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Fraction word problem

May 17, 2011

Recently, a student had to tackle this word problem

The square above measures 10 inches. × 10 inches. and
the rectangle inside it measures 6 1/2 inches. × 31/4 inches
How many square inches is the colored area?
This requires the student to
  • multiply mixed numbers
  • subtract mixed numbers
  • work with area
  • understand how to find the coloured bit by subtraction of areas.

So it is a multi-step word problem.

He didn’t understand it,  he said. My first “help” was this:
“Let’s say we change those fractions to whole numbers 6 and 3. Can you mark those in the image? Would you be able to solve the problem now?”

The strategy I used is:
If you can’t solve the problem at hand, change it and make it easier. Then try to solve the easier problem.

He was able to mark 6 and 3 on the sides of the rectangle (that is inside the square). But he said he couldn’t solve it. He said it’s not possible to find the area of the colored area!

Then I asked him, “Is there anything you CAN find? Is there anything you CAN solve using this information?”

And this is another great strategy in solving any problem (whether math or not): If you can’t find the answer to the question in the problem, solve what you CAN solve. That might lead you to find the answer to your question somewhere along the way!

I said, “Well, we CAN find the area of the square. It is 100 square inches. We CAN find the area of the rectangle inside it.”

THEN immediately after I said that, he saw it: “OHH! SUBTRACT!” And on he went to multiply the mixed numbers in the problem. So the story had a happy ending!

Here’s the complete solution:

First multiply the mixed numbers 6 1/2 and 3 1/4 to find the area of the rectangle. Keep in mind they need to be changed into fractions before multiplying.

6 1/2 × 3 1/4 = 13/2 × 13/4= 169/8 =

Change this into a mixed number. Here one needs to use long division to divide 169÷ 8 = 21 R1. This tells us the whole number part is 21. The remainder,1, tells us how many 8th parts are “left over.”

So, 169 / 8 = 21 1/8  square inches. So this is the area of the rectangle.

Now, the area of the surrounding square is simply 10 in x 10 in. = 100 square inches.

And lastly, the area of the colored area is found by subtracting 100 − 21 1/8 = 78 and 7/8 square inches.


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