Mixture Problems

Mixture problems come in a few disguises — acid solutions, coffee blends, nut mixes, coin jars, investment splits — but they are all the same problem. Something is combined from parts of different strength or value, and you track one quantity across the combination: the pure amount of a substance, or the total dollar value.

The Word Problems five-step method still applies. What this lesson adds is a chart, much like the one in Distance, Rate, and Time, that makes the “before equals after” balance almost fill itself in.

The Mixture Formula

and are the quantities (volumes, weights, counts) of the two parts, and are their concentrations or unit values, and is the final concentration or value of the whole mixture. Each product is the pure amount contributed by that part. The equation says the pure amounts add up.

The Mixture Chart

The Pure-amount column adds down: the two ingredient rows sum to the mixture row. That is the equation.

Percent-Solution Mixtures

Two solutions of different concentration are combined. Concentrations are decimals: .

The final concentration must be between the two starting concentrations, or no positive amount works.

Adding Pure Substance or Pure Water

• Pure water dilutes: concentration , so its term is , but it still adds to the total quantity.
• Pure substance (100% acid, salt, alcohol) strengthens: concentration , so its term is just its amount.

Dry Mixtures with Prices

The same structure, with “value per unit” in place of concentration. A blend of /lb and /lb coffee:

where is the target price per pound of the blend.

Coin and Ticket Problems

The “concentration” is the value of each item. For a jar of nickels and dimes with nickels and dimes:

Write in terms of using the count equation, substitute into the value equation, and solve one linear equation.

Worked Example A: Two Acid Solutions

How many liters of a acid solution must be added to liters of a acid solution to make a solution?

Chart — let be the liters of solution:

Equation:

Answer: liters of the solution. Check: L of acid, and L. ✓

Worked Example B: Diluting with Water

A chemist has liters of a alcohol solution. How much pure water must be added to dilute it to ?

Let be the liters of water. Water contributes alcohol.

Answer: add liters of water. Check: L of alcohol in L total is . ✓

Worked Example C: A Coffee Blend

A shop wants pounds of a blend worth per pound, mixing a /lb coffee with a /lb coffee. How many pounds of each?

Let be the pounds of coffee; then is the coffee.

Answer: lb of the coffee and lb of the coffee.

Worked Example D: A Coin Problem

A jar has coins, all quarters and dimes, worth . How many of each?

Let be the number of quarters; then is the number of dimes.

Answer: quarters and dimes. Check: . ✓

Common Mistakes to Avoid

• Using percents instead of decimals. is in the equation.
• Forgetting that water or a diluent still adds to the total quantity. Its concentration term is , but the mixture row’s quantity is still .
• Setting the final concentration outside the range of the two ingredients. Mixing and can only produce something between and .
• Mislabeling the second quantity. If the total is fixed at and one part is , the other is , not another free variable.
• Adding concentrations directly. ; you multiply each concentration by its amount first.
• Dropping a unit-value term in a coin problem. Every coin type contributes (count)(value); a missing term throws off the total.

Where This Shows Up Later

• Systems of Equations. Coin, ticket, and investment problems with two genuinely independent unknowns are naturally two-equation systems (count and value).
• Applications of Linear Equations. Simple-interest “split investment” problems are mixture problems where the “concentration” is an interest rate.
• Rational Equations. A few dilution problems that ask for a final concentration after repeated dilution lead to fractional equations.
• Weighted averages and statistics. The mixture balance is exactly a weighted average: the final concentration is the amount-weighted mean of the parts.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. How many liters of a acid solution should be mixed with liters of a solution to get a solution?

Problem 2. How much pure water must be added to liters of a salt solution to make it ?

Problem 3. A nut mixture worth /lb is made from cashews at /lb and peanuts at /lb. For lb of mixture, how much of each?

Problem 4. A jar of nickels and dimes has coins worth . How many of each?

Problem 5. How much pure antifreeze must be added to quarts of a antifreeze mixture to raise it to ?

Problem 6. A and a fertilizer solution are combined to make gallons of a solution. How many gallons of each?

Problem 7. A theater sells tickets, adult at and child at , for total receipts of . How many of each?

Problem 8. How many liters of pure water must evaporate from liters of a salt solution to make it ? (Salt stays; water leaves.)

Problem 9. A grocer mixes /lb candy with /lb candy to make lb worth /lb. How much of each?

Problem 10. is split between an account paying and one paying . The total annual interest is . How much is in each account?

Problem 11. How much of an copper alloy must be melted with kg of a copper alloy to produce a copper alloy?

Problem 12. A jar has three times as many dimes as quarters and is worth . How many of each?

Quick Reference

The decimals-and-parentheses algebra is Linear Equations; if you prefer to clear the decimals first, that is the Linear Equations with Fractions move. Split-investment problems overlap with the interest work in Applications of Linear Equations, and the chart is the same one from Distance, Rate, and Time. The rest of the Algebra lessons are there when you want more.