MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
Mixture problems involve combining different quantities at different rates. When a problem involves mixtures, amounts and prices, plug the values into this mathematical sentence: (amount x price of ingredient 1) + (amount x prices of ingredient 2) = mixture amount x price of mixture. Using this "formula", use the information from the word problem to fill in the given values. Solve for the unknown value.
If you guys are coffee drinkers or when you become coffee drinkers, you'll find that a lot of times coffees are actually mixed from different types of beans. And different types of beans are more expensive than others. That's why you come into problems like this that are actually taken from the real world.
4 pounds of brand x coffee which costs $7 per pound, were combined with 3 pounds of brand y coffee which costs $14 a pound. Find the cost per pound of the mixture. So think about this before you start. I'm combining a $7 coffee with a $14 coffee and I want to find the price.
It's probably going to be somewhere in between 4 and 17. It might even be right smack in the middle. I would say it would be exactly in the middle, if I was adding equal quantities. But since I'm adding a little more of the cheaper coffee, it might be slightly weighted way from the middle between 14 and 7. Let's see what I'm talking about.
So when I'm doing problems like this, where I have mixtures, amounts and prices, I'm going to try to use this idea. Amount times price of ingredient 1 plus amount times price of ingredient 2. It's going to be equal to my mixture amount, times the price of the mixture. And that's good because that's what I'm trying to find. It says find the cause of the mixture.
That should be my only variable, my only letter, I'm going to call it m, for mixture. Let's do this all out. The amount times price of my first ingredient here we go. I have 4 pounds it costs $7. There's my first product. Then I'm adding, combining this next product; 3 pounds $ 14.3. Now I have to figure out my last piece which is how much I have of the mixture. Well if I put 4 pounds plus 3 pounds I'm going to end up with 7 pounds of this mix. I need to solve for m, which is going to be the price of my mixture. This is the hardest part you guys, and if you don't understand this quite yet, don't be too hard on yourself.
This is probably the first time or the first few days of your math class, where you've seen these types or problems. It's really difficult to go from words to equations like this. So if you can't do this on your own yet, don't worry. I promise you'll get it.
Let's go through and solve this 28 plus 3 times 14 is 42, that's going to be equal to 7m. Continue by simplifying this side you get 70 is equal to 7m which tells you m equals 10.
Don't forget to go back and label this with units. What does m stand for again? You got to make sure you are answering the question. M stands for the price of my mixture and that's good, the costs per pound of the mixture. So I would say $10 per pound.
And that makes sense, because before we started, I thought my mixture should cost somewhere between 7 and 14. So I know that this answer makes sense. It's always a good idea when you do a word problem like this, to stop at the end, and make sure you have a value for your answer, that makes sense in the real world context.
So before I let you guys go and practice your problems, try using this formula; amount times price of ingredient 1, plus amount times price of ingredient 2, is going to equal your amount times price of your mixture.
