Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 7 Probability and Statistics
Problem
Solution
1) What two numbers must be added to this set to make the mean equal to 20 and to make median equal the mode?
22, 19, 17, 25, 15
  1. Put the numbers in order to simplify:
        15 17 19 22 25
  2. Compute the sum of the 2 terms necessary to make the mean 20:
        (15 + 17 + 19 + 22 + 25 + 2X)/7 = 20
        (98 + 2X)/7 = 20
        98 + 2X = 140
        2X = 140 - 98 = 42
        This means the 2 numbers must add to 42.
  3. Divide 42 by 2 makes both added numbers 21: 21 and 21
  4. Another solution would be when one of the 2 added numbers duplicates either 19 or 22 (the middle 2 numbers ) and the other makes it add to 42. The only solution that fits this is
        19 and 23
2) How many ways are there to arrange the letters in the word FOOTY without duplicates? Using the rules for the permutation of the letters of words with repeated letters we have:
    5! = 60
    2!
3) A dart lands on a random spot on this target made of concentric circles. What is the probability that it lands on a shaded region?
Express your answer as a fraction in lowest terms.
Each marked segment is 2 inches long.
 This is the area of the shaded region divided by the area of the total circle.
  1. Area of the circle:
        A = 72
        A = 49
  2. Area of the shaded ring:
        As = 52 - 32
        As = (25 - 9) = 16
  3. Area of the shaded dot in the middle:
        Ad =
  4. Probability =
        (16+1) = 17/49
          49

Problem
Solution
4) There are two bags. The first bag contains 2 blue marbles and 1 yellow marble. The other bag contains 2 blue marbles, 1 yellow marble, and 1 red marble. You draw one marble randomly from each bag. What is the probability that the colors match?
Express your answer as a fraction in lowest terms. 		The easiest way to solve this problem is to list all the 2-marble possibilities and count the matches. There are 12 possible 2-marble combinations (3x4=12). Using R for red, B for blue, Y for yellow.
Then Bag 1 = BBY     Bag 2 = BBYR
Here they are:
Bag 1Bag
2
Pair?
BBYES
BBYES
BYNO
BRNO
BBYES
BBYES
BYNO
BRNO
YBNO
YBNO
YYYES
YRNO
There are 5 out of 12 possibilities that make matches, therefore the probability of a match is 5/12
5) A red six-sided and a blue six-sided dice are rolled. What is the probability that the number on the red die is divisible by the number on the blue die?
Express your answer as a fraction in lowest terms. 		Well, here are the ones that are evenly divisible:
  • All 6 red numbers are divisible by a blue 1: 6
  • All red numbers are divisible by the same blue number: 5
  • A blue 2 divides evenly into 4 and 6: 2
  • A blue 3 divides evenly into 6: 1
This is 14 out of 6x6 = 36 possibilities = 14/36 = 7/18