Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 7 Measurement
Problem
Solution
1) Hans, an exchange student, calls home and tells his mom that it was 24 degrees Fahrenheit that morning but that it is now 60 degrees Fahrenheit. She's a little confused because she doesn't know the Fahrenheit scale, only Celsius. To explain how big a change this is to her he does some converting. What does that change in temperature correspond to in Celsius if the formula is
    F = 9/5C + 32?
Give your answer to the nearest degree in Celsius. 		First, turn the equation around to be an equation for the conversion of Fahrenheit to Celsius:
    C = (5/9)(F - 32)
  1. The first temperature: C = (5/9)(24 - 32) = (5/9)(-8) = -4.44 deg. Celsius
  2. The second temperature: C = (5/9)(60 - 32) = (5/9)(+28) = +15.56 deg. Celsius
  3. The change is 15.56 - (-4.44) = 20 deg. Celsius

2) A glass has a height of 12 cm and an inside diameter of 6 cm. Another glass has the same volume but its radius is a third, or about 33.3%, larger. What is the height of the other glass to the nearest centimeter?
  1. Compute the volume of the first glass:
        V = 32 x 12 = 108
  2. The radius of the second glass is 1.333 x 3 = 4 cm.
  3. From this volume, compute the height of the second glass:
        108 = 42 x h
  4. h = 108 / 16 = 108/16 = 6.75 cm. =
        7 cm. rounded

Problem
Solution
3) Tania enters a 26.2 mile marathon. She checks out the course one weekend and it takes her about 4 hours and 24 minutes to run the course. She ran at an average rate of 10 minutes per mile for the first hour, then increased her speed to 9 minutes per mile for the next 36 minutes. She repeated this 10 minutes per mile for an hour, then speeding up to 9 minutes per mile for 36 minutes and slowing back down to 10 minutes per mile for an hour. What was her average speed for the remainder of the course? Give your answer in tenths of a minute per mile.
  1. Compute the distance Tania ran in that first hour:
        36 minutes = .6 hour, so the first 2 10 min/mile + 9 min/mile cycles takes 2(1.6) = 3.2 hours
  2. The remainder of the marathon was run in 4.4 - 3.2 hours = 1.2 hours
  3. In the 10 min/mile segments she ran 60 min / (10 min/mile) = 6 miles
  4. In the 9 min/mile segments she ran 36 / (9 min/mile) = 4 miles.
  5. So, for 1 cycle of running at 10 minute miles and then 9 minute miles she covers 6 + 4 = 10 miles
  6. After 2 of these cycles she has covered 20 miles and has 6.2 miles left to run
  7. Her average speed for this last stretch was 1.2 hour / 6.2 miles =
    .19355 hour/mile x 60 =
    11.613 min/mile = 11.6 min/mile
4) A water tank has the shape of a rectangular prism with a base of 2000 sq. cm. This tank is being filled at the rate of 12 liters per minute. Find the rate at which the height of the water in the tank increases. Express your answer in millimeters per second. 1 cm3 = 1 milliliter of water. The problem is to find the height of 12 liters of water in the 2000 sq. cm. base tank.
  1. Since 1 cm3 of water = 1 milliliter of water (1/1000 liter) and we put 12 liters in the tank in 1 minute, then we put in 12000 milliliters of water in the tank in 1 minute
  2. That 12000 milliliters is 12000 cm3 of water
  3. Since the tank has a base of 2000 sq. cm., the height of the water is 12000 cm3 / 2000 cm2 = 6 cm.
  4. It raises the height of the water in the tank 6 cm per minute which is 60 mm/60 seconds = 1 mm/sec
5) Two runners, Jo and Doyle, run on the same oval track. Jo does 2 laps around the track in 26 minutes. Doyle does 5 laps around the track in 35 minutes. If Jo and Doyle both cross the finish line marker on the track now, when would they both be together at the finish line on the track again if they both kept running?
  1. Jo runs one lap in 13 minutes
  2. Doyle runs one lap in 7 minutes
  3. The least common multiple of 13 and 7 is
    91 minutes = 1 hour 31 minutes