Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 6 Number Sense


Problem
Solution
1) Sally is thinking of a three digit number and she tells you this. All the digits are different. Arranging the digits in one order creates an odd number divisible by 5 and 3. Arranging the digits in another order makes a number divisible by 2 and 5. Arranging the digits in a third order creates a number which factors as the product of two primes. The number is the biggest one you can make with the three digits. What is the number? 1. Well, let's start with the last clue:
the number is the biggest one you can make with the 3 digits.
That means they are in numerical order, highest to lowest because if you swap the position of any two, the number is no longer the biggest.

2. If you can make a 3-digit number out of it that is divisible by 5,
    then the number must end in either a 5 or a 0.

3. If you can arrange the digits so that it makes an odd number
    divisible by 5 and 3 then the last digit in this arrangement must be a 5.
    So there is a 5 here!

4. If there is both a 7 and a 5 in the number,
    then the final number must be one of these:
    765, 750, or 975.

5. If there is a 1 and a 5 in the final number,
    then it can only be 510
    (because 5 cannot be the last if there is a 1).

6. Examine these 4 possibilities to come up with the final number
Possible
number		 Odd,divisible
by 5 & 3		Even, divisible
by 2 & 5		factors into
2 primes
765765X 
750705750057 factors:
19 & 3
975975X 
510105510015: factors:
3 & 5
= Either 750 or 510

Problem
Solution
2) A recipe for fruit punch says to use 8 ounces of orange juice for every 3 ounces of grape juice. Nicky made 99 ounces of fruit punch. How many more ounces of orange juice than grape juice were used? 1. Out of 11 total parts, the orange juice is 8 of those parts.
2. Similarly, the grape juice is 3 of those parts.
3. Compute the number of ounces of orange juice in 99 ounces and the number of ounces of grape juice in 99 ounces and subtract =
    Orange juice: (8 X 99)/11 = 72 ounces.
    Grape juice:  (3 X 99)/11 = 27 ounces.
    72 - 27 = 45 ounces

3) Lance's sandwich shop sold 4 kinds of sandwiches yesterday. For every bologna sandwich sold, 3 turkey sandwiches were sold. For every turkey sandwich sold 1.5 ham sandwiches were sold. 30 roast beef sandwiches were sold. If only whole sandwiches were sold and bologna sandwiches made up 10% of the total number of sandwiches sold, how many bologna sandwiches were sold?		 1. If bologna sandwiches make up 10% of the total then turkey sandwiches make up 30%.
2. If there are 1.5 ham sandwiches for every turkey sandwich, then ham sandwiches make up 45% of the total.
3. What's left is roast beef sandwiches which are
    (100 - 10 - 30 - 45) = 15% of the total.
4. Compute the total number of sandwiches and take 10% to get the number of bologna sandwiches =
    if 15% (roast beef) is 30 sandwiches then the total number of sandwiches is 200.
    bologna is 10% of this = 20 sandwiches.

Problem
Solution
4) M is a mystery number. Take 2/3M, divide it by 0.1, multiply it by 1.2, and finally multiply it by 1/8. If the resulting number is N, what is M/N? Careful! They are not asking for the values of M or N, just their ratio.
1. The equation is (2/3) M x (1/10) x 1.2 x (1/8) = N
2. Reduce this equation and compute M/N
    (2 x M x 1.2 x 10)/(3 x 8) = N
    24M/24 = N
3. M/N = 1
5) Rory draws a number line representing his street. His school is at 0. His house is at 4. Sam's house is at -12, Tina's is at -7, and Demi's is at 14. They each take turns being the first to leave their house and bike to the other three before arriving at school together. Compare Demi's shortest possible trip to Sam's shortest possible trip, whose is longer and by how many units? 1. Compute the distance Sam rides to first Tina and then Demi and then back to Rory's house and then to the school =
    Sam:   5 + 21 + 10 + 4 = 40
    Demi: 10 + 16 + 5 + 7 = 38
2. Subtract the two: Sam is longer by 2 units.