Finding Percentages

Question

Brent has a diagram that he needs to place into a report. It is only 3.5" tall x 4" wide, but it needs to be about twice that big to make the impact he wants. Brent has use of a photocopy machine that can enlarge the height and width of an image by 20%. If he repeatedly uses the machine to enlarge each subsequent image, what is the minimum number of times he will have to make a copy to get the image to at least 7" tall? A. 1 B. 2 C. 3 D. 4 E. 5

Answer 1

A common first line of thought would be that in order to double the height (increase it by 100%), you would have to enlarge it by 20% 5 times.

However, thinking more accurately, you are not adding 20% of the original height with each iteration. Instead, you are adding 20% of the height that is now there, which means it should take fewer than 5 enlargements.

To calculate how many, we recognize that each enlargement multiplies the existing height by 1.2. This will result in exponential growth. To set up the equation, reason like this: 3.5" is the original height, and to multiply repeatedly by 1.2, we use exponentiation to raise 1.2 to the power we need:

n: # of 20% enlargements h(n): height of the image after n enlargements

h(n) = 3.5·1.2ⁿ , for n = 0, 1, 2, 3, etc. Finally, set h(n) = 7 and solve. This would require logs, and would result in an exact answer that was not a whole number, so instead we just need to find the first positive integer power of 1.2 that exceeds 2: 1.2¹ =1.2 , 1.2² = 1.44 , 1.23³ = 1.728 , 1.24⁴ = 2.0736.

So our answer is 4.