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Percentage of Decrease

Date: 11/20/97 at 23:41:07
From: Eugene
Subject: What is the percent?

If last year I collected 500 cans and this year I only collected 350 
cans, what would be my percentage of decrease for this year?

Date: 12/19/97 at 09:09:13
From: Doctor Sonya
Subject: Re: What is the percent?

Hi Eugene,

The word 'percent' means out of 100, so 40% means 40 out of 100.  

Another way to say it is to take 40 things out of a hundred things, 
and that group of 40 things will be 40%.  

A percentage is also a fraction whose denominator is 100. 

Another thing we have to remember is that to express one quantity as a 
percentage of another, both quantities must be in the same units.  
This is just like trying to compare two fractions. It's hard to say 
which is bigger, 253/8876 or 344/6543, but if we give them both a 
common denominator, we'll be able to tell really easily. We'll do the 
same kind of thing with percentages, but instead of finding a common 
denominator, we'll turn them both into the same percentage.

To express 350 cans as a percentage of 500 cans, we are actually 
making 500 cans to a base of 100 to be our 100%.

We want the 500 cans to be 100%, so how do we make 500 into 100?  
We divide by 5.

  500 cans divided by 5 = 100

Now divide 350 by 5.

  350 cans divided by 5 = 70

We can now say that 500 cans is 100% of what you collected last year.

This year you collected 350 cans, which is 70% of what you collected 
last year. Another way to say this is that 350/500 = 70/100, but we 
like the ratio 70/100 better, because it's a percent too. To find the 
percent decrease, we just subtract:

  100% - 70% = 30%

This means that 70% (350 cans) is 30% less than what you collected 
last year (100%, 500 cans).

To see this more clearly, perhaps we can compare two sets of 
different numbers. 

For example:

Say you collected 500 cans last year and 350 cans this year (you 
collect 150 fewer.)

Say I collected 900 cans last year and 540 cans this year (I collect 
360 fewer.)

Although I collected more cans than you did, I also had a larger 
decrease in the number of cans I collected. Does that mean that I am a 
better or worse can collecter?

The best form of comparison is to change what we collected to 

Let's do what we did above with both of our can totals. We'll have to 
make both of our original numbers of cans be 100% (not at the same 
time, of course).

You originally collected 500 cans. To make your original number of 
cans be 100%, we have to divide by 5, just as we did above.

If  500 cans = 100%, then 350 cans = 350/500 x 100%  = 70%.

Notice here that instead if dividing 350 by 5 as I did above, I set up 
a  ratio:

  350/500 = x%/100%

and then solved for x% by multiplying both sides by 100%.  

Compared to your 500 cans last year, you only collected 70% of last 
year's total this year.

Your Percentage decrease is therefore

   = (100 - 70)%
   = 30%

Your percentage decrease is 30%.

Now we'll make my original number of cans 100%:

If 900 cans = 100%, then 540 cans = 540/900 x 100%  = 60%

Compared to my 900 cans last year, this year I only collected 60% of 
my last year's total.

My Percentage decrease is

   = (100 - 60)%
   = 40%

My percentage decrease is 40%.

So when we compare our two percentages, I did not collect as many as 
you, and I also had a bigger decrease in my collection of cans, while 
your percentage decrease was only 30%.

Remember that we are comparing the percentages to ourselves as well as 
comparing then against one another.

I am able to collect more cans. When you total up both years, I 
collected more than you. However, my collection this year was much 
less than last year: I only collected 60% of my last year's total. So 
in this example, I have a decrease of 40%. 

You, however, did better than I, as you were able to collect 70% of 
your last year's total. Comparing your last year's total and this 
year, you did better since your decrease was only 30%.

We have just used percent decrease to compare our can collecting 
abilities.  I bet the recycling center will be thrilled :).

Got the idea ?
Happy Calculating.
Try out this on other numbers.

-Drs. Sonya and Dr. Lim,  The Math Forum
Associated Topics:
Middle School Fractions

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