Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 7 Measurement
Problem
Solution
1) Trent has made 10 cups of punch by using three times as much lemonade as grape juice. Yaara drinks 1 cup and then realizes that Trent needs a full 10 cups of the punch. She tried to help by adding 1 cup grape juice to the pitcher. What is the ratio of lemonade to grape juice in the punch, using whole numbers in both parts of the ratio?
  1. If Trent used 3 times as much lemonade as grape juice then he used 7.5 cups of lemonade and 2.5 cups of grape juice.
  2. Yaara removed (1/10) of the punch. So the remaining punch has = (9/10)x 7.5 = 6.75 cups of lemonade
  3. She also removed (1/10) of the grape juice, so the remaining punch has .9 x 2.5 = 2.25 cups of grape juice.
  4. After she adds the cup of grape juice, the punch contains 3.25 cups of grape juice and 6.75 cups of lemonade.
  5. The ratio of lemonade to grape juice is 6.75/3.25 =
    63⁄4 = 27⁄4 = 27⁄13
    31⁄4     13⁄4

Problem
Solution
2) Sally and Yvonne wanted to make a rectangular toy box out of plywood. At the lumber yard, they realized that they had forgotten the dimensions of the box. Sally remembered that one of the sides had an area of 96 square centimeters. Yvonne remembered that the other sides had areas of 252 and 168 square centimeters. What are the dimensions of the box?
  1. Call the 3 dimensions of the box a, b and c
  2. The 3 sides are  a x b,   a x c,   and b x c
  3. The factors of the 3 sides' areas are:
         96 = 2, 3, 4, 6, 8, 12, 16, 24, 32, 49
        252 = 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126
        168 = 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84
  4. Reduce these to only those factors shared by at least one other side:
         96 = 2, 3, 4, 6, 8, 12, 24
        252 = 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
        168 = 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 84
  5. The dimensions must be combinations of these factors.
    Here are the possible combinations:
          96 sq. cm,: (24x4) (12x8)
        252 sq. cm.: (84x3) (42x6) (21x12)
        168 sq. cm.: (84x2) (42x4) (28x6) (24x7) (21x8) (14x12)
  6. For the 96 sq. cm. side, (24x4) doesn't work because the other 2 sides would have to be (42x4) and (42x6), resulting in 4 dimensions, so the only solution for the 96 sq. cm. side is (12x8)
  7. With the 96 sq. cm. side resolved, the only solution for the 252 sq. cm. side is (21x12), leaving the 168 side as (21x8)
  8. The 3 dimensions, therefore, are 8 x 12 x 21 cm.

Problem
Solution
3) A man walks for 15 minutes to catch a bus. He rides the bus for 5 miles. The bus is going 30 miles per hour. How long does his total journey take? Answer in minutes.
  1. The bus ride takes 5 miles divided by 30 miles/hour =
    1⁄6 hour = 10 minutes
  2. The total trip takes 15 + 10 = 25 minutes
4) It takes Bobby 1 hour and 15 minutes to mow his grandma's lawn, see figure. If Bobby mows at the same rate, how long will it take him to mow his uncle's field which has an area of 5000 square feet? Express your answer in hours and minutes rounded to the nearest minute.
 Careful! There are mixed dimensions!
  1. The area of Grandma's lawn is
    (converting to feet and using area subtraction):
    56x60 - 30x30 = 2460 sq. ft.
  2. Bobby's mowing rate is 2460 sq. ft. / 75 min =
    32.8 sq. ft./min
  3. Time for uncle's field =
    5000 sq. ft / (32.8 sq. ft/min) =
    152 min = 2 hours 32 min
5) Toni needs 3 gallons of paint. Two brands, Color Fine and Purity, have the same price in the store, but she has a coupon for 15% off on the Purity brand. Sales tax is 10%. If the original price is $26 per 1 gallon can, how much will she save overall if she buys the Purity brand? (Note: the sales tax is added after the discount is taken.)
  1. 15% off is a multiply by .85

  2. $26 x .85 = $22.10 x 3 = $66.30 x 1.10 (tax) = $72.93

  3. Without the coupon, $26 x 1.10 = $85.80

  4. Toni saves 85.80 - 72.93 = $12.87