May – 2021 Challenge

1. A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $800 cost equally, but each friend’s share of the cost increased by $20. How many friends were in the group originally?

Solution:

Let n be the number of friends originally in the group.
The original share, in dollars, for each friend was originally 800/n.
When two friends decided not to go on the trip, the share, in dollars, for each friend is now 800/(n – 2). Since this share represented a $20 increase over the original share, then 800/n + 20 = 800/(n-2). Multiplying each side by n(n-2),gives

800(n-2) + 20n(n-2) = 800n
800n – 1600 + 20n² – 40n = 800n
40n – 80 + n² – 2n = 40n

Therefore, n² – 2n – 80 = 0
After factoring, this becomes (n+8) (n–10) = 0

The solutions of this equation are −8 and 10. Since a negative solution makes no sense for the number of people in a group, the number n of friends originally in the group was 10.

2. A dairy farmer uses a storage silo that is in the shape of the upright circular cylinder, shown below. If the volume of the silo is π72 cubic metres, what is the diameter of the base of the cylinder, in metres?

Solution:

Volume of the cylinder = (π r²)h = 8 π r² = π 72,
r² = 9, therefore, radius (r) = 3 and diameter (d) = 6 metres.

3. Jeremy deposited x pounds in his investment account on January 1, 2001. The amount of money in the account doubled each year until Jeremy had 480 pounds in his investment account on January 1, 2005. What is the value of x?

Solution:

Let x = pounds deposited in the investment account on January 1, 2001.
The situation can be represented by the equation x(2^4) = 480, where the 2 represents the doubling each year and the 4 represents the 4 years between January 1, 2001, and January 1, 2005.

Then, x(2^4) = 480 gives 16x = 480 and x = 30 pounds.