Problem of the Week
Problem A and Solution
Balancing Act
Problem
Clare wants to know the mass of a box of chocolates. She has several boxes of chocolate, some
weights and a scale. She is sending one box to her mother for Mother’s Day and needs to know
how much the postage will cost to mail her package. Clare was able to discover the following:
How much does one box of chocolates weigh?
Solution
If we look at the first diagram we see that 3 pentagons have a total mass of 24 kg.
Therefore, the mass of 1 pentagon is 24 kg ÷ 3 = 8 kg.
Knowing that each pentagon has a mass of 8 kg, we can say that the total mass
of 2 pentagons is 2 ⇥ 8 kg =16 kg.
From the scale shown on the second diagram we see that the mass of 4 boxes of
chocolates is equal to the mass of 2 pentagons which is 16 kg.
To find the mass of one of the boxes of chocolates we take our 16 kg for 4 boxes
of chocolates and divide by 4. Thus, the mass of one box of chocolates is
16 kg ÷ 4 = 4 kg.
Alternatively, we can see that the mass of 2 pentagons is equal to the mass of 4
boxes of chocolate.
So, the mass of 1 pentagon is equal to the mass of 2 boxes of chocolates. Since
1 pentagon weighs 8 kg, this means 2 boxes of chocolates would weigh 8 kg.
Therefore, 1 box of chocolates weighs 8 kg ÷ 2 = 4 kg.
Therefore Clare would need postage sufficient to cover the cost of a 4 kg box of
chocolates.
Teacher’s Notes
This type of problem lends itself nicely to an algebraic solution, which is a
technique students would see in later mathematics. Algebraically, we could solve
the problem this way.
Let p represent the mass of one pentagon.
Let c represent the mass of one box of chocolates.
From the diagram we know that:
3p = 24 (equation 1)
and
2p = 4c (equation 2)
From equation 1, we can divide both sides by 3 and get:
3p 24
=
3
3
1
3p
◆
1
3
◆
=
8
⇢
24
⇢
1
3
◆
p = 8 (equation 3)
Now, we substitute the value of p from equation 3 into equation 2 and get:
2(8) = 4c
16 = 4c (equation 4)
Now, from equation 4, we can divide both sides by 4 and get:
16 4c
=
4
4
4
⇢
16
⇢
1
4
◆
1
=
4◆c
1
4
◆
4=c
Since c represented the weight of one box of chocolates, the mass is 4 kg.