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32 plums and peaches bought for $52. plums cost $2, peaches cost $1. How many plums were bought

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2 Answers

If amount of plums is x (question of the problem) then amount of peaches is (32 - x)
Pedro paid for plums $(2 · x) and for peaches  $1 · (32 -x).
All purchase 2x and (32-x) costs $52
2x + (32 - x) = 52
2x - x + 32 = 52
          - 32   - 32
x = 20 plums.

Hi, Susan.

This type of story problem is called a "mixture" problem because you're mixing two things (plums and peaches).  Whenever you have mixture problems, consider two ideas to create two equations:
Quantity
Value

The Quantity just states the relationship about the amounts of items you have.  In this problem it's about how many plums and peaches we have altogether.

The Value is putting a value on each item and writing an equation from this.  In this problem we will be putting the value of $2 on plums, $1 on peaches, and $52 for the total value.

Okay, enough discussion.  Let's assign variables to represent our plums and peaches.
Let x = the number of plums
Let y = the number of peaches

Quantity:  x  +  y  =  32
Value:  2x   + 1y  =  52  (technically you do not need the number 1 in front of the y)

To solve this, you may use substitution or elimination...

Substitution:

Solve one of the equations for a variable, and place its expression into the other equation.

x + y = 32
  - y     - y                                                           .
x    =  - y + 32

2x + 1y = 52
2(- y + 32) + 1y  = 52

Solve this by simplifying the left side and going on...

- 2y  + 64  + 1y  = 52
-1y  + 64  = 52
       - 64    -64
-1y     =   - 12

-1y   =  -12
 -1         -1  

y = 12 (Remember y represented peaches.)

So, if we have 12 peaches, we have 20 plums (because 12 + 20 = 32).

Elimination:

Add the equations vertically to cancel out one of the variables.  If nothing cancels, then multiply one (or both) of the equations by something to do that...

 x  +  y  = 32
2x  + 1y  = 52

If I add these vertically, I end up with 3x + 2y = 84... this does no good.  So let's multiply the top equation by negative one, and try again...

-1(x   + y   =  32) becomes:  - x   -  y   =   -32

- x  -  y  =   - 32
2x  +  y  =     52

Now when I add these two equations vertically, I get:  x = 20 (the y's cancel out)

Remember x represents plums, so we have 20 plums and 12 peaches (20 + 12 = 32).

------------------

If you have not learned about solving equations in two variables yet, there is a third way to do this using only one variable in representing both the plums and the peaches.

Solving this in one variable:

Let x = the number of plums
Let 32 - x = the number of peaches

Still assign value to each expression listed to make this equation:
2(x) + 1(32 - x) = 52

Now solve.

2x + 32 - x = 52
x  + 32  =  52

x  + 32  =   52
    - 32     - 32
  x       =   20

We still got the same answer of 20 plums and 12 peaches. 

I hope this helps you with understanding ways to solve this problem.

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