Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 7 Measurement
Problem
Solution
1) John wants to know how tall Steve is, but Steve won't tell him. On a sunny day, John measures Steve's shadow and finds it to be 90 inches long. He quickly has a friend measure his own shadow. John's shadow is 105 inches long. If John is 72 inches tall, how tall is Steve? Round to the nearest inch. Set up the ratio of John's actual height to his shadow and set that equal to Steve's actual height (call that S) and his shadow length:
    72/105 = S/90
    S = 90x72/105 = 61.71 = 62 inches

2) Sharon takes a trip and travels at the speed limit on the different roads within seconds of getting on the road. The table shows her trip. How far did she travel? Express your answer to the nearest tenth of a mile.
  1. 10 min x 25 mph = 1⁄6 hour x 25 mi/hr = 25/6 mi = 41⁄6 miles
  2. 20 min x 39 mph = 1⁄3 hour x 30 mi/hr = 30/3 mi = 10 miles
  3. 20 min x 60 mph = 1⁄3 hour x 60 mi/hr = 60/3 mi = 20 miles
  4. 10 min x 30 mph = 1⁄6 hour x 30 mi/hr = 30/6 mi = 5 miles
Total distance = 41⁄6 + 10 + 20 + 5 =
391⁄6 miles = 39.167 = 39.2 mi
3) In March, Grandma gave Amy $15 on her 10th birthday. Amy starts saving for a trip to Disney Land. That March she also decides to add $5.75 to her savings every month. If she can cover tickets and transportation with her savings, they will go on her 15th birthday. Grandma decides she will contribute the remainder needed for Amy's 15th birthday present. On her 15th birthday they find out that their tickets will cost $125 and transportation costs $260. If Amy has already added in the $5.75 for March, how much does Grandma need to contribute so that they will go?
  1. The number of months Amy saves for her Disney Land trip is
        12x5 + 1 = 61 months
  2. Amy will have saved 61 x $5.75 = $350.75
  3. The Disney Land trip costs $125 + $260 = $385
  4. Amy has the original $15 + her savings = $365.75
  5. Grandma needs to contribute $385 - 365.75 = $19.25

Problem
Solution
4) Michael, Ted, and Ron are going to paint Ron's bedroom together. Ron knows it would take him 3 hours, Ted thinks he can do it in 2 hours and Michael thinks it would take him 4 hours. If they are right, how long will it take them to do the job together? Express your answer to the nearest minute.
  1. Assume the area to be painted is some easy number to work with, like 100 sq. feet.
  2. Ron can paint 100 sq. ft in 3 hours so he works at a rate of 33.33 sq. ft./hour
  3. Ted works at a rate of 100/2 = 50 sq. ft./hour
  4. Michael works at a rate of 100/4 = 25 sq. ft./hour
  5. Together, in one hour, they can paint 33.33 + 50 + 25 = 108.33 sq. ft.
  6. Since they have only 100 sq. ft. to paint, it takes them:
        100 sq. ft/108.33 sq.ft/hour = 0.9233 hour =
    55 min
5) Leslie builds a paper model pyramid with a square bottom. The square bottom has side length 6 centimeters. The triangular sides are all identical and have height 4 centimeters. Her class is going to make a big scale model of her pyramid for the Ancient History fair using paper over a frame. All dimensions will be increased by a scale factor of 50. How many square meters of paper will be contained in the four triangular sides? Express your answer to the nearest hundredth of a square meter.
  1. Multiply all dimensions by 50 and convert to meters:
    base = 6x50 = 300 cm = 3 m.
    height = 4x50 = 200 cm = 2 m.
  2. Compute the area of one triangle:
    Area = 3x2/2 = 3 sq. m.
  3. Multiply by 4 to get all 4 sides =
    4x3 = 12 sq. m.