Question

Nelson claims that when the length and the breadth of a rectangle change by +20% and -20% respectively, the area of the rectangle will remain unchanged. Do you agree? Explain your answer.

Answer 1

Answer: No, the area will not stay the same.

The new area is smaller than the old area.

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Step-by-step explanation:

Let's look at an example.

Consider a 10 by 30 rectangle. Its area is 10*30 = 300 square units.

Now let's change the length and width.

• Increase the "10" by 20% to get 10*1.20 = 12
• Decrease the "30" by 20% to get 30*(1-0.20) = 30*0.80 = 24

The rectangle is now 12 by 24. The new area is 12*24 = 288 square units.

In summary:

• old area = 300 square units
• new area = 288 square units

This is one counterexample, of infinitely many, that proves the area does not stay the same. Instead, the area shrinks.

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This section will go over a more formal proof using algebra.

• L = length
• W = width

old area = length*width = LW

Let's say the length increases by 20% and the width decreases by 20%.

• L becomes 1.20L after the 20% increase.
• W becomes 0.80W after the 20% decrease (since 100%-20% = 80%)

Then,

new area = (new length)*(new width)

new area = (1.20L)*(0.80W)

new area = (1.20*0.80)*(LW)

new area = 0.96*(LW)

new area = 0.96*(old area)

This shows that the new area is 96% of the old area. There has been a 4% reduction (since 100% - 96% = 4%)

Notice how 96% of 300 = 0.96*300 = 288. See the previous section above.