if 5 apples and 8 bags of chips cost $29, and 3 apples and 4 bags of chips cost $15, what is the price of an apple? and what is the price of a bag of chips

Question

if 5 apples and 8 bags of chips cost $29, and 3 apples and 4 bags of chips cost $15, what is the price of an apple? and what is the price of a bag of chips

Answer 1

Hi Jamis,

This is easy -- call the price of 1 apple = x, the price of 1 bag of chips = y, transform the word problem into equations (that's two equations:

5x + 8y = 29 and 3x + 4y = 15), and solve those two equations for x and y, such as by elimination of first x, then substitute back into either equation to get y.

Just a note: why do we have both word problems and equations for the same thing? It's because usually you think of a problem in terms of words. But you can't solve problems in terms of words, your brain doesn't work that way!

I like to think of it this way -- a float for the citrus parade (in Florida) passes by, with about 10,000 oranges on it. Then, a float with about 10,000 apples passes by the other way. And then you have to guess whether the number of oranges *exactly* matched the number of apples. It's like comparing .... You get the picture. But if you saw a piece of paper with "9,998 oranges" and another paper with "10,021 apples" on it, comparing them is a piece of cake (figuratively speaking). Even in your head, you could say "... there are 23 more apples than oranges!" Well, that's why you don't try to solve word problems directly, it's like looking at the floats going by.

SO, you change them into mathematical equations, which say the same thing in math language.

Maybe more importantly, the math statements keep track of what you KNOW already (as numbers) and what you DON'T know already (as variables). That allows you to concentrate on finding the value for each variable, in turn (usually, the answer to the problem!).

After that, the rules on what you can do (to the equations) are pretty simple:

1) If you add something to one side, do the same addition to the other side.

2)  If you multiply one side of an equation by something, do it to the other side too.

3) If you square one side of an equation, square the other side, too.

and so on.

4) You can turn an equation around (exchange right and left sides).

4) You can add or subtract *entire equations* from each other. Just keep things from the left side on the left side, and things from the right side on the right side.

Hope this gives you some insight, and so on.

Cheers,

--S. de Riel

Answer 2

Hi Jamis,

 

In this question, it is asking you to find 2 unknowns.  How much does a bag of chips cost?  How much does an apple cost?

 

One concept in algebra is that you must have the same number of equations as unknowns in order to find a solution.  You have 2 unknowns, so you must write 2 equations.

 

For example, you can write the unknowns as follows:

 

A = $ Apple

B = $ Bag of Chips

 

To write the words in math form:

 

5A + 8B = 29

3A + 4B = 15

 

You now have 2 equations and 2 unknowns so you can solve.  There are multiple ways to solve this.  One way would be to solve the first equation for A, and then plug that into the 2nd equation to solve for B:

 

1) 5A + 8B = 29

2) 5A = 29 - 8B

3) A = 29/5 - (8/5)B

 

Now plug our answer for A into the 2nd equation and you will have an equation in terms of B only.  1 equation and 1 unknown so we can solve.

 

1) 3A + 4B = 15

2) 3*(29/5 - (8/5)B) + 4B = 15

3) 87/5 - (24/5)B + 4B = 15

4) 87/5 - (4/5)B = 15

5) -(4/5)B = 15 - 87/5

6) -(4/5)B = -12/5

7) B = -(12/5) * -(5/4)

8) B = 3

 

Now you know that a bag of chips costs $3.  You can now take this knowledge of B and plug it into the equation you found for A to find that A = 1. 

 

And always remember to do a few simple checks if you can.  Plug A and B into either of your original equations, and make sure that you get the right result.

 

For example:

5A + 8B = 29

5(1) + 8(3) = 5 + 24 = 29

 

This checks out.  Also, in problems like this, you can make sure your answer makes sense.  For example, if you forgot a minus sign and ended up with a negative cost, you can recognize that it should not be possible for anything to cost a negative number.

 

Hope this helps.