Mixing Two Concentrations | The Cross Method (Alligation) and Working Back to the Concentration

Finding the mixing ratio with the cross method (alligation)

When you mix a liquid of concentration A with a liquid of concentration B to get a concentration T between them, you find the mixing ratio by crossing over the “differences from the target”. Because the numbers are written in an X shape, this is called the cross method; because you can think of it as balancing a pair of scales, it is also known as the balance method or alligation.

1. 1

   Line up the three concentrations

   Say you want to make 300 mL of 6% (T) from a 10% liquid (A) and a 4% liquid (B).

2. 2

   Cross the differences

   A’s share is T − B = 6 − 4 = 2, and B’s share is A − T = 10 − 6 = 4. The ratio is 2 : 4 = 1 : 2.

3. 3

   Split the total by the ratio

   Divide 300 mL in the ratio 1 : 2 and mix 100 mL of the 10% liquid with 200 mL of the 4% liquid.

4. 4

   Check the answer

   The amount of active ingredient is 100 × 10% + 200 × 4% = 10 + 8 = 18 mL. 18 ÷ 300 = 6%, so it checks out.

You can only make a target concentration that lies between the two liquids. With 10% and 4% liquids, no mix will ever go below 4% or above 10%. To make it stronger, you need a stronger liquid.

Diluting with water is the same as mixing with a “0% liquid”

Water can be treated as a liquid with 0% concentration, so diluting with water can also be written as a cross-method calculation. To take 6% bleach down to 0.05% with water, the bleach’s share is 0.05 − 0 = 0.05 and the water’s share is 6 − 0.05 = 5.95, a ratio of 0.05 : 5.95 = 1 : 119 (120-fold). For 500 mL, that is about 4.2 mL of bleach and about 495.8 mL of water.

If you are only diluting with water, thinking in terms of a dilution factor is simpler (How to Calculate Dilution Ratios). The cross method earns its keep when you mix two liquids neither of which is water, such as correcting a liquid that is too strong with a weaker one, or mixing anhydrous ethanol with 40% spirit. Enter the two concentrations and the target in the tool’s “Mix two liquids” and it gives the ratio and amounts straight away.

Working back to the concentration of a mixed liquid

To find what concentration a liquid you have already mixed ended up at, add up the amounts of active ingredient and divide by the total amount.

• Mixing 200 g of 8% salt solution with 300 g of 3% salt solution gives 16 + 9 = 25 g of salt in 500 g in total, so 25 ÷ 500 = 5%
• Adding 10 mL of 6% bleach to 990 mL of water gives 6 × 10 ÷ 1000 = 0.06% (600 ppm), a 100-fold dilution
• 70 mL of 99.5% ethanol with 30 mL of water gives 99.5 × 70 ÷ 100 = about 69.7%

This is handy for checking that a disinfectant or liquid fertiliser you mixed by eye isn’t too strong. The tool’s “Check a solution’s concentration” works back from the stock concentration and the amounts of stock and water you added to the final concentration, ppm and dilution factor.

Converting between % and ppm

ppm means parts per million, and 1% = 10,000 ppm. Writing very low concentrations in % produces strings of zeros after the decimal point that are easy to misread, so ppm is widely used for disinfectants and water quality.

To go from % to ppm multiply by 10,000; from ppm to % divide by 10,000. The cross method and the back-calculation formula give the same result whether you work in % or convert everything to ppm. How to make chlorine disinfectant by ppm is covered in How to Dilute Sodium Hypochlorite.

Don’t confuse percent by weight with percent by volume

A percentage concentration can be a proportion by weight (w/w%, mass percent) or by volume (v/v%, vol%). Salt-solution problems and sugar water normally use weight %, while ethanol uses volume %. Apply the cross-method ratio to amounts in the same unit as the original percentages (grams for weight %, mL for volume %).

• Measuring a salt-solution ratio in mL is slightly off because salt solution is heavier than water. For accuracy, weigh it in grams
• For dilute water solutions such as bleach and liquid fertiliser, the difference between weight and volume is tiny, so measuring in mL gives almost the same result
• Ethanol and water shrink in volume when mixed, so vol% calculations are slightly off (see How to Dilute Anhydrous Ethanol to 70%)

For making disinfectant or liquid fertiliser at home, measuring and calculating in mL is fine. Only pay attention to matching the units when an exam question or experiment calls for precision. Converting between molarity (mol/L), commonly used in the lab, and mass percent, and how to make a 0.1 mol/L solution, are explained in How to Calculate Molecular Weight and Moles.

FAQ

How do I find the ratio for mixing two liquids of different concentrations?

Use the cross method (alligation): amount of strong liquid : amount of weak liquid = (target − weak concentration) : (strong concentration − target). To make 6% from 10% and 4%, it is (6−4) : (10−6) = 1 : 2.

How do I calculate the concentration after mixing?

Add up the amount of active ingredient in each liquid (amount × concentration) and divide by the total amount. For 200 g of 8% salt solution and 300 g of 3% salt solution, (16 + 9) ÷ 500 = 5%.

Can I make a 12% liquid from 10% and 4% liquids?

No. The concentration of a mixture always lies between the two liquids (here, 4 to 10%). To get 12%, you need a liquid stronger than 12%.

What percentage is 1 ppm?

1% = 10,000 ppm, so 1 ppm is 0.0001%. 500 ppm is 0.05% and 1000 ppm is 0.1%.

Can I use the cross method when diluting with water?

Yes. Treating water as a 0% liquid, the ratio for taking a 6% stock to 0.05% comes out as 0.05 : 5.95 = 1 : 119 (120-fold). But if you are only diluting with water, thinking in terms of a dilution factor (stock concentration ÷ target concentration) is simpler.