Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Algebra
Problem
Solution
1) Greg's father is 40 years old and Greg is 14 years old this year. Greg notices that each year the ratio of his age to his father's age gets smaller. In how many more years will that ratio be 1:2 or 1/2? Let Y = the number of years after which Greg's age will be half his father's age.
  1. Greg's age will be 14 + y
  2. His father will be 40 + y
  3. Twice Greg's age will be his father's age, so the equation is:
        2(14 + y) = 40 + y
        28 + 2y = 40 + y
        y = 40 - 28 = 12 years
2) Sally usually gets $10 a week for her allowance. Sally asks her parents to change her allowance so that she gets 1 penny on the first day and on each next day double the amount of the previous day. Her parents agree to pay her in pennies for two weeks. Her parents want to know at the end of the two weeks which allowance is better for her and how much better? Compute the difference between the two allowances at the end of the two weeks.
  1. Since there are 14 days in 2 weeks, Sally gets 214 cents - 1 cent for the first day = $163.84 - .01 = $163.83
  2. Her usual 2-week allowance is $20.00
  3. The difference is $143.83

3) A batch of brownies uses 3 eggs and weighs 25 oz. A cake takes 4 eggs and weighs 35 oz. If Wanda has 22 eggs and enough of all the other ingredients how many batches of brownies and how many cakes should she make so that the total weight of baked goods is as large as possible?
  1. Brownies result in 25/3 oz. per egg = 81⁄3 oz/egg
  2. Cakes result in 35/4 oz. per egg = 83⁄4 oz/egg
  3. She should maximize the number of cakes.
  4. 22/4 = 5 cakes with a remainder of 2 eggs, but you can't make a batch of brownies with 2 eggs, so you must back off 1 cake to
        4 cakes (using 16 eggs) +
        2 batches of brownies (using 6 eggs)
    for a total of 22 eggs.

Problem
Solution
4) Over five years, Lady Simone wishes to give $1,000,000 to her favorite charity. Her plan is to give them more each year. The last or fifth year will be the biggest donation. The fourth year she will give 1/10 what she gives the fifth year. The third year she will give 1/10 what she gives the fourth year. If this pattern continues how much will she give the last year to the nearest dollar?
  1. Let x = the amount she gives in year 5
  2. Then the amount she gives in year 4 is .1 x
  3. The amount she gives in year 3 is .01 x
  4. The amount she gives in year 2 is .001 x
  5. The amount she gives in year 1 is .0001 x
  6. All these donations add to 1,000,000, so the equation is
        .0001 x + .001 x +.01 x + .1 x = 1,000,000
        x (1 + .1 + .01 + .001 + .0001) = 1,000,000
        x (1.1111) = 1,000,000
        x = 1,000,000 / 1.1111 = $900,009
5) The table shows some data on sales of tickets for a charity event. Paul assumes the data is linear. If that is true, what is the price of 9 tickets?
  1. Assume the price change between the 5 ticket price and the 10 ticket price is linear.
  2. Set up the ratio of the 5-to-9 ticket price to the 5-to-10 ticket price:
        ($80.25-$44) / 5 = (x - $44) / 4 =
        $36.25/5 = (x - $44) / 4
        (4/5)$36.25 = x - $44
        $29 = x - $44
        x = $73