Jon M. answered • 01/20/21
Honor's Student with 1 Year of Tutoring Experience
Hi Alia!
To solve this problem, you'll need to set up a system of equations. You may have heard of this before, but if not, a system of equations uses two or more equations with two or more variables, something like (x+y=1, x-y=1). We can actually solve these two equations together, in order to find a specific value for x and y. I like to think of this as x+y=1 and x-y=1 equations of lines, and if you graph these lines they will intersect at a specific point, this point is the solution to your system.
In your example, you can start by listing the information that you are given, namely number of nickels + number of quarters = 38 and price of nickels + price of quarters = $4.50. To write these into algebra that we can work with, we can call n the number of nickels and q the number of quarters. This will directly translate the first equation to be n+q=38. The next equation will use the monetary value of each coin to say 0.05n+0.25q=4.50.
If your class allows calculators, some will let you plug those two equations in and spit out answers, but solving them by hand isn't too painful. Ill show you how I would do it and walk you through it after:
1) n+q=38
2) 0.05n+0.25q=4.50
3) n=38-q
4) 0.05(38-q)+0.25q=4.50
5) 1.9-0.05q+0.25q=4.50
6) 0.2q+1.9=4.50
7) 0.2q=2.6
8) q=13
9) n+(13)=38
10) n=25
final: 13 quarters, 25 nickels
In this solution, I used something called substitution. To set this up, I manipulated the equation in step one to have one variable on one side (I chose n) and got the equation in step 3 by subtracting q from both sides. I can now substitute this into the other equation to get step 4. Steps 5-8 are simplifying this equation down to get a value for q. Once I have a value for q, i can plug this value into the equation in step 1 (I could have chose either but equation 1 seemed easier) and then I simplify down to get a value for n.
In short, because these problems come up a lot. You'll want to set up two equations, one with both your variables adding up to the total number of coins, and another equation that's similar but with the value of each variable as the coefficients adding up to the total price. Then you can solve the equations with substitution.
Hope this helps!
