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The admission fee at an amusement park is $2.00 for children and $6.80 for adults.

The admission fee at an amusement park is $2.00 for children and $6.80 for adults. On a certain day, 285 people entered the park, and the admission fees collected totaled 1458 dollars. How many children and how many adults were admitted?

number of children equals
number of adults equals

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1 Answer

Okay, here's how you figure this out:  you use what they call systems of equations.  We know that the the number of children plus the number of adults is 285.  After all, EVERYBODY is either an adult or a child.  We also know that 2 times the number of children plus 6.8 times the number of adults adds up to 1458. 

Now here's how we use the system.  First, let's assign a variable to children and to adults.  Let's say the number of adults is X and the number of children, Y.  In that case, we know the following:

X + Y = 285     AND WE KNOW:    6.8x + 2y = 1458

Now what we will do is solve for one of these variables.  Either X or Y will do.  I decided to solve for X.  Therefore, what I will do is use the substitution method to get X by itself.  How do I do that?  BY FINDING A WAY TO EXPRESS Y IN TERMS OF X. 

Since X + Y = 285 in one of the equations, I can take this equation, subtract X from each side, and then get Y = 285 - X  

Now that I have solved for Y in the first equation, I can SUBSTITUTE that value of Y in for "Y" in the second equation and BINGO, I have an equation entirely written in terms of X.

6.8x + 2y = 1458 will now turn into....    6.8x + 2(285 - X) = 1458

See what I just did?  I plugged the (285 - X) value of Y in for the Y value of the second equation.  Now I can distribute:  6.8x + 570 - 2x = 1458 is the new value of the equation after I do this.  Then I combine like terms on the left side:

4.8x + 570 = 1458   Now I'll subtract 570 from each side:

4.8x = 888  And now I divide each side by 4.8:

X = 185.

So there you go.  The value of X, the number of adults, is 185.  Since there were 285 people, the number of children must therefore have been 100. 

For best results, make sure your values work in both equations:

X + Y = 285      Now plug and chug to get 185 + 100 = 285.  A true statement

6.8x + 2y = 1458  Now plug and chug to get 6.8(185) + 2(100) = 1458.  Another true statement.  

There were, therefore, 185 adults and 100 children.