Concentration — finding it, diluting and mixing

Salt is at the top in the middle, water at the bottom left. Whole solution, bottom middle, is the two added together. Concentration, top right, divides the salt by the whole solution and multiplies by 100. The card top left is there for comparison: it divides the salt by the water alone.

Changing the salt or the water updates the whole solution and the concentration.

The divisor is the whole solution, not the water

Concentration says what percentage of the whole solution is salt. Here 20 g of the 200 g is salt, so it is 10%. As a formula:

concentration (%) = salt ÷ (salt + water) × 100

Dividing by the 180 g of water instead gives 11.1%, the card top left. It is the salt measured against the water, which is a different number from the concentration.

The weaker the solution, the smaller the gap between the two. Changing the numbers on Board 1:

• 15 g salt, 485 g water (500 g in all) — concentration 3.0%; divided by water, 3.1%
• 20 g salt, 180 g water (200 g in all) — concentration 10.0%; divided by water, 11.1%
• 40 g salt, 180 g water (220 g in all) — concentration 18.2%; divided by water, 22.2%

The first line is the make-up of 500 g of a 3% solution: the salt is 3% of 500 g, which is 15 g, and the water is the remaining 485 g. With 15 g of salt put into 500 g of water instead, the whole is 515 g, so the concentration comes out at 2.9%.

The third line doubles the salt from 20 g to 40 g. The concentration does not double to 20%; it is 18.2%. Adding salt also adds the same amount to the whole it is divided by.

Diluting with water: the salt stays the same

Adding water does not change how much salt is dissolved. Only the weight of the whole changes. On Board 1, with the water raised from 180 g to 280 g (the same as adding 100 g of water), the salt is still 20 g, the whole becomes 300 g, and the concentration drops to 6.7%.

The other way round, how much water brings it down to a given concentration, is on Board 2.

Top left is the starting concentration (Solution %); bottom left is the weight of the starting solution. With the target concentration entered top right, the water to add appears in the middle of the bottom row. The example dilutes 200 g of a 10% solution to 4%, in three steps:

1. Salt in it — 10% of 200 g is 20 g. Diluting does not change these 20 g
2. Diluted total — the weight of which 20 g is 4%: 20 ÷ 4 × 100 = 500 g
3. Water to add — 500 g minus the starting 200 g, which is 300 g

Step 2 is the concentration formula solved for the whole. Concentration = salt ÷ whole × 100, so whole = salt ÷ concentration × 100. The salt is fixed, so the target concentration fixes the weight of the whole.

Halving the concentration to 5% doubles the whole to 400 g, so the water to add is 200 g. Dividing the concentration by some number means multiplying the whole by the same number.

Mixing two solutions: the salt and the whole add up separately

Board 3 calculates the concentration of 300 g of a 10% solution mixed with 200 g of a 5% solution.

The top and bottom of the left column are the weights, and the top and bottom of the right column the concentrations. The top and bottom of the middle column are the salt in each: 300 g at 10% holds 30 g of salt, 200 g at 5% holds 10 g.

After mixing, the salt is 30 + 10 = 40 g (Total salt, middle right) and the whole is 300 + 200 = 500 g (Mixture total, middle left). Mixture %, in the centre, is 40 ÷ 500 × 100, which is 8.0%.

That is not 7.5%, the average of 10% and 5%. There is more of the 10% solution, so the result sits closer to 10%. A concentration is the result of a division, so concentrations cannot simply be added or averaged. What adds up is the weight of salt and the weight of the whole.

With both weights on Board 3 set to 250 g, the result is 7.5%. The plain average of two concentrations only comes out for equal weights.

With B % set to 0, the mix is 300 g of the 10% solution and 200 g of water: 6.0%. Water fits into the same calculation as a 0% solution.

How much of each to mix for a target concentration

The reverse question — how much of the 10% and the 5% solution make 500 g at 8% — is answered by the cards on Board 4.

The card on the left has four fields: "A conc. %", "B conc. %", "Target %" and "Amount". The line underneath reads "A 300 + B 200": 300 g of the 10% solution and 200 g of the 5% one, the same pair as on Board 3.

The split comes from the differences in concentration. 8% is 2 below 10% and 3 above 5%.

• Every gram of the 10% solution brings 0.02 g more salt than 8% would need
• Every gram of the 5% solution brings 0.03 g less salt than 8% would need

When the surplus and the shortfall cancel exactly, the mixture is 8%. A × 2 = B × 3 holds when A and B are in the ratio 3 to 2, and 500 g split 3 to 2 is 300 g and 200 g.

More is used of the solution closer to the target, and the amounts are in the inverse ratio of the differences. As a formula:

amount of A = amount × (target − B %) ÷ (A % − B %)

Here that is 500 × (8 − 5) ÷ (10 − 5) = 300 g.

The card on the right sets A to 0%, which is water. Making 500 g at 4% from a 10% solution gives "A 300 + B 200": 300 g of water and 200 g of the 10% solution. That is the answer from Board 2.

A mixture can only land between the two concentrations. With "Target %" on the left card set to 12, the line shows "—": no mix of 10% and 5% is stronger than 10%.

With both concentrations and the target set to 8, the card says "Any ratio gives 8%". That is not "impossible"; it means the amounts are not pinned down to one answer.

Other numbers

Board 2, starting from 200 g of a 10% solution, with different targets:

Getting to one tenth, 1%, takes ten times the weight, 2,000 g. The water to add is nine times the starting solution, 1,800 g.

The water needed is not proportional to how far the concentration drops. From 10% to 5% takes 200 g; from 5% down to 1% takes another 1,600 g.

Board 4, left card, keeping the amount at 500 and changing "Target %":

The closer the target is to 10%, the more of A. At 7.5%, halfway between the two, the amounts are equal. The 6% pair can be checked on Board 3: with A amount set to 100 g and B amount to 400 g, the mixture is 6.0%.

On the right card, a target of 2 gives "A 400 + B 100" and a target of 1 gives "A 450 + B 50". So 100 g of the 10% solution plus 400 g of water makes 500 g at 2%.

What this calculation covers, and what it does not

Concentration on this page is percent by mass: weights are divided. In a kitchen, 1 mL of water weighs close to 1 g, so a measuring jug marked in mL works for water in grams. For other liquids, mL and g are not the same.

The alcohol strength on a drinks label (ABV) is percent by volume. Mixing alcohol and water gives slightly less volume than the two added together, so the "add up the whole" step on this page does not carry over to volume percentages as it is.

Salt only dissolves up to a limit: about 36 g in 100 g of water at 20 °C. On Board 1, 36 g of salt and 100 g of water gives 26.5%: at 20 °C, this corresponds to a saturated solution of about 26.5% by mass. The board does not include this limit, so more salt keeps giving higher numbers that cannot actually be made.

Read next — Scaling a recipe — the factor from servings or from an ingredient It calculates one factor from the number of servings or from one ingredient, multiplies every ingredient by it, and shows why that keeps the proportions the same.

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