Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 7 Algebra
Problem
Solution

1) The sum of three numbers is 160. The second number is 1⁄5 the first number. The third number is 17 more than the first number. What are the three numbers?
  1. Calling the first number N, the equation is:
    N + N/5 + (N+17) = 160
  2. Combining terms:
    2N + N/5 = 143
    (11/5)N = 143
    11N = 715
    N = 715/11 = 65
  3. The numbers are: 65, 65/5 = 13, and 82

2) My machine combines numbers using a rule. Find the missing number.
 Call the first input A (the one on the left) and the second one B (the top input).
  • Method 1: Use the 2nd and 3rd inputs.
    Looking at the 2nd and 3rd inputs the output differs by 10 (19-9) with the A and B inputs swapped between the 2 inputs. The difference between the 2 inputs is 5 which is half of the difference of the outputs. AB for both these machines is 14 which is halfway between 9 and 19, suggesting that the machine is multiplying the 2 inputs and subtracting their differences. Using this information, it is either AB + (B-A) or AB +(A-B). Using the B-A option on the second machine yields
        7x2 + (7-2) = 14 + 5 = 19.
    Using this formula on the last machine gives us:
        ? = 8x5 + (5-8) = 40 - 3 = 37

  • Method 2: Use the 1st, 4th and 5th inputs:
    All 3 of these inputs have 5 as the B input. The A inputs are 3,6 and 8. The differences in the outputs are:
      From A = 3 to 6 the outputs differ by 12: 29 -17 = 12
      From A = 6 to 8, this is 2/3 of the difference between the A = 3 to 6 inputs, so the output should be 2/3 of 12 greater which is 8. Add 8 to the 4th output (29) gives you 37

Problem
Solution

3) In the following sequence, how many squares are needed to make the 100th figure?
 Well, it's not an arithmetic sequence (the difference in squares between successive figures is not a constant), so we have to go row-by-row:
The figure to the left is color-coded as to the 3 separate parts that make up the whole:
  • Bottom row: yellow
  • Top row: black
  • Middle rows: red
Calling the figure number "N" and the number of associated squares "S":
  1. Bottom row: In the bottom row each figure is N+2 squares, so
    S = N+2
      (3, 4 and 5)
  2. Top row: In the top row each figure has N - 1 squares so
    S = N-1.
      (0, 1, 2)
  3. Middle rows: The rows in the middle are squares (N+1) squares on a side, so S = (N+1)2.
      (4, 9, 16)
  4. So the equation for the Nth figure is:
    Sum = (N+2) + (N-1) + (N+1)2
  5. Combining terms and expanding the square term (using the distributive property):
    Sum = 2N + 1 +(N2 + 2N + 1)
  6. Simplifying: Sum = N2 + 4N + 2, so
  7. The sum of the squares in the 100th figure are:
    Sum = 1002 + 400 + 2 = 10000 + 402 = 10402 squares
4) Frannie has a certain amount of money in her spending jar. Beginning at the start of the year, she takes the same amount out every week. After 8 weeks she has $1182 in the jar. After 24 weeks there is $802 in the jar.
How much money was in the jar at the start of the year?
How much does she take out each week?
  1. From the 8th week to the 24th week she spent
      ($1182 - $802) = $380 dollars.
  2. Divide this by the number of weeks (24 - 8) = 16, gives us:
      $380 / 16 = $23.75 per week.
  3. From the first week through the 7th week she spent 8 times this, so
    First week = $1182 + 8($23.75) = $1372
5) A shirt is discounted 20% off. Miguel has a coupon that will give him an additional 30% off. What is the total discount on the original price? Express your answer as a percent. Problems like this are best solved using an example starting price.
Let's use $100.
  1. The first discount is 20% off, so the remainder is 80% of the original price = $80. The discount is $20.
  2. The second discount is 30% off this price = 30% of $80 = $24.
  3. The total discount is $20 + $24 = $44 which is 44%