Chapter 5.2, Problem 25E

To determine

To find: How many problems of each point value are on test.

Answer to Problem 25E

The required answer for number of 5 points problem is 8 and number of 2 points problem is 30.

Explanation of Solution

Given information:

A math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 points.

Calculation:

Total points $=100$

Number of problems $=38$

Let, number of 5 points problem be $x$

And number of 2 points problem be $y$

Therefore,

  $\begin{array}{l}x+y=38\\ \Rightarrow x=38-y\end{array}$

(Equation 1)

And,

  $\begin{array}{l}5x+2y=100\\ \Rightarrow 5\left(38-y\right)+2y=100\end{array}$

(Putting the value of x from equation 1)

  $\Rightarrow 190-5y+2y=100$

(Simplify)

  $\Rightarrow 190-190-3y=100-190$

(Subtracting 190 on both sides)

  $\begin{array}{l}\Rightarrow -3y=-90\\ \Rightarrow -3y\left(-1\right)=-90\left(-1\right)\end{array}$

(Simplify and multiplying -1 on both sides)

  $\begin{array}{l}\Rightarrow 3y=90\\ \Rightarrow \frac{3y}{3}=\frac{90}{3}\end{array}$

(Divide by 3 on both sides)

  $\Rightarrow y=30$

(Simplify)

Therefore from equation 1 we get;

  $\begin{array}{l}x=38-y\\ x=38-30\end{array}$

(Putting the value of y)

  $x=8$

Therefore the number of 5 points problem is 8 and the number of 2 points problem is 30.