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Mixture Problems

Lesson Plans > Mathematics > Algebra > Word Problems
 

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I have $3.12 in pennies and dimes. I have 78 coins in all. How many are pennies?

I have three more dimes than nickels. In all I have $1.50. How many coins do I have in all?

I have a can of nuts that is 45% peanuts. I have another can that is 20% peanuts. If I want to have 5 cups of nuts which contains 30% peanuts, how many of each can should I use?

Bob's milk jug has 1 cup of cream for every 10 cups of milk. Sam's milk just has 1 cup of cream for every 6 cups of milk. If I want to have a container with 0.5 cups of cream and 3.5 cups of milk, how much should I take from each jug?

I have three times as many 20 dollar bills as 5 dollar bills. In all, I have $390. How many 20 dollar bills do I have?

I have ten more chickens than pigs. The animals have a total of 44 legs. How many of each are there?

I have drawn two more pentagons than squares. In all, I have drawn 91 vertices. How many shapes have I drawn in all?

When the aliens invaded earth, 60% of the aliens had four eyes, and 40% had six eyes. In all, there were 576 eyes. How many aliens were there in all?

N of my students have three textbooks. 2N have four textbooks. Three students have five textbooks. In all there are 180 textbooks. How many students do I have?

In my classroom, some of my students have two pencils, and some have three pencils. There are 30 students in all, and 86 pencils. How many students have three pencils?

Solution A is 5% salt, and Solution B is 15% salt. If there are 5 gallons of solution A, how much of solution B needs to be added to it in order to make a 7.5% salt solution?

Lesson Plan/Article

Mixture Problems

This lesson contains several worksheets you can use in your classroom to give your students extra practice with different types of mixture problems. Before assigning them, you may wish to do some of the practice problems in the slideshow.

I remember the first time I was introduced to "mixture" problems in high school algebra class - I thought they were very confusing and difficult. I think most - if not all - my classmates felt the same way. Mixture problems can be very difficult, because they typically involve holding two types of quantities in your mind at the same time. For example, if you're talking about a mixture of coins, you're talking about the number of each kind of coin, but you're also talking about the value of each kind of coin.

This kind of problem may benefit from setting up a table of values. As in the example below:

Example: I have 20 coins. Some of them are dimes and some are quarters. I have a total of $3.95. How many quarters do I have?

Solution: Set up a table with rows representing types of coins, and columns representing number of coins and value of coins.

number of coins
value of coins
quarters
q
0.25q
dimes
20 - q
0.1(20 - q)

Now you can use the values column to write an equation for the total: 

0.25q + 0.1(20 - q) = 3.95, which leads to q = 13.

The same process works if it's a mixture of animals in a field:

Example: I have ten more chickens than pigs. The animals have a total of 44 legs. How many of each are there?

Solution: Again, set up a table. This time with columns "number of animals" and "number of legs"

number of animals
number of legs
pigs
p
4p
chickens
p + 10
2(p + 10)

Thus, 4p + 2(p + 10) = 44, which gives 4 pigs and 14 chickens.

Similarly, if you're talking about solutions (for example, a saline solution), you're talking about the total amount of liquid, but you're also talking about the amount of solute. Even though you don't have more variables, adding these extra types of quantities makes it much more complicated.

Example: Solution A is 5% salt, and Solution B is 15% salt. If there are 5 gallons of solution A, how much of solution B needs to be added to it in order to make a 7.5% salt solution?

Solution: Create a table for each solution.

Amount of solution
Amount of salt
Solution A
5
0.05(5)
Solution B
x
0.15x
New solution
5 + x
0.25 + 0.15x

Since the new solution must be 10% salt, we can write an equation: 0.075(5 + x) = 0.25 + 0.15x, which leads to x=
5
3
gallons.

In the same way, problems in which someone invests money in two different bank accounts with different interest rates is a type of mixture prooblem.

The worksheets here are designed to give students more practice if they're struggling with these types of problems.

I've split it up into similar types of problems on each worksheet, starting with a worksheet of coin problems, a worksheet of solution problems, a worksheet of miscellaneous mixture problems.

Lesson by Mr. Twitchell

Handouts/Worksheets

Mixture Problems - Coins

Solve each problem below.

  1. I have both nickels and pennies. I have 5 more nickels than pennies. I have a total of 73 cents. How many of each kind of coin do I have?

     
  2. I have twice as many dimes as pennies. I have a total of $2.52. How many dimes do I have?

     
  3. I have the same number of dimes as I have nickels. In all, I have $2.10. How many coins do I have in all?

     
  4. I have $137 in ones and fives. I have 49 bills in all. How many are fives?

     
  5. I have the same number of ten dollar bills and twenty dollar bills. If I had one more of each, I would have $420. How many bills do I have?

     
  6. I have twice as many dimes as nickels. I have 33 coins in all. How much money do I have?

     
  7. I have a total of 20 coins. They are all dimes and quarters. I have $2.60. How much money do I have in dimes?

     
  8. I have pennies, nickels and dimes. The number of nickels is 1 more than the number of dimes. The number of pennies is two more than the number of dimes. I have $1.99. How many of each coin do I have?

     
  9. I have dollar bills and quarters totaling $34.25. If I have a total of 41 bills and coins, how much money do I have in quarters?

     
  10. I have 32 coins. Half of them are nickels, a quarter of them are pennies, and the rest of dimes. How much money do I have?
     

Mixture Problems - Coins: Answer Key

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Mixture Problems - Miscellaneous

Solve the following problems.

  1. A farmer has both pigs and chickens. There are twice as many chickens as pigs. In all, there are 104 legs. How many chickens are there?

     
  2. On a piece of paper I have drawn some triangles. I have also drawn some squares. There are five more squares than triangles. In all, there are 111 vertices (corners). How many shapes have I drawn in all?

     
  3. In a certain classroom, there are 34 students. Each math student has three textbooks, and each history student has two textbooks. In all there are 92 textbooks. How many math students are there?

     
  4. In a garden there are 50 flowers. Some have five petals, and some have eight petals. In all, there are 319 petals. How many flowers have five petals?

     
  5. In Clownville, all clowns either wear two silly shoes or a rubber nose. The number of silly shoes and rubber noses combined is 3590. There are ten more silly-shoe clowns than rubber-nose clowns. How many clowns are there in all?

     
  6. In a monster movie, the "Mike" monsters have one eye, and the "Sully" monsters have two eyes. There are seven more Sully monsters than Mike monsters. In all, there are 59 eyes. How many Sully monsters are there?

     
  7. On a web there are 132 legs. There are 6 times as many ants on the web as spiders. How many of each are there?

     
  8. Two thirds of my friends each have a car and a bicycle. The other third of my friends each has just a bicycle. If there is a total of 450 tires (not including spares) how many friends do I have?

     
  9. I have sixteen bags of buns, some of which are bags of hotdog buns, and some of which are bags of hamburger buns. There are 8 hotdog buns per bag, and six hamburger buns per bag. If there is a total of 118 buns, how many of them are hotdog buns?

     
  10. I have several dice. Some are four-sided dice, some are six-sided dice, and some are twelve-sided dice. In all there are 86 sides. There is one fewer four-sided than six-sided, and two more twelve-sided than six-sided. How many of each are there?

Mixture Problems - Miscellaneous: Answer Key

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Mixture Problems - Combining Mixtures

Solve the problems below.

  1. A solution is 10% saline. If 20 gallons of a 5% solution are added, it will be 7% saline. How many gallons were there originally?

     
  2. A solution is 10% acid. If 20 liters of pure water is poured into it, the resulting mixture will be 8% acid. How much of the solution was there originally?

     
  3. Solution A is 8% acid. Solution B is 14% acid. How much of each should be used in order to have 10 liters of 10% acid solution?
     
  4. One cookie recipe has 1 cup of chocolate chips for every 7 cups of batter. Another cookie recipe has 1 cup of chocolate chips for every 3 cups of batter. If I have 15 cups of the first recipe, and I want to have 1 cup of chocolate chips for every 4 cups of batter, how much of the second recipe should I add?

     
  5. I have two buckets containing red and blue marbles. In bucket A, 40% of the marbles are red. In bucket B, 30% of the marbles are blue. If I want to have 200 marbles, with 120 of them blue, how many should I take from each bucket?

     
  6. One classroom has 20 students, and is made up of 40% girls. Another classroom is made up of 70% girls. To have a classroom that is evenly split between boys and girls, how many students from the second classroom should be added to the first?

     
  7. Solution A is x% salt. Solution B is 40% salt. If a new solution is made by adding two gallons of solution A and three gallons of solution B, the result is 30% salt. Find the value of x.

     
  8. Solution B has twice the concentration of salt as solution A. If ten liters of solution B are added to fifteen liters of solution A, the result is 7% salt. What is the concentration of solution B?

     
  9. I want to invest $40,000. I will invest part of it at 4% interest, and part at 7.5% interest. If the interest for the two accounts combined is 5%, how much did I invest at each rate?

     
  10. A bucket of gravel contains 20% pebbles, 10% rocks, and the rest is sand. There are 9 gallons of gravel in the bucket. How much pure sand must be added in order to have 15% pebbles?

     

Mixture Problems - Combining Mixtures: Answer Key

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