Middle School Math Club Problems Sept & Oct 2011
1. Each letter represents a different number. Find the values of P, O, and F that will
make this puzzle true.
2. A woman got a job at a starting salary of $35,000 a year. If she received an 8 percent
raise each year, how much would her salary be at the beginning of the tenth year?
Round your answer to the nearest dollar.
3. Mike and Michelle live in different towns 56 km apart. They decide to ride their
bicycles to the halfway point between their two towns. Mike can ride his bike at 8
km an hour, and Michelle rides at 6 km an hour. They want to meet at noon. At what
time should each of them leave home?
4. A contractor is asked to build a new set of townhouses in attached clusters of
different sizes. He created plans for one-, two-, and three-house clusters, as shown
in the diagram below. The builder used computer software to draw the line
segments used to represent the houses. How many line segments are needed to
draw 4 houses? 10 houses? 47 houses? n houses? Explain your strategy.
5. Becky wants to make a long-distance call to her friend Sarah from a pay telephone.
She has $5.00 in change. The call costs $0.90 for the first three minutes and $0.24 for
each additional minute. How long can Becky talk to Sarah?
6. Christopher Columbus has just spent 20 pieces of silver. He bought a hat, a cape, and
a spyglass. The spyglass cost 9 more pieces of silver than the cape. The spyglass and
cape together cost 16 more pieces than the hat. How much did he pay for each item?
7. What three consecutive counting numbers have a sum that is 20% of the product of
the three numbers?
8. Your new summer job is at the local swimming pool. One hot summer day, 242
people are at the pool. You notice that twice as many children as adult females are at
the pool. Also, 10 fewer adult men than adult females are at the pool. Determine the
number of children, female adults, and male adults that are at the pool. 10A total of
3 Holstein cows and 3 Guernsey cows give as much milk in 3 days as 2 Holstein cows
and 4 Guernsey cows in 4 days. Which breed of cow is the better milk producer,
Holstein or Guernsey?
1.
F = 0, P = 8, and O = 9.
F must be O, since P + F = P in the ones column. P must be an even number, since it is the double of the O's. O
is 1 more than P, since when the O's are summed to P in the tens column, only 1, "100," could have been
regrouped into the hundreds column.
Another strategy is to use a system of equations. From the hundreds column, we have P + 1 = O, and from the
tens column, O + O = P + 10. Substitute the first into the second equation to get (P + 1) + (P + 1) = P + 10 and
solve for P = 8 and O = 9.
3. Mike should leave at 8:30 A.M., and Michelle should leave at 7:20 A.M. They must travel 28 km
each. Mike will travel the distance in 28/8, or 3.5 hours. Michelle will travel the distance in 28/6, or
4 2/3, hours, which is 4 hours and 40 minutes.
6. The amounts of silver would be 2 for the hat, 4 1/2 for the cape, and 13 1/2 for the spyglass.
The total cost of the hat, h, the cape, c, and the spyglass, s, is, h + c + s = 20 pieces of silver.
The cost of the cape plus the spyglass is 16 more pieces of silver than the cost of the hat: c + s = h + 16 pieces
of silver. Thus, h + (h + 16 pieces of silver) = 20 pieces of silver, which gives h = 2 pieces of silver and c + s =
18 pieces of silver.
Also, s = c + 9 pieces of silver, so c + (c + 9 pieces of silver) = 18 pieces of silver, which gives c = 4 1/2 pieces
of silver and s = 13 1/2 pieces of silver.
8. The numbers are 126 children, 63 adult women, and 53 adult men.
A guess-and-revise method is shown on the chart below.
Alternatively, we notice that each time the number of children increases by 2, the total number of people in the
pool increases by 4.
To increas the number in the pool from 190 to 242, we need to add (242 - 190)/4 = 13 increments of 2 children,
or 26 children, to the initial 100.
Finally, using a system of equations, if we let F equal the number of adult females, C equal the number of
children, and M equal the number of adult males, we have the following:
Substituting for C and M in the first equation, we get F + 2F + (F - 10) = 242. When we solve for F, we can
find C and M.
19. 3, 4, and 5. Begin by picking a counting number, such as 7. If 7 is the first of three counting numbers, then
7 + 8 + 9 would need to be 20% of (7)(8)(9). The sum is 24, whereas 20% of the product is 100.8. Select three
different numbers greater than 7, 8, and 9, such as 12, 13, and 14. As seen in the table below, greater values
produced a greater difference between the sum and 20% of the product. Therefore, try values less than 7, such
as 4, 5, and 6, and continue until the correct values are determined. Using the following table, column 1 must
equal column 3.