Problems for middle school enrichment
Tempted by Triangles
How many different triangles are in the figure below?
Extension: Note that this is a 10 by 10 by 10 triangular grid. What if it were an N by N by N triangular grid, then how
many triangles?
One-half of a class garden is planted in tomatoes. The class plants broccoli on one-fourth of the ground that remains.
After that, the class plants lettuce in one-half of the remaining ground. The rest of the garden is planted in carrots. Draw
a map of the garden that shows how the land might be divided for the produce. What percent of the garden is planted in
carrots?
What is the sum of 20 percent of 30 percent of 40 percent of 80 and 1/4 of 2/3 of 1/6 of 72? Write your answer as a
mixed numeral in lowest terms.
Mom baked a cake for the entire family. Dad ate 1/6 of the cake. Brother ate 1/5 of what was left. Sister ate 1/4 of what
was left after that. Spot the dog ate 1/3 of what was left after that. Baby sister ate 1/2 of what was left after that. How
much of the original cake was left for Mom to eat? Who ate the most cake?
Our middle school needs new game balls for next year. Footballs cost $80.00 each, baseballs are $22.50 each, and
basketballs are $68.00 each. The school needs three times as many baseballs as basketballs, half as many basketballs as
footballs, and twelve footballs. How much will the school spend for new game balls?
Seattle has many coffee drinkers. Recently, the city council proposed a 10¢ tax on each fancy coffee drink served in the
city, and estimated that the tax would raise $7 million per year. About how many fancy coffee drinks are served per year
in Seattle?
If a 14" pizza has a 1" crust, what is the area of the crust? What percent of the pizza is crust?
One-eighth is the same part of one-sixth as one-third is of what number?
Beverages are available in three sizes: 12-ounce cans cost $0.80, 16-ounce bottles are $0.99, and 2-liter bottles cost $1.69
each, including tax. Extra empty cups that hold 8 ounces are $0.10 each. Since Sonja and her friends do not want to drink
out of the same container, they want to purchase drinks so that each person gets 8 ounces and so that no one drinks
directly from a 2-liter bottle. (Hint: 1 liter is approximately 1 quart.) What is the most economical way to purchase drinks
for two people? Three people? Four people?
Which pizza gives you more for your money, a 12-inch-diameter round pizza for $10.95 or an 8-inch-square pizza for
$7.49?
The third exit on a highway is located at milepost 40, and the tenth exit is located at milepost 160. A service center on the
highway is located three-fourths of the way from the third to the tenth exit. At what milepost would you expect to find
this service center?
The Hot Java Coffee Company advertises its small cup of decaffeinated coffee as being 96 percent caffeine-free as
compared with a small cup of caffeinated coffee. If a small coffee is 8 ounces, how many ounces of decaffeinated coffee
would you have to drink to get the equivalent amount of caffeine found in one small cup of regular coffee?
Solutions
18.75 percent. Positions on the garden map may vary, but relative sizes should not. T stands for tomatoes, which
take up 1/2 of the garden. B refers to broccoli, which is occupying 1/4 × 1/2, or 1/8, of the garden. L stands for
lettuce, which is taking up 1/2 × 3/8, or 3/16, of the available space. The fraction 3/16 converted to a percent is
18.75 percent. Thus, the carrots (C) are also occupying 3/16 of the garden, or 18.75 percent.
3 23/25. The expression "20 percent of 30 percent of 40 percent of 80" can be written as (0.2)(0.3)(0.4)(80), or
1.92, or 1 and 92/100. The expression "1/4 of 2/3 of 1/6 of 72" can be simplified to 2. Their sum is 1 92/100 + 2 = 3
92/100, or 3 23/25.
1/6 was left for Mom. Everybody eats the same amount of cake.
(1/6)(1) = 1/6 eaten; 1 – 1/6 = 5/6 of the cake that is left.
(1/5)(5/6) = 5/30 = 1/6; 5/6 – 1/6 = 4/6
(1/4)(4/6) = 4/24 = 1/6; 4/6 – 1/6 = 3/6
(1/3)(3/6) = 3/18 = 1/6; 3/6 – 1/6 = 2/6
(1/2)(2/6) = 2/12 = 1/6; 2/6 – 1/6 = 1/6
$1773. Working backward through the clues, we see that the school needs twelve footballs. Half as many
basketballs would be six basketballs. Three times as many baseballs as basketballs would be eighteen baseballs.
The school will spend 12 × $80 + 6 × $68 + 18 × $22.50 = $960 + $408 + $405 = $1773.
Seventy million coffee drinks are consumed. Since the tax is 1/10 of $1.00, 10 drinks have to be consumed for
each dollar of tax revenue, which is equivalent to dividing $7 million by $0.10 per drink. Students can research the
population of Seattle to find the average number of drinks per person.
26.5 percent. A 14" pizza has a radius of 7"; its area is 49π square inches. Without crust, the area is 36π square
inches. So the area of the crust is 13π square inches. The percent is 13π/49π, or 0.265.
4/9. Set up a proportion; 1/8 is to 1/6 as 1/3 is to x:
Two friends share one 16-ounce bottle + 1 cup; three friends share two 12-ounce cans + 1 cup; four friends share
one 2-liter bottle + 4 cups. The chart shows the details.
The 12-inch pizza. You end up paying $0.097 per square inch for the round pizza, instead of $0.117 as for the
square pizza, or roughly $0.10 versus $0.12.
Milepost 130. The third and tenth exits are 120 miles apart. Since 3/4 of 120 is 90, the service station is 90 miles
past exit 3 on the way to exit 10, or at milepost 40 + 90 = 130. A diagram is helpful.
200 ounces. One of the 96 percent decaffeinated small cups contains 4 percent caffeine. Therefore, it would take
25 cups of decaf coffee to have 100 percent of the amount of caffeine found in one regular small cup of coffee.
The problem asks for the solution in ounces; since there are 8 ounces in a small cup, 25 × 8 = 200 oz.