A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after hours? Find a linear expression to represent the number of members at the end of the hour.

Understand the Question The rally begins with 120 members, and 9 members leave every hour. To find the remaining members at any hour, subtract the total number of members who have left from the initial . By calculating the remaining count for and hours, we can establish a pattern and form a general linear expression for hours. Step 1 · Calculate Members Remaining After 1, 2, and 3 Hours Initial number of members Number of members dropping out each hour Members remaining after hour: Members remaining after hours: Members remaining after hours: Step 2 · Find Linear Expression for Hours Observing the pattern: After hour: After hours: After hours: Let represent the number of members remaining at the end of the hour: Answer > Members remaining after hours are , and the linear expression for the hour is . Common Mistakes Adding Instead of Subtracting: Writing instead of . Since members are dropping out, the total count decreases each hour. Off-by-one Variable Assignment: Using instead of . At the end of the hour, exactly members have left, so the term is simply .

Introduction to Linear Polynomials | Exercise 2.3

Question 2

A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after $1, 2, 3, \dots$ hours? Find a linear expression to represent the number of members at the end of the $n^{\text{th}}$ hour.

Solution

Understand the Question

The rally begins with 120 members, and 9 members leave every hour.
To find the remaining members at any hour, subtract the total number of members who have left from the initial $120$ .
By calculating the remaining count for $1, 2,$ and $3$ hours, we can establish a pattern and form a general linear expression for $n$ hours.

Step 1 · Calculate Members Remaining After 1, 2, and 3 Hours

Initial number of members $= 120$ Number of members dropping out each hour $= 9$ Diagram 1

Members remaining after $1$ hour:

\begin{aligned} 120 - 9 \times 1 &= 120 - 9 \\ &= 111 \end{aligned}

Members remaining after $2$ hours:

\begin{aligned} 120 - 9 \times 2 &= 120 - 18 \\ &= 102 \end{aligned}

Members remaining after $3$ hours:

\begin{aligned} 120 - 9 \times 3 &= 120 - 27 \\ &= 93 \end{aligned}

Step 2 · Find Linear Expression for $n$ Hours

Observing the pattern:

After $1$ hour: $120 - (9 \times 1)$
After $2$ hours: $120 - (9 \times 2)$
After $3$ hours: $120 - (9 \times 3)$

Let $M_n$ represent the number of members remaining at the end of the $n^{\text{th}}$ hour:

\begin{aligned} M_n &= 120 - 9 \times n \\ &= 120 - 9n \end{aligned}

Answer

Members remaining after $1, 2, 3, \dots$ hours are $111, 102, 93, \dots$ , and the linear expression for the $n^{\text{th}}$ hour is $120 - 9n$ .

Common Mistakes

Adding Instead of Subtracting: Writing $120 + 9n$ instead of $120 - 9n$ . Since members are dropping out, the total count decreases each hour.
Off-by-one Variable Assignment: Using $(n - 1)$ instead of $n$ . At the end of the $1^{\text{st}}$ hour, exactly $1 \times 9$ members have left, so the term is simply $9n$ .

Question 2

More questions in Exercise 2.3