Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 6 Number Sense


Problem
Solution
1) The digits 2, 3, 4, 5, 6, 7, and 9 can be placed into the boxes to make the equation true. Each one is used exactly once. Notice the multiplier is a decimal. What is the product?
 The actual multiplication looks like this →
1. The last 2 digits of the bottom result must both be zeroes for there to be a whole number result.
2. That means the product of the top number and the multiplier must end in 0. There are only 3 possibilities:
    (5 x 4=20), (6 x 5=30) and (5 x 2=10).
All of which involve a 5. Therefore one of those ending digits must be a 5.
3. No multiplier that ends in a 4, 6 or a 2 is going to produce an integer result, so the second digit of the multiplier must be the 5. That means the last digit of the top number must be a 2,4 or 6.
4. Assume the multiplier is .25 (the top number is divisible by 4) and examine the possibilities.
5. Note: the intermediate calculations can use any digits.
6. Fill out the squares in the problem with your answer.
7. The final calculation is :

    The product is 94
2) Ben wants to make a lot of crepes. He has 3 cups of milk left and 1 and 1/2 cup of flour. He has plenty of eggs and everything else he needs. The recipe calls for 3/4 cups milk and 3/8 cup flour and yields 6 crepes. What is the maximum number of crepes he can make with the ingredients he has? 1. Divide the number of cups of milk by the amount of milk the recipe calls for =
    3/(3/4) = (3 x 4)/3 = 4
2. Divide the number of cups of flour by the amount of flour the recipe calls for =
    (1 1/2)/(3/8) = (3/2) x (8/3) = 4
3. Take the minimum of these and multiply by
    the number of crepes the recipe makes = 4 x 6 = 24 crepes.

Problem
Solution
3) In a class of 30 students, exactly 5/6 have cell phones, exactly 2/5 have pocket calculators, and exactly 3/10 have both. How many of the students have neither? 1. Compute the number of students that have cell phones:
    (5/6) x 30 = 25,
    the number that have calculators (2/5) x 30 = 12
    and the number that have both (3/10) x 30 = 9.
2. Compute the number who have cell phones but no calculators as the number who have cell phones - the number that have both =
    25 - 9 = 16.
3. Add this to the number that have calculators =
    16 + 12 = 2.
4. Subtract this from the total number of students =
    30 - 28 = 2 students
4) The sum of two positive integers is 30 and their product is 216. Find the positive distance between the two numbers. 1. factor 216 into primes = 3 x 3 x 3 x 2 x 2 x 2
2. Notice that there are only 2 distinct primes in this list.
3. Combine the factors in groups of 2 or 3 until you find 2 that add to 30 = 12 (3x2x2) and 18(3x3x2)
4. Their difference is 6
5) Emmalee wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table as shown below. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What are the hidden numbers behind each card? List them in order: what number goes with 44, what number goes with 59 and what number goes with 38?
 Here are the primes up to 100:
    2 3 5 7 11 13 17 19 23 29 31 37 41
    43 47 53 59 61 67 71 73 79 83 89 97
1. Start with the largest number, 59, and add primes to it, starting at the lowest. Compute the sum and subtract the first card's value from it =
    (59 + 2) - 44 = 17.
2. If this is prime, do the same for the 3rd card =
    61 - 38 = 23
3. Repeat this process, going up through the primes until the number you add to 44 and the number you add to 38 are both prime.
4. The 3 numbers (left to right) are 17, 2, and 23