Some problem situations include a minimum and maximum between which a value may fluctuate. To solve these problems, it may be necessary to calculate the least and/or greatest amount of something based on the other information in the problem.

Example 1

At Washington High School, there are 46 homeroom classes. In each class, there are between 18 and 22 students. Determine a reasonable minimum and maximum number of total students at Washington High School. To calculate the total number of students at Washington High School, s, use the equation 46h = s, where h represents the number of students in each homeroom.

Minimum:
46(18) = 828 students

Maximum:
46(22) = 1,012 students

Example 2

Jacqueline is reading a novel for her English class. The novel has 178 pages. If Jacqueline reads between 20 and 30 pages per day, what is the minimum number of days it will take for her to finish the novel? To calculate the minimum number of days for Jacqueline to finish the novel, d, use the equation 178 ÷ p = d , where p represents the number of pages she will read per day.

Minimum:
178 ÷ 30 = d   Use 30 for p.
5.93 = d

The minimum number of days it will take Jacqueline to finish the novel is 6.

Example 3

Coach Partain had $350 in his awards account. He bought trophies for the upcoming track meet. Each trophy cost $65. The amount of money remaining in the account, a, can be represented by the function a = 350 – 65t, where t represents the number of trophies that he purchased. What is the maximum number of trophies Coach Partain can purchase for the upcoming track meet?

Source: Jim Thorpe Trophy 191, Carl Malamud, Flickr

To calculate the maximum number of trophies Coach Partain can purchase, t, use the equation a = 350 – 65t, where a represents the amount of money remaining in the account.

Maximum:
a = 350 – 65t
0 = 350 – 65t   Use 0 for a since the maximum amount Coach Partain spends will deplete the account to $0.
-350 = -65t   Solve for t
5.4 ≈ t

The maximum number of trophies Coach Partain can purchase is 5.