Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 5 Algebra
Problem
Solution
1) Jim likes to text message. The cell phone family plan his parents have allows for 250 text messages per month for $5.00 as part of the plan and additional text messages cost 15 cents per message both incoming and outgoing. What is the total number of messages Jim can send and receive for less than $10.00 if no one else in his family texts? 1. If Jim wants to keep the cost of text messages under $10 then the amount he can spend on messages above 250 is $5
2. At 15 cents per message compute the number of messages he can send or receive above 250: 500/15 = 33.333 (round down to 33)
3. Add these to the number of base messages to get the total number of messages under $10: 250 + 33 =
    283 messages.
2) Hannah has some pencils in her pencil case. She loans 1/6 of the pencils to Tommy and then 3 pencils to Marc. On her way home from school Hannah loses 50% of her remaining pencils through a hole in her bag. When she gets home she realizes that she only has 6 pencils left. How many pencils did Hannah start the day with? Method 1: Work backwards:
1. If she lost 50% of the remainder from a hole in her bag and the number left is 6, then before she lost the pencils through the hole she must have had 12 pencils.
2. Just before that she gave 3 pencils to Marc, so before Marc she must have had 15 pencils.
3. Before that she loaned 1/6 of her pencils to Tommy, so what's left is 5/6 of the original number of pencils.
    5/6 of X = 15
    X = (6 x 15)/5 = 90/5 = 18
    So, the original number of pencils is 18

Method 2: Use guess-and-check:
Guess ## original pencils  -1/6    -3  -50%
12420178.5
21815126

Method 3: Write an equation:
1. Let P = original number of pencils.
2. Then the number of pencils before the hole appears is
    (P - P/6 - 3)
3. Then 50% of that is 6 pencils.
4. Write the equation for this and solve.
    (P - P/6 - 3)/2 = 6
    (5P/6 - 3)/2 = 6
    5P/6 - 3 = 12
    5P/6 = 15     5P = 90     P = 18

Problem
Solution
3) Pictured is a sequence of growing chairs. The first chair is made of 6 squares. How many more squares are in the 8th chair in the sequence than in the first?
 Method 1:Write out the sequence of numbers
1. The sequence of numbers is:
    6   10   14   18   22   26   30   34
2. In this sequence, 4 is added to each element to get the next one.
3. The 8th chair has 34 squares.
4. Subtract the original number of squares in the first chair to get the answer: 34 - 6 = 28

Method 2:Use the equation for the nth term of an arithmetic sequence.
1. The equation for the nth term of an arithmetic sequence is:
    An = A1 + D (n - 1), where:
    An is the nth term (the one you're looking for)
    A1 is the first term
    D = the number added to each term
2. Write out the equation with the parameters
    of this problem and solve for An:
    An = 6 + 4(8 - 1) = 6 + 4 x 7 = 34.
    Subtract the number of squares in the first chair
    to get the answer = 34 - 6 = 28
4) A boy lives on the same road as his school and the road is straight. He bikes to school. When he reaches halfway he notices he has dropped his math book. He bikes back 400 yards to pick it up. He is now 1300 yards from school. How far is his house from school? Solution: 
S----------B---------H-----------------------S
|          |<--400-->|
|          |<--------------1300 ------------>|
|          |         |<----- 900 ----------->|
|------------------1800 ---------------------|
Home      book     half                school
1. The distance from half way to school: 900 yards
2. The whole distance from home to school:
    900 x 2 = 1800 yards.
5) Mr. Jones takes three Uber rides with the same cab driver. The table shows how many miles he traveled and what the fare was. Determine what the base fare is for a trip that is 6 miles long.
 1. The fare increase for every 2 miles is
    $14.50 - $8.50 = $6.00
2. The fare for a 6 mile trip is $14.50 + $6.00 = $20.50