Olympiad Common Problems Homework - 5th Grade

1). Problem type: Find missing number of a list given the average.
Problem:
The average (mean) of 3 numbers is 21.
If the smallest number is removed, the average of the remaining 2 numbers is 27.
What is the value of the number that was removed? 		Solution:
1. The sum of the 3 numbers is _____.
2. The sum of the 2 remaining numbers is _____.
3. Subtract #2 from #1 to get the 3rd number = ______
2). Problem type: Areas of composite shapes
Problem:
What is the area of this garden?
(dimensions are in feet)
 Solution:
Cut it up into rectangles:
1. Determine the length of the unlabeled horizontal side (bottom) by using the 3 horizontal measurements given.
    Length = _____ ft.
2. Determine the length of the unlabeled vertical side by using the 3 vertical side measurements (you need all 3).
    Length = ____ ft.
3. Now, you can cut the garden into 3 rectangles, either with vertical or horizontal cuts.
4. With all the sides measured, you should be able to determine the area of each of the 3 parts. Then add the areas together.





Total area = ______ sq. ft.

3.) Problem type: Arithmetic sequence:
Problem:
Louis is stacking textbooks and starts forming a pattern. If the pattern below repeats, how many books will there be in the 7th stack?
 Solution:
1. Write out the total number of textbooks under each stack
2. Determine D, the number of textbooks added at each stack:
    D = ____
3. 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


    a7 = _____
4). Problem type: Guess-and-check
Problem:
There were 36 heads and 104 legs in a group of horses and riders. How many riders were in the group?
Solution:
(Make sure the number of horses and riders adds to 36)
Guess ## Horses# riderstotal
heads		total
legs
19273690
2    
3    
4    
5    

# riders = _____
5). Problem type: Permutations
Problem:
Four friends, Amber, Becky, Callie, and Darcy went to a party. If Callie was the third to arrive, how many different orders could the other three friends have arrived in? 		Solution:
The fact that Callie was the third to arrive makes this a 3-person permutation problem. The fact that Callie arrived 3rd is irrevelant. How many ways can you order 3 things or persons?


Number of ways = _____

6.) Problem type: Working a problem backwards
Problem:
A chocolate bar is separated into several equal pieces. If one person eats 1/4 of the pieces, and a second person eats 1/2 of the remaining pieces, then there are six pieces left over. Into how many pieces was the original bar divided? 		Solution:
1. If the second person ate half the remaining pieces and 6 were left, then there were ____ pieces before he ate his half.
2. If the first person ate 1/4th of the bar, then that person left ____ of the bar.
3. That fraction of the bar is what the second person got.
4. The bar was separated into ____ pieces.
7). Problem type: Turning words into equations:
Problem:
Trina, who is 10, wants to know how old her mother is. Her mother says two times the difference between our ages is the same as the sum of our ages plus 14 more. How old is her mom? 		Solution:
Use T for Trina and M for Mom.
1. Write the expression for "2 times the difference in our ages" =
    _____________________.
2. Write the expression for "the sum of our ages +14".
    _______________________
3. Set these two expressions equal to each other and solve




    M = _____
8). Problem type: Cartesian coordinates
Problem:
A square has three vertices (1,3), (7, 1) and (9, 7).
What is the fourth vertex?

Solution:
1. Plot the 3 points on the provided graph.
2. Find the offset from the 2nd point (7,1) to the 3rd point (9,7).
(Subtract the 2nd point's x and y coordinates from the 3rds)
Offset = (___,___).
3. Apply this offset to the 1st point (1,3) to get the 4th point = (___,___).