At the fair, 5 friends each bought an $8 meal voucher and one ice-cream voucher. The total cost of the vouchers was $55. What was the price of an ice-cream voucher?

Question

a) Choose a variable to represent the price of an ice-cream voucher. Write an equation to model this problem.

b) Solve the equation using the distributive property.

c) Verify the solution. Explain your thinking in words.

Answer 1

Hi! I'm more than happy to help, but for future reference, I think this question would be better categorized as an Algebra 1 problem. As confusing as the name is, Linear Algebra is actually a college course typically taken after Calculus 2.

a) Let's call the variable representing the price of an ice-cream voucher x. We don't know what it is yet, so we'll just refer to it as x for now, and then use the information given to us to solve for what the value of x, a.k.a. the price of an ice-cream voucher, actually is.

From the information given to us, we have that each friend bought one meal voucher and one ice-cream voucher. So if we were to write an expression for the cost of the vouchers each individual friend bought, it would just be the cost of the one meal voucher ($8) plus the cost of the one ice-cream voucher (x- this is the definition of the variable that we defined at the start) that every person bought. Written mathematically, this is 8 + x.

Now, if we want an expression for the total cost of all of the vouchers that the friends bought collectively, we would just multiply the cost of the vouchers that each individual friend bought (8 + x) by the number of friends (5). Written mathematically, we have 5*(8 + x). We have to be careful with this multiplication, as we want to multiply 5 by the sum cost of the vouchers that each individual friend bought, which is the sum of the $8 a meal voucher costs and the x dollars an ice-cream voucher costs. This is why we keep the sum in parentheses for now.

Finally, to get our equation, we just recognize that the total cost of the vouchers bought by all of the friends together is $55- this is given to us in the question. We just created an expression for that exact quantity, based on the voucher prices and how many vouchers were bough- so we can write that these quantities are the same with an equation, specifically 5*(8+x) = 55.

b) Now we need to solve this equation for x. They tell us to use the distributive property, so our first step will be to distribute the 5 to each term in the parentheses on the left hand side of the equation.

5*(8 + x) = 55

5*8 + 5*x = 55

40 + 5x = 55

Now we want to isolate all of the terms with x in them to one side. Here, we just need to subtract 40 from the left side to make the 5x term alone on the left hand side. To make sure that the equality between both sides holds, we make sure to subtract 40 from the right side too.

40 + 5x - 40 = 55 - 40

5x = 15

Now we just need to divide 5x by 5 to get x entirely alone on the left side. Again, we make sure to do the same thing to the right side of the equation, so that both sides are still equal to each other.

(5x)/5 = (15)/5

x = 3

By solving the equation, we get that the x, the price of an ice cream voucher, is $3.

c) To verify that the price of an ice cream voucher is actually $3, we can plug it into the expression we got for the total cost of all of the vouchers that the friends bought, and see if it equals the $55 that we expect.

If we plug x = 3 into our equation:

5*(8 + x) = 55

5*(8 + 3) = 55

5*(11) = 55

55 = 55

When we plug x = 3 in, we see that our expression on the left for the total cost simplifies to 55, and our equation holds. This is a good sign that our value for the price of an ice cream voucher is correct.