

Rule of alligation

\frac{q_{\text{cheaper}}}{q_{\text{dearer}}} = \frac{p_2 - m}{m - p_1}

p_1, p_2the cheaper and the dearer price
mthe mean price, between $p_1$ and $p_2$

Worked example

Coffee at ₹520 per kg is mixed with coffee at ₹400 per kg to give a mixture worth ₹450 per kg. In what ratio are they mixed, and how much of the cheaper kind goes with 10 kg of the dearer?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 3 · Q59Easy

Example 2 · Ratio, Proportion and Variation · Mixtures and Alligation

How much rice of ₹4 per kg should be mixed to 15 kg of rice of ₹10 per kg, so as to make a mixture worth ₹6.50 per kg?

Giving each kind its own distance

In the cross, the cheaper kind's share is the DEARER price minus the mean. If the mean is close to the cheaper price, the cheaper kind is the larger part.

A mean outside the two prices

The mean of a two-kind mixture must lie between the two prices. A mean outside them means a price was misread or a selling price was used where the cost price belongs.

Drill 3 more on the alligation cross for a mixing ratio

Concept 3 of 3: Mixtures sold at a profit or a loss

Alligation balances COSTS. When the stem gives the price the mixture is sold at and the profit made, that price is not the mixture's cost, so the cost price must be recovered first; only then does it belong in the middle of the cross.

Definition

Cost price of the mixture: $CP = \frac{SP}{1 + \frac{r}{100}}$ for a profit of r%, and $\frac{SP}{1 - \frac{r}{100}}$ for a loss of r%.
Mixing ratio: put that CP in the middle of the alligation cross.
Gain or loss % of a mixture: find the mean cost first, then compare the selling price with it.
A missing price or quantity: use the CP as the mean price and work as on the mean-price concept.
Fast route: a 20% profit means $SP = \frac{6}{5}CP$ , so CP is $\frac{5}{6}$ of SP. Common fractions: 10% is $\frac{11}{10}$ , 25% is $\frac{5}{4}$ .

Cost price of the mixture

CP = \frac{100 \times SP}{100 + r} \qquad \text{gain \%} = \frac{SP - CP}{CP} \times 100

SPthe selling price per unit
rthe profit percentage

Worked example

Rice at ₹48 and ₹68 per kg is mixed and the mixture is sold at ₹70 per kg for a 25% profit. In what ratio were the two kinds mixed?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 2 · Q61Moderate

Example 3 · Ratio, Proportion and Variation · Mixtures and Alligation

A shopkeeper has two types of pulses: one costing ₹60/kg and the other costing ₹90/kg. He mixes them in a specific ratio and sells the mixture at ₹84/kg, thereby earning a

20\%

profit. In what ratio did he mix the two types of pulses?

Putting the selling price in the cross

A mixture of ₹48 and ₹68 rice sold at ₹70 cannot use 70 in the cross: it lies above both prices. The cross needs the cost price, here ₹56.

Taking the profit off the selling price

A 25% profit means SP is 1.25 times CP, so

CP = SP \div 1.25

. Taking 25% off the selling price (

SP \times 0.75

) gives a cost that is too low.

Drill 10 more on mixtures sold at a profit or a loss

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Mean price of a mixture
Mean price
$m = \frac{q_1p_1 + q_2p_2}{q_1 + q_2} \qquad q_1(p_1 - m) + q_2(p_2 - m) = 0$
The alligation cross for a mixing ratio
Rule of alligation
$\frac{q_{\text{cheaper}}}{q_{\text{dearer}}} = \frac{p_2 - m}{m - p_1}$
Mixtures sold at a profit or a loss
Cost price of the mixture
$CP = \frac{100 \times SP}{100 + r} \qquad \text{gain \%} = \frac{SP - CP}{CP} \times 100$

Watch out for (6)

Averaging the prices without the quantities
→ Mean price of a mixture
Mixing price units
→ Mean price of a mixture
Giving each kind its own distance
→ The alligation cross for a mixing ratio
A mean outside the two prices
→ The alligation cross for a mixing ratio
Putting the selling price in the cross
→ Mixtures sold at a profit or a loss
Taking the profit off the selling price
→ Mixtures sold at a profit or a loss

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