Pat's Money System of Inequalities

Question

Pat has x quarters and y dimes, and, in addition, has no more than two dollars. Write inequalities that represent this situation and then graph all points in the coordinate plane that satisfy this condition.

Answer 1

The key to answering this problem lies in understanding what information is present (or isn't). We know that Pat has a mix of dimes and quarters and that regardless of how many of each coin he has, their total worth is no more than $2.00. However, what the problem doesn't tell us is how many coins he has in all. Thus, it's possible that he may have 6, 7, or even 10 total coins depending on which combinations of quarters and dimes work. To find these combinations, we must first discover the limits of the number of each type of coin he can have that meet this requirement... and to find THAT, we must develop an equation that models this situation. Since a quarter is worth $0.25 and a dime is worth $0.10, the equation can be written as follows:

0.25x + 0.1y ≤ 2

Note that in lieu of an equal sign, a "less than or equal to" sign is used. It's used here because the problem states that Pat has "no more than" $2.00. "No more than" means that while Pat can't have more than $2.00 total in coins, he can have exactly $2.00 if the coin combination allows it (which is why the "less than or equal to" sign is used instead of "less than".) The benefit of having this sign in a standard form inequality is that the limits of this inequality are always the x and y intercepts (that is, Pat can't have more than x quarters maximum if he has 0 dimes and he can't have more than y dimes maximum if he has 0 quarters.) Thus, by finding the intercepts, we can find the limits of how many of each coin he can have. We'll find from plugging in 0 for each variable that the x-intercept is (8,0) and the y-intercept is (0,20). Once these values are plotted onto the graph, simply draw a straight line connecting the two dots and voila, the graph is done.

Since it's impossible to have a negative number of quarters or dimes, we know that the only points that can satisfy the equation and yet make sense are the intersections between whole positive values of x and y that lie on or underneath the graph of the inequality (on graph paper, they are more visible since the intersections are already drawn). With that said, no further thinking is required at this point. Just label these intersections as needed to satisfy the problem. ;)