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### Calculating Percentage Increase

```Date: 07/09/2004 at 15:38:52
From: Sharon
Subject: (no subject)

If my salary increased from 45k to 65k over a period of 10 years, what
percentage increase did I receive per year?

```

```
Date: 07/10/2004 at 12:24:57
From: Doctor Ian
Subject: Re: (no subject)

Hi Sharon,

It depends on how you're defining things.  To show you what I mean,
suppose you're making \$45K in 1990.  10% of that is \$4.5K.  If you
receive an increase of \$4.5K per year for 10 years,

1990   \$45,000
1991   \$49,500         add 10% of \$45,000
1992   \$54,000         add 10% of \$45,000
1993   \$58,500         add 10% of \$45,000
1994   \$63,000
.
.
2000   \$90,000

But suppose each year, you receive an increase amounting to 10% of
what you made that year.  Now each increase is a little bigger than
the last:

1990   \$45,000
1991   \$49,500         add 10% of \$45,000
1992   \$54,450         add 10% of \$49,500
1993   \$59,895         add 10% of \$54,450

What's going on in this case is that each year, the previous salary is
multiplied by 1.10 (100% plus 10%), so after 10 years the salary would be

10
45,000 * 1.10   = 45,000 * 2.59

= 116,550

you just take the total difference and divide by 10 to get the yearly
increase,

65,000 - 45,000   20,000
--------------- = ------ = 2,000
10            10

and then you figure out what percentage that is of the original
salary:

2,000    ?
------ = ---
45,000   100

2    ?
-- = ---
45   100

2*100 = 45*?

200
--- = ?
45

which comes out to about 4.4%.  That is, if you increase your original
salary by 4.4% OF THE ORIGINAL SALARY for 10 years, you end up with \$65K.

In the second case, the situation is this:

10
45,000 * ?   = 65,000

We can start by dividing each side by 45,000:

10
?   = 65,000 / 45,000
10
?   = 1.44

Now what we want to know is:  What number, raised to the 10th power,
gives us 1.44?  It turns out that

10
1.037  = 1.438

so if you increase your original salary each year by 3.7% OF THE
PREVIOUS YEAR'S SALARY, after 10 years you end up with \$65K.

Which method you prefer might depend on whom you're trying to convince
of what.  :^D

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Fractions

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