Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 5 Number Sense
Problem
Solution

1) What is the time exactly half way between 9:37 pm and 2:43 am? 		1. Compute the number of minutes from 9:37 to midnight:
    2 hours + 23 minutes = 120 + 23 = 143 minutes.
2. Compute the number of minutes from midnight to 2:43 am
    = 120 + 43 = 163 minutes.
3. Add these and divide by 2: (143 + 163)/2 = 153 minutes.
4. If this is greater than #1 above, subtract that from
    it and convert back to hours and minutes: 0 hours 10 minutes.
5. The half way time is 12:10 am.

2) A bowling ball and a bag together cost $88. The ball costs three times as much as the bag. How much does the bowling ball cost? 		This is an algebra problem!
Let B = cost of the bag.
Then:
3B + B = $88.
4B = $88
B (cost of the bag) = $22.
Cost of the bowling ball = 3 X $22 = $66.

3) Kyle baked at least two dozen cookies but fewer than 60 cookies. If he divides the cookies evenly among 7 plates, 4 cookies are left over. If he divides the cookies evenly among 6 plates, 5 cookies are left over. How many cookies did Kyle bake? 		Make the following table:
Cookies
per plate		times 7 +4 times 6 +5
43225
53935
64641
75347
86053
96759
The number of cookies that appears in both columns is 53 cookies.

Problem
Solution
4) The sum of two numbers is 16 and their product is 48. What is the ratio of the two numbers when expressed as an improper fraction in lowest terms ? 1. Make a table of numbers whose product is 48:
  • 8 x 6 = 48
  • 2 x 24 = 48
  • 3 x 16 = 48
  • 4 x 12 = 48
2. Find two numbers in the table above whose sum is 16: 4 and 12
3. Divide the larger one by the smaller one: = 3/1 = 3
4. Reduce to lowest terms: = 3
5) Four friends have their birthdays in the same month. This year the squares containing their birthdates on the calendar touch each other such that they make a larger square. Two of their birthdates are multiples of six. The horizontal sums and the vertical sums are prime. What is the sum of the four birthdates? OK, a tough problem! The place to start is with the horizontal sums of 2 consecutive days of the month that add to a prime number. Here they are:

    (1+2) (2+3) (3+4) (5+6) (6+7) (8+9) (9+10) (11+12)
    (14+15) (15+16) (18+19) (20+21) (21+22) (23+24)
    (26+27) (28+29) (29+30) (30+31)

2. Now, with this list, the dates under the pairs are these dates +7, so write out the pairs (one pair above the other) that fit this requirement:

  (1+2) 
  (8+9) 		  (2+3) 
  (9+10) 		  (8+9) 
(15+16) 		  (11+12) 
  (18+19) 		  (14+15) 
  (21+22) 		  (21+22) 
  (28+29) 		  (23+24) 
  (30+31)

3. Select out of this list those that add vertically to primes:

  (2+3) 
  (9+10) 		  (11+12) 
  (18+19)

4. Finally, Select out of this list those in which 2 of the 4 dates are multiples of 6:
  (11+12) 
  (18+19)
5. Sum the 4 dates = 11+12+18+19 = 60