SOLUTION: This seems pretty easy, but I'm a little confused with this question: There is $3.50 in nickels and dimes, if there are already 50 coins; how many of each type of coin is there?

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: This seems pretty easy, but I'm a little confused with this question: There is $3.50 in nickels and dimes, if there are already 50 coins; how many of each type of coin is there?       Log On


   

Question 83855: This seems pretty easy, but I'm a little confused with this question:
There is $3.50 in nickels and dimes, if there are already 50 coins; how many of each type of coin is there?

A coffee merchant has coffee beans that sell for $9 per pound and $12 per pound. The two types are mixed to create 100lb of a mixture that will sell for $11.25 per pound. How much of each type of bean should be used in the mixture?
Also if I could and explanation on how I would figure out how to decide how much of each bean should be used.
Thank you for any help that I can get.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of nickels, y= # of dimes
Since we know there is 50 coins, the sum of the coins is 50
x%2By=50
And we also know that all of the coins are worth $3.50, so we have this equation
0.05x%2B0.1y=3.5

100%280.05x%2B0.1y%29=100%283.5%29 Multiply both sides by 100 to make every number whole

5x%2B10y=350

So we now have the system of equations

x%2By=50
5x%2B10y=350

Now lets solve this system by addition

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=50
5%2Ax%2B10%2Ay=350

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 5 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 5 is 5, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -1 like this:

5%2A%281%2Ax%2B1%2Ay%29=%2850%29%2A5 Multiply the top equation (both sides) by 5
-1%2A%285%2Ax%2B10%2Ay%29=%28350%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
5%2Ax%2B5%2Ay=250
-5%2Ax-10%2Ay=-350

Notice how 5 and -5 add to zero (ie 5%2B-5=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%285%2Ax-5%2Ax%29%2B%285%2Ay-10%2Ay%29=250-350

%285-5%29%2Ax%2B%285-10%29y=250-350

cross%285%2B-5%29%2Ax%2B%285-10%29%2Ay=250-350 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-5%2Ay=-100

y=-100%2F-5 Divide both sides by -5 to solve for y



y=20 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=50 to solve for x

1%2Ax%2B1%2820%29=50 Plug in y=20


1%2Ax%2B20=50 Multiply



1%2Ax=50-20 Subtract 20 from both sides

1%2Ax=30 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2830%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=30 Multiply the terms on the right side


So our answer is

x=30, y=20

which also looks like

(30, 20)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=50
5%2Ax%2B10%2Ay=350

we get



graph of 1%2Ax%2B1%2Ay=50 (red) 5%2Ax%2B10%2Ay=350 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (30,20). This verifies our answer.



So there are 30 nickels and 20 dimes


---------------------------------------------------------------------
Let x=beans that sell for $9, y=beans that sell for $12

x%2By=100 here is the sum of the 2 beans which equals the target weight
9x%2B12y=11.25%2A100 "The two types are mixed to create 100lb of a mixture that will sell for $11.25 per pound"

So we have the system of equations

x%2By=100
9x%2B12y=1125


Now lets solve this system by addition

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=100
9%2Ax%2B12%2Ay=1125

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 9 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 9 is 9, we need to multiply both sides of the top equation by 9 and multiply both sides of the bottom equation by -1 like this:

9%2A%281%2Ax%2B1%2Ay%29=%28100%29%2A9 Multiply the top equation (both sides) by 9
-1%2A%289%2Ax%2B12%2Ay%29=%281125%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
9%2Ax%2B9%2Ay=900
-9%2Ax-12%2Ay=-1125

Notice how 9 and -9 add to zero (ie 9%2B-9=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%289%2Ax-9%2Ax%29%2B%289%2Ay-12%2Ay%29=900-1125

%289-9%29%2Ax%2B%289-12%29y=900-1125

cross%289%2B-9%29%2Ax%2B%289-12%29%2Ay=900-1125 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-3%2Ay=-225

y=-225%2F-3 Divide both sides by -3 to solve for y



y=75 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=100 to solve for x

1%2Ax%2B1%2875%29=100 Plug in y=75


1%2Ax%2B75=100 Multiply



1%2Ax=100-75 Subtract 75 from both sides

1%2Ax=25 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2825%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=25 Multiply the terms on the right side


So our answer is

x=25, y=75

which also looks like

(25, 75)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=100
9%2Ax%2B12%2Ay=1125

we get



graph of 1%2Ax%2B1%2Ay=100 (red) 9%2Ax%2B12%2Ay=1125 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (25,75). This verifies our answer.


So we need 25 pounds of coffee beans that sell for $9 a pound and 75 pounds of coffee beans that sell for $12 a pound to make a 100lb mixture that will sell for $11.25 per pound.