# Bob’s Parking Problem….Answer to Yesterday’s Question:

by on November 11th, 2009

The answer to yesterday’s question.  Try the problem first if you haven’t done so already and click more to see if you got the answer right.

A particular parking garage is increasing its rates by 15 percent per month. Bob decides to reduce the number of days he uses the garage per month so that the amount he spends at the garage per month remains unchanged. Which of the following is closest to Bob’s percentage reduction in the number of days he uses the garage each month?

A. 10%

B. 11%

C. 12%

D. 13%

E. 14%

The first important piece of this question is setting up the initial question.  Let’s call x the old number of days, X the new number of days, and r the old rate.

In order for the old price to equal the new price,

xr = 1.15Xr   (A quick way to represent at 15% increase is to just multiply by 1.15.)

By canceling the r variable, we are left with

x = 1.15X.

The wording in the question is tricky. To find the “percentage reduction,” we must find “what percent of X is x” and then subtract from 1 to find the “reduction.” To do this, we can rearrange the equation to read:

X = (1/1.15) * x

In essence, this reads X is (1/1.15) of x. Now we are only interested in (1/1.15). First let’s get rid of the decimal by multiplying by 100 to get 100/115. When we subtract 1 – (100/115), we get 15/115.

Great! We know 15/115 is our answer, but the choices are in percents. This is where approximation comes in.

Let’s reduce 15/115 to 3/23. We can estimate this fraction by changing the numerator and denominator slightly. By converting the fraction to 3/24 (1/8), we know we are making it smaller. (Increasing a denominator reduces a fraction.) So, 3/23 must be larger than 12.5% (1/8), which is a common and important conversion to memorize.

(A), (B) and (C) are out because they are smaller than 12.5%.

Moving in the opposite direction, we can then look at 3/21 (1/7). 1/7 = 14.3%. This is another conversion with which you should be familiar. Now we know the answer is between 12.5% and 14.3%

Since 3/23 is closer to 3/24 than it is to 3/21, we assume that, among answer choices (D) and (E), 13% will be correct, since 13% is closer to 12.5% than 14.3%.

Knowing your conversions is important, but there are other ways to go about this. You may multiply 3/23 by 4 to get 12/92. Add 10% to top and bottom and you get 13.2/101.2. Any of these steps could key out that the percent will be closer to 13% than 14%. Adjust the fractions to find ranges, compare to other well-known fractions, and generally speaking, if it seems like there must be an easier way, there usually is.

• assume:

100\$x20 days /mo = 2000
115 x X days = 2000

=> x = 400/23

%change: (20-x)x100/20
= (20-400/23)x5
=(460-400)/23 x 5
=300/23
Solve : 13.0xxx.... = 13% (rounded)

• Raghu, I agree with your method, same way I calculated this and much easier than the above explained method.

• Assume the original charge was \$60 for 30 days, \$2/day. New charge is 60 * .15 = 69.

This means he needs to cancel 4 days approx. So percent reduction in number of days = (4 / 30)*100 rounding ---->13%

• I think the last explanation is better and faster.

• I did easy plug - \$10 to start and 10 day for the increase and plug the other numbers

\$10.00 x 11.5 days = \$115

\$11.50 x 10 days = \$115 (cost increase 15%)

therefore change in days is 1.5/11.5 = 3/23 = 13%

• isn't the increase is 15% per month rather then 15% per day. wouldn't that make a difference in solving this problem.

• 1 - .15 +.0225 (a rapid way to estimate)
1 - .1275

12.75 percent.

1 - x + x^2

• Hey..these questions can be answered quickly with the help of fractions....

The price is increased by 15% i.e. 3/20.

So, the price has become 23/20 times the earlier price.

Hence, the consumption must be 20/23 times the earlier consumptions so that the expenditure remains same.

20/23 times the earlier consumption

i.e. 3/23 less than earlier consumption

i.e. 13% less than earlier consumption