Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 6 Measurement
Problem
Solution
1) A square has an area of 10 square meters. Its side lengths get tripled to create a new square.
What is the area of the new square? 		Method 1: Analysis:
Since the sides were tripled, the area has been increased by the square of this =
    3 x 3 = 9
    The area is now 9 x 10 =     90 sq. m.

Method 2: Brute force!
1. Take the square root of 10 = 3.16228 m.
2. Multiply this by 3 = 9.48683 m.
3. Square it = 90 sq. m.
2) Esmeralda is buying 6000 batteries. The mini pack has 30 batteries for $70.50, the family pack has 500 for $1165. The jumbo pack costs $2844 and has 1200 batteries. Which is the best buy? Divide the cost by the number of batteries for each option.
    Mini = 70.50/30 = $2.35
    Family = $1165/500 = $2.33
    Jumbo = $2844/1200 = $2.37
    The best is Family
3) A truck drives from a factory to a warehouse and then takes the same route back to the factory. On the trip to the warehouse, the truck's average speed was 30 miles per hour. On the trip back to the factory, the truck's average speed was 60 miles per hour. What is the truck's average driving speed for the whole trip? Careful! The answer is not (30 + 60) / 2!

Assume the trip is 60 miles.
1. Compute the time for the trip to the warehouse = 60/30 = 2 hours.
2. Compute the time for the return trip = 60/60 = 1 hour.
3. Add these 2 together and divide 2 X the distance by this time =
    2 X 60/ (1 + 2) = 120/3 = 40 miles/hour.

Problem
Solution
4) A road race is taking place for bicyclists. Susan put a blue marker post every 2600 meters along the route. Ravi places a red marker post every 1800 meters. How far from the start is the first place re a blue post and a red post will be together? This is a least common multiple (LCM) problem.

The LCM of 2 numbers is their multiple divided by their greatest common factor (GCF):
    LCM(a,b) = a x b / GCF(a,b)
1. To simplify the arithmetic, divide both distances by 100 = 26 and 18
2. Find the greatest common factor of these 2 numbers = 2
3. Multiply the 2 reduced numbers together and divide by this GCF =
    26 x 18 / 2 = 234
4. Multiply by 100 to get the true distance = 23400 meters.
5) A bowl is full of blue, red, and green jelly beans. There are three times as many blue as there are red. There are also two times as many green as there are blue. Altogether there are 210 jelly beans in the bowl. How many blue jelly beans are in the bowl? Method 1: Count them up from red:
Assume you have 1 red jelly bean.
1. If there are 3 times as many blue as red, then there are 3 blue ones.
2. If there are 2 times as many green as blue, then there are 6 green
3. Compute the sum of 1 + these 2 numbers =
    1 + 3 + 6 = 10
4. Divide the number of jelly beans by this sum =
    210/10 = 21
5. Multiply the number of blue beans, above by this factor to get the total blues =
    3 x 21 = 63

Method 2: Use algebra.
  1. Let R = # of red, B = # of blue and G = # of green marbles
  2. Then
    B = 3R and
    G = 2B = 2(3R) = 6R
  3. Write the equation:
    R + B + G = 210
    R + 3R + 6R = 210
    10R = 210, so R = 21
    Therefore B = 3R = 63