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Sample submitted solution:
From: Mihai, age 11
School: Mary Johnston Public School,
Waterloo, Canada
I will present the answer to question 2
first, because it will help answer all the
other questions.
Notation: I will denoted Squares by s, Circles
by c and Triangles by t.
2. Which coin is worth the least and
how many does it take to equal each of the
other coins?
The squares are worth the least. It takes
3s to equal 1c and it takes 7s to equal
1t. I found this out by experimenting with
the vending machine, and by finding out
how many coins of each type are needed to
buy a package of zoogs. We get the equations:
(1) 8s = a package of zoogs,
(2) 3c - 1s =a package of zoogs
(3) 2t - 2c =a package of zoogs.
I will use equations (1) and (2) to find
the relation between 1s and 1c. Because
the left hand sides of both equations (1)
and (2) equal one package of zoogs they
equal each other, so we can make the following
equations:
3c - 1s = 8s
3c = 9s (added 1s to both sides)
1c = 3s (divided both sides by 3).
Now that we know the relationship between
1s and 1c, we can find out the relationship
between 1s and 1t by using the equations
(1) and (3). Again the left hand sides of
both equations equal a package of zoogs,
so we can make the following equation:
2t - 2c = 8s
2t - 6s = 8s (substituted 2c for 6s)
2t = 14s (added 6s to both sides)
1t = 7s ( divided both sides by 2)
3. How many of the least valuable coins
would it take to buy a package of Zoogs?
As pointed out in the solution for 2, the
least valuable coin is the square, and it
takes 8s to buy a package of zoogs.
1. Buy a package of Zoogs. What is the
fewest number of coins needed to purchase
a package of Zoogs using exact change? Make
sure you list types of coins as well.
The smallest number of coins to buy a package
of zoogs without change is 2. Indeed, we
now know that a package of zoogs = 8s =
7s + 1s = 1t + 1s.
4. Buy a package of Glorps. What is
the fewest number of coins needed to purchase
a package of Glorps using exact change?
Include the types of coins in your answer.
How many of the least valuable coins would
it take to buy a package of Glorps?
The fewest number of coins needed to buy
a package of glorps is 3, namely 1s + 1c
+ 1t. Indeed, by experimenting with the
vending machine, we found out that a package
of glorps costs 11s. This equals 7s + 3s
+ 1s = 1t + 1c + 1s.
6. What is the fewest number and types
of coins needed to purchase a package of
Mushniks using exact change?
The fewest number of coins needed to buy
a package of mushniks is 4, namely 2t +
2s. Indeed, by experimenting with the vending
machine, we found out that a package of
mushniks costs 16s. This equals 14s + 2s
= 2t + 2s.
5. Another Earthling arrives on the
next shuttle and wants to try some Mushniks.
Explain how you figured out the relationship
between each coin.
Answer: I would show to the other Earthling
my solution to questions 2 and 6 (as presented
above), so that he/she knows how to buy
mushniks.
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