Chapter 5.2, Problem 25E

To determine

To write:the number of problems based on the point value.

Answer to Problem 25E

Therefore, there are eight $\text{5-point}$ problems and thirty $\text{2-point}$ problems

Explanation of Solution

Given:

The total marks of the test is $100$ points

There are $38$ problems.

The questions are categorized as $5$ points or $2$ points.

Calculation:

  $25.$

Let:

  $x\to$ number of $5$ point problems

  $y\to$ number of $2$ point problems

Setup the equations.

The number of problem is $38$ :

  $x+y=38\leftarrow \text{Eq}\text{.1}$

The total worth of the test is $100$ :

  $\text{5x+2y=100 }\leftarrow \text{Eq}\text{.2}$

Solve for one variable in using eq. $\text{1}$ :

  $\text{y=38-x}$

Substitute on eq. $\text{2}$ :

  $\text{5x+2}\left(38-x\right)=100$

Simplify & solve for $x$ :

  $\begin{array}{l}5x+76-2x=100\\ 3x=24\\ \text{x=8}\end{array}$

Solve for $\text{y}$ :

  $\begin{array}{l}\text{y=38-8 }\\ \text{y=30 }\end{array}$

Therefore, there are eight $\text{5-point}$ problems and thirty $\text{2-point}$ problems

Conclusion:

Therefore, there are eight $\text{5-point}$ problems and thirty $\text{2-point}$ problems