Mixtures and Alligations: The Cross Method and Replacement

The alligation rule

When two ingredients with values (prices, concentrations, averages) d and c are mixed to reach a mean value m, with c < m < d:

quantity of cheaper : quantity of dearer = (d − m) : (m − c)

The cross diagram makes it quick:

   cheaper c       dearer d
          \         /
             mean m
          /         \
      d − m       m − c

Example. Rice at ₹40 and ₹55 per kg mixed to get ₹45: (55 − 45) : (45 − 40) = 10 : 5 = 2 : 1.

It works for any average. Boys averaging 70 and girls averaging 80 give a class average of 74, so boys : girls = (80 − 74) : (74 − 70) = 3 : 2.

Mixing in a given ratio

Weighted average of the parts. Tea at ₹120 and ₹180 mixed 2 : 1 costs (240 + 180)/3 = ₹140 per kg.

Changing a mixture: hold one part fixed

Adding water doesn't change the amount of milk; adding pure acid doesn't change the amount of water.

Example. 40 L of milk and water in the ratio 3 : 1 has 30 L milk and 10 L water. For a 3 : 2 ratio, water must be 20 L, so add 10 L of water.

Example. 60 L of solution is 20% alcohol (12 L). To make it 25% alcohol by adding x litres of pure alcohol: (12 + x)/(60 + x) = 0.25 gives x = 4 L.

Repeated replacement

If x litres are removed from a vessel holding V litres of liquid and replaced with water, n times over, the original liquid left is

V × (1 − x/V)ⁿ

Example. 80 L of milk, 8 L removed and replaced by water twice: 80 × 0.9² = 64.8 L of milk.

Mixing for profit

To sell at the cost price of pure milk and still make a profit, mix in water so the cost of the mixture is lower.

Example. Milk costs ₹50/L. Selling the mixture at ₹50 with a 25% profit means the mixture must cost 50/1.25 = ₹40 per litre. By alligation with water (₹0): milk : water = 40 : 10 = 4 : 1.

Alloys

Compare the fraction of one metal. Equal amounts of alloys with copper : zinc = 2 : 3 (copper 0.4) and 3 : 7 (copper 0.3) give copper 0.35, so copper : zinc = 7 : 13.

Shortcuts and tricks

The standard way is to write an equation for the quantity that matters (cost, milk, acid). Alligation and the "fixed part" idea skip the algebra.

Alligation

Example: tea at ₹60/kg and ₹90/kg is mixed to make tea worth ₹70/kg. Find the ratio.

• Standard way: 60x + 90y = 70(x + y) gives 20y = 10x, so x : y = 2 : 1.

• Shortcut: cheaper : dearer = (90 − 70) : (70 − 60) = 20 : 10 = 2 : 1.

Hold the unchanged part fixed

Example: 50 L of milk and water in the ratio 4 : 1. How much water makes it 2 : 1?

• Standard way: there is 40 L of milk and 10 L of water. (10 + x)/40 = 1/2 gives x = 10 L.

• Shortcut: milk stays at 4 parts, and 2 : 1 = 4 : 2, so water goes from 1 part to 2 parts. One part is 10 L, so add 10 L.

Repeated replacement

Example: 4 L is taken from 40 L of milk and replaced with water, twice.

• Standard way: after the first step there is 36 L of milk. The second 4 L removed is 90% milk, 3.6 L, leaving 32.4 L.

• Shortcut: milk left = 40 × (1 − 4/40)² = 40 × 0.81 = 32.4 L.

Profit from free water

Example: a seller adds water and sells at the cost price of milk for a 20% profit.

• Standard way: the mixture must cost 100/120 of the milk price, then use alligation with water at 0.

• Shortcut: the profit comes from the water, so water is 20% of the milk: milk : water = 100 : 20 = 5 : 1.

Exam tips

• Always put the mean in the middle of the cross; the ratio comes out in cheaper : dearer order.

• With replacement problems, track the original liquid, not the water.

• Express every concentration as a fraction of the same total.