Question

Mrs. Townsend is giving you a test worth 100 points and it contains 40 questions. There are two-point and four-point questions on the test. How many of each type of question are on the test?

Answer 1

`10` four point questions and
`30` two point questions

Explanation:

Let's say
`f =`
`\text{number of four point questions}` and
`t = \text{number of two point questions}`. The total number of questions is
`40`, therefore
`t + f = 40`. We know the test is worth
`100` points so we can write
`2t + 4f = 100`. We have a system of equations:
`t + f = 40 \text{ and } 2t + 4f = 100`. We can rewrite
`t` as
`t = 40 - f`. We rewrite this in the equation
`2t + 4f = 100`. Since
`t = 40 - f`, we have
`2 \cdot (40 - f) + 4f = 100`. Distributing & simplifying gives

`2 \cdot (40 - f) + 4f = 100\\(80 - 2f) + 4f = 100,\\\text{Remove parentheses} 80 - 2f + 4f = 100,\\80 + (-2f) + 4f = 100\\80 + 2f = 100,\\\text{Subtract 80 from both sides} 2f = 20,\\\text{Divide 2 by both sides} f = 10.\\`

Since we determined
`t = 40 - f` and
`f = 10`, subtracting
`10` from
`40` will give
`t = 30`.

Therefore, there are
`10` four point questions and
`30` two point questions.

Hope that helps!