August 19, 2008

Bar diagram problem

This was asked of me as of today:
Please solve this using the bar/block diagram method. My friends and I are stumped....

Desmond had 480 more oranges than pears. After selling half of his oranges and half of his pears, he had four times as many oranges as pears left. Find the number of pears he had at first.
Thank you!
This problem is from a Primary School Leaving Examination (PSLE) paper. PSLE is the final examination for primary school students in Singapore. So, you would expect to see these kind of problems in Singapore Math.

My first attempt for solving this was like this:
pears and oranges bar diagram

It shows the difference being 480. The red lines are halving the quantities of pears and oranges.

But I quickly noticed this was way off. The amount of pears needed to be way less than the amount of oranges.

My second attempt was like this:
Pears and oranges bar diagram
It was a little better, but half the pears looked more like 1/3 of the half the oranges. So the amount of pears needed to still be less.

Pears and oranges bar diagram
This is the final diagram that solves the problem. The main idea is that the parts "match" - that you can see the relationship 1 to 4 in the amount of pears versus oranges, or in the halved amounts.

Actually, the CRUCIAL point of this problem - that there are 1/4 as many pears as there are oranges - can be understood without any kind of bar diagrams. You see, if that is true of the halved quantities (4x as many oranges as pears), then the same is true of the original quantities as well!

So without ever drawing anything, one can figure that there are 4 times as many oranges as pears. The DIFFERENCE is 480. The difference is also 3/4 of the oranges. So if 3/4 of the oranges is 480, then 1/4 of the oranges is 480 ÷ 3 = 160 (which is the number of pears). And the total number of oranges is then 4 x 160 = 640. That solves the problem then without the usage of bar diagrams or algebraic equations.

Bar diagrams can be of enormous help in visualizing the relationships between PARTS in the problem. BUT, to draw one is not necessarily easy because to get it right (like in my case) actually REQUIRES understanding something about the problem.

So, let the bar diagrams be a tool that helps you understand the problem ALSO in this sense: while you try to draw one, let the misdrawn ones guide your thinking towards the right ideas.
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