Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 6 Number Sense
Problem
Solution
1) A small country prints only two kinds of stamps, 7 cent stamps and 13 cent stamps. One letter requires 65 cents postage. How many of each stamp should you ask for so that you can mail the letter and spend the least amount of money?
# 7-cent
stamps		# 13-cent
stamps		  Total  
0565

2) Two competing music stores hold sales. Musico has a sale on CDs: Buy 5 for the regular price of 3. Songs Unlimited has the sale: Buy 7 CDs for the regular price of 4. If the sale price in both stores for 35 CDs is $210, what is the ratio of the regular price of a single Musico CD to the regular price of a single Songs Unlimited CD? 		You must work backwards from the sale price for each store to the regular price.
Write equations using X for the regular price:
1. Musico: 3/5 of the regular price is $210 for 35 CDs.
2. Songs Unlimited: 4/7 of the regular price is $210 for 35 CDs.
3. Compute the regular price of 35 CDs for each store and then divide by 35 to get the price of a single CD from each store:
  Musico:
  (3/5) X = $210;    X = $350/35 = $10 per CD
  Songs Unlimited:
  (4/7) X = $210;     X = $367.50/35 = $10.50 per CD
4. Express these as a fraction using whole numbers: 10/10.5 = 20/21

Problem
Solution
3) The prime factors of one number are 2, 3, and 7. A second number has the prime factors 2, 7, and 13. If their least common multiple is 9,828 what is the smallest the first number could be? Difficult problem!
1. The greatest common factor of the two numbers is 7.
2. The smallest the second number can be is all its prime factors
    multiplied together = 2 x 7 x 13 = 182.
3. The relationship between the LCM and the GCF of 2 numbers is:
    LCM(A,B) = (A x B) / GCF(A,B)
    where A and B are the two numbers.
4. Assuming B is that smallest value for the second number, solve the above equation for the first number =
    9828 = 182 A / 7
    A = (9828 x 7) / 182 = 378

4) Jessica is camping with her family. She needs to get water for the camp. She needs 1 1/2 gallons. She has a pot that holds 5/8 gallon. The spring where she gets water is 1/5 mile from camp. How many miles will she walk while getting the water using only the 5/8 gallon pot? Express your answer as a mixed number. 		1. Convert the number of gallons Jessica needs
    to an improper fraction in 8ths = 12/8
2. Divide this numerator by the numerator of the size of her pot = 12/5 = 2 2/5. This is the number of trips she must make.
3. Round this number up (because you can't take part of a trip) and multiply by 1/5 mile x 2 (going and coming back)
    for the number of miles she must walk =
    3 x (1/5) x 2 = 6/5 = 1 1/5 mile

5) Your very rich charity minded aunt makes you a proposal business person to business person. If you donate your weekly $5.00 allowance to the local food bank for the next 6 years, she will pay you $16 on your birthday for the next 6 years. The $16 is 1/10 of 0.1% of an amount held in trust for you. What is the total amount held in trust for you and is this more than your donated allowance? 		1. Compute the amount of your donated allowance for 6 years using 52 weeks in a year = 52 X 5 X 6 = $1560.
2. Convert the 0.1% to a fraction = 0.1/100 = 1/1000
3. Take 1/10 of this = 1/10000
4. Divide the $16 by this fraction:16 X 10000 = $160,000
5. Is this more than your donated allowance? Yes