Question type

Mixture problems on the SAT

In a mixture problem the amount of the substance is conserved, not the percent: $c_{1} v_{1} + c_{2} v_{2} = c (v_{1} + v_{2})$ . Three liters at 12% and five at 20% hold $0.36 + 1.00 = 1.36$ liters of juice in 8 liters, which is 17%. Averaging 12% and 20% gives 16%, which is right only when the amounts are equal.

These questions mix two solutions, alloys or blends with different concentrations, or add water to dilute one, and ask for an amount or the resulting concentration.

The standard version is Medium and the Hard version puts the unknown on both sides of the equation. There is no Easy version.

Desmos helps

Why students miss it

Averaging the percents. The mixture's percent is a weighted average: it leans towards the larger amount.

Forgetting that adding water changes the total but not the amount of acid, salt or juice.

In evaporation questions, the salt stays and only the water leaves.

How to spot it

Two percentages "by volume" or "by mass" being combined.
"How many liters must be added", "evaporate", "pure water".
The same structure in money: two ticket prices and an average takings per ticket.

The method

Track the substance

For each part, amount of substance = concentration × total.

Combining two parts: $c_{1} v_{1} + c_{2} v_{2} = c (v_{1} + v_{2})$ .
Adding pure water: substance unchanged, total grows.
Evaporating water: substance unchanged, total shrinks.

With Desmos

Once the equation is written, Desmos solves it for the unknown amount.

Type the balance equation, for example 0.08*10+0.2x=0.12(10+x).
The solution appears as a vertical line at x = 5.

Desmos solves the equation you give it. The setup, substance on both sides, is the part to get right.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#994Data

A café has 10 gallons of lemonade that is 8% syrup by volume. How many gallons of lemonade that is 20% syrup by volume must be added so that the resulting lemonade is 12% syrup by volume?

Pick an answer, then check it.

Show the answer and explanation

Balance the syrup. Adding

x

gallons of 20% gives total volume

10 + x

0.08 (10) + 0.20 x = 0.12 (10 + x)

0.8 + 0.20 x = 1.2 + 0.12 x

0.08 x = 0.4

, so

x = 5

gallons.

2Question 2 of 2#991Data

A laboratory has 8 liters of a solution that is 45% acid by volume. How many liters of pure water must be added so that the resulting solution is 30% acid by volume?

Pick an answer, then check it.

Show the answer and explanation

Adding water changes the total but not the acid. Acid stays at

0.45 (8) = 3.6

L, and you need

3.6 = 0.30 (8 + x)

. So

8 + x = 12

, giving

x = 4

liters of water.

Questions students ask

How do you solve mixture problems on the SAT?

Write the amount of the substance in each part as concentration × total, add them, and set the sum equal to the final concentration × the final total.

Why isn't the answer the average of the two percentages?

Because the parts are different sizes. The result is a weighted average and is closer to the concentration of the larger part.

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