Hands-on math!

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

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, write the equation for the sum:
        _______________________________
  2. Solve this for N:
        N = _______________________________
  3. Given the first number, N, the second and third numbers are:
        _______ and _________
        making the 3 numbers _____, _____, and _____

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.
    The second and third inputs are 2 and 7 but swapped between the A and B inputs.
    1. Find the difference between the outputs and notice where the product of the 2 inputs falls.
    2. Using the difference between the 2 inputs, find the equation that the machine is using and apply it to the last set of inputs:     ________________

  • Method 2: Use the 1st, 4th and 5th inputs Using the difference between the outputs of the 1st and 4th sets, form a ratio of this difference between their A values (3 and 6) A difference to the 4th to 5th ones and apply that to the output:
        ____________________

Problem
Hint

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:
Calling the figure number "N" and the number of associated squares "S":
  1. Bottom row: Find the expression for S that gives you the number of squares in the bottom row:
        S = ________
  2. Top row:Find the expression for S that gives you the number of squares in the top row:
        S = __________
  3. Middle rows: Notice that the rows in between the top and bottom row are squares. Find the expression for S that gives you the number of squares in these rows:
        S = ___________________
  4. Combine these terms into a term for the total number of squares in any row N:
        S = _______________________
  5. Solve for the 100th term:
        S100 = ____________________

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
        ______ dollars.
  2. Divide this by the number of weeks gives you the amount she spent each week:
        __________________
  3. From the first week through the 7th week she spent _____ times this, so
    First week = _______________

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
        $_________
  2. The second discount is 30% off this price = $______
  3. The total discount is $_____ which is ____%