A Mathematics exercise has 100 questions. The Mathematics teacher asks Matthew to answer each question numbered \( 5 n-2 \). How many questions did Matthew answer?

To determine how many questions Matthew answered, let's break down the problem step by step. **Given:** - There are 100 questions numbered from 1 to 100. - Matthew is asked to answer each question numbered \( 5n - 2 \), where \( n \) is a positive integer. **Objective:** Find the number of values of \( n \) for which \( 5n - 2 \) falls within the range of 1 to 100. **Steps:** 1. **Set up the inequality:** \[ 1 \leq 5n - 2 \leq 100 \] 2. **Solve for \( n \):** - **Lower bound:** \[ 5n - 2 \geq 1 \\ 5n \geq 3 \\ n \geq \frac{3}{5} \] Since \( n \) must be a positive integer, the smallest possible value for \( n \) is **1**. - **Upper bound:** \[ 5n - 2 \leq 100 \\ 5n \leq 102 \\ n \leq \frac{102}{5} \\ n \leq 20.4 \] Since \( n \) must be an integer, the largest possible value for \( n \) is **20**. 3. **Count the number of valid \( n \) values:** \[ n = 1, 2, 3, \ldots, 20 \] There are **20** such integers. 4. **Verification:** - For \( n = 1 \): \[ 5(1) - 2 = 3 \] - For \( n = 20 \): \[ 5(20) - 2 = 98 \] Both results are within the range of 1 to 100. **Conclusion:** Matthew answered **20** questions. **Answer:** Matthew answered 20 questions.