Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 5 Algebra
Problem
Solution
1) A girl walks a straight path from her home to go play in the park. When she is 1/3 of the way along, she realizes she has dropped her mittens. She walks back 120 meters to retrieve them. She is now 420 meters from home. How far is she from the park? 1. The distance she walks back to get her mittens + the distance from home after she retrieves her mittens is 1/3 of the distance to the park. 1/3 of the distance to the park is 120 + 420 = 540 meters.
2. So the distance from home to the park is 3 X 540 = 1620 meters.
2. Since she was 420 meters from home when she picked up her mittens she was 1620 - 420 = 1200 meters from the park.
2) Find the area of this rectangle. Its perimeter is 20 feet. Round to the nearest hundredth.
1. The equation for the perimeter is:

    2(3X - 4) + 2X = 20 ft.
    6X - 8 + 2X = 20
    8X = 28 ft.
    X = 3.5 ft.
    3X - 4 = 6.5 ft.
2. Compute the area given these measurements:
    Area = 3.5 x 6.5 = 22.75 sq. ft.
3) Every week, Amy helps her neighbor clean their home and does extra chores around their house. She is paid a fixed amount for the cleaning and an additional hourly amount for the extra chores. Based on the table below, how much is the hourly amount for the extra chores?

Hours of extra chores3712
Total amount earned  $58  $102  $157 
 The only data points you need are the first 2.
1. The extra amount she earned in week #2 is
    $102 - $58 = $44.
2. The extra hours she worked chores in the second week over the first week is 7 - 3 = 4 hours.
3. Divide the first amount by the difference in hours to get the hourly rate = $44/4 = $11/hour.
4. Check it using the 3rd week:
The fixed amount = $58 - $33 = $25, so in the 3rd week:
    $25 + 12 x $11 = $25 + $132 = $157. It checks!

Problem
Solution
4) When a one-digit number is multiplied by 8, it increases by 42. What is the one-digit number? Use X for the one-digit number.
1. Expression for "one-digit number is multiplied by 8" = 8X.
2. The expression for "it increases by 42" = X + 42.
3. Set these 2 expressions equal to each other and solve.
    8X = X + 42
    7X = 42
    X = 6
5) Louis is stacking textbooks and starts forming a pattern. If the pattern below repeats, how many books will there be in the 7th stack?
 This is an arithmetic sequence.
Find the number added at each stack: 3
Method 1: Compute the number of textbooks in each stack:
    1     2     3       4       5       6       7
    1     4     7     10     13     16     19

Method 2: Use the equation for finding the nth element of an arithmetic sequence:
    an = a1 + d (n-1) where:
    an = the nth element (the one you're looking for)
    a1 = the first element and
    d = the number of textbooks added to each stack = 3
    a7 = 1 + 3 (7 - 1) = 1 + 18 = 19