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Mixture problems
Acid and water, juice and soda, two kinds of coffee beans: mixture problems freeze a lot of test-takers, but they all follow one line of arithmetic. Track the pure part, and the rest is adding.
Track the pure part
| Liters | Juice % | Pure juice | |
|---|---|---|---|
| Drink A | 6 | 10% | 0.6 L |
| Drink B | 4 | 35% | 1.4 L |
| Mix | 10 | ? | 2 L |
Every mixture has two things to count: the total amount and the pure part inside it (the juice, the acid, the salt). The pure part of each ingredient is its amount × its percent. When you pour them together, the amounts add and the pure parts add. Here: 2 L of juice in 10 L of drink, so the mix is 2 ÷ 10 = 20% juice.
The mix lands in between
The mixture is always weaker than the strongest ingredient and stronger than the weakest. And it sits closer to the ingredient you used more of: 20% is closer to 10% because there were 6 liters of the 10% drink.
How to solve them
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1
Amount × percent = the pure part
The percent tells you how much of every unit is pure. Multiply it by the amount to get the pure part in liters, ounces or pounds.
pure part = amount × percent
Next step: Mixing twoExample8 liters of a 15% salt solution. How much of it is salt?
- 15% = 0.15.
- 8 × 0.15 = 1.2.
Answer 1.2 liters of salt
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2
Mixing two: add the amounts, add the pure parts
Find the pure part of each ingredient, add them, and divide by the total amount. Never average the two percents unless the amounts are equal.
Next step: Find the amountExample2 liters of a 10% solution and 8 liters of a 30% solution. What percent is the mix?
- Pure parts: 2 × 0.10 = 0.2 and 8 × 0.30 = 2.4.
- Together: 2.6 liters pure in 10 liters.
- 2.6 ÷ 10 = 0.26 = 26%, not 20%.
Answer 26%
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3
How much to add? One equation
Call the unknown amount x. The pure parts going in must equal the pure part of the mix. Use whole percents (15, not 0.15): it is the same equation with fewer decimals.
a × p + x × q = (a + x) × r
Next step: Water and pureExampleHow many liters of a 45% solution must be added to 9 liters of a 15% solution to get a 25% solution?
- 9 × 15 + x × 45 = (9 + x) × 25.
- 135 + 45x = 225 + 25x, so 20x = 90.
- x = 90 ÷ 20 = 4.5.
Answer 4.5 liters
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4
Water is 0%; pure chemical is 100%
Adding water makes the mix bigger without adding any pure part. Adding pure chemical adds its whole amount to the pure part.
Next step: Price blendsExampleHow much water must be added to 6 liters of a 40% solution to make it 30%?
- Pure part: 6 × 40 = 240 (in liter-percents). Water adds none.
- 240 = (6 + x) × 30, so 6 + x = 8.
- x = 2.
Answer 2 liters of water
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5
The same method works for prices
Blending two products at different prices is a mixture too: pounds × price per pound gives the cost of each part. Add the costs, then divide by the total pounds.
Next step: The in-between checkExample3 pounds of nuts at $8 a pound are mixed with 5 pounds at $4 a pound. What is the price per pound of the mix?
- Costs: 3 × 8 = 24 dollars and 5 × 4 = 20 dollars.
- Together: 44 dollars for 8 pounds.
- 44 ÷ 8 = 5.50.
Answer $5.50 a pound
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6
Check that the answer lands in between
Before you choose, check the result against the two ingredients. It costs two seconds and throws out most wrong answers.
BetweenA 10% and a 35% drink, mixed
The mix is between 10% and 35%
Nearer the bigger amount6 L at 10% with 4 L at 35%
20%: closer to 10%
Equal amountsThe same amount of 20% and 40%
Exactly halfway: 30%
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Averaging the percents
2 L of a 10% solution and 8 L of a 30% solution:
Wrong(10 + 30) ÷ 2 = 20%
Right(0.2 + 2.4) ÷ 10 = 26%
Treating water as 100%
Adding x liters of water:
Wrongpure part x × 100
Rightpure part x × 0 = 0
Forgetting that the mix grows
5 L of 40% plus x liters of water makes 25%:
Wrong5 × 40 = 25x
Right5 × 40 = 25(5 + x) → x = 3
Percent of the wrong total
3 L of pure juice in 12 L of drink:
Wrong3 ÷ 9 ≈ 33%
Right3 ÷ 12 = 25%
5 questions at test level
Pick an answer, then read why. No timer, no account.
How many ounces of pure alcohol are in 20 ounces of a 35% alcohol solution?
Show the answer
C. Pure part = amount × percent = 20 × 0.35 = 7. 13 is the part that is not alcohol, and 5.7 divides 20 by 3.5. There are 7 ounces.
Four liters of a 25% solution are mixed with six liters of a 50% solution. What is the percent of the mixture?
Show the answer
B. Pure parts: 4 × 25 = 100 and 6 × 50 = 300 (liter-percents), so 400 in 10 liters, and 400 ÷ 10 = 40. 37.5% averages the two percents, which ignores that there is more of the 50% solution. The mixture is 40%.
How many liters of water must be added to 12 liters of a 30% solution to make a 20% solution?
Show the answer
C. The pure part stays 12 × 30 = 360. The new mix has 360 = 20 × (12 + x), so 12 + x = 18 and x = 6. 18 is the size of the new mix, not the water added. Add 6 liters.
A grocer mixes 6 pounds of coffee at $9 a pound with 4 pounds at $14 a pound. What is the price per pound of the mixture?
Show the answer
A. Costs: 6 × 9 = 54 dollars and 4 × 14 = 56 dollars, so 110 dollars for 10 pounds, and 110 ÷ 10 = 11. $11.50 averages the two prices as if the amounts were equal. The mixture costs $11.00.
How many liters of a 70% solution must be added to 6 liters of a 20% solution to get a 40% solution?
Show the answer
B. 6 × 20 + 70x = 40(6 + x), so 120 + 70x = 240 + 40x. Then 30x = 120 and x = 4. Check: 120 + 280 = 400, and 400 ÷ 10 = 40. Add 4 liters.
9 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 9 cards known.
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