Word Problems with Percents
If the value of a painting increases by 15 percent every year, what was it worth 5 years ago? Hold up: why would you want to sell it now? In a few more years, you could be a millionaire. Patience is key.
|Ratios, Percentages, and Proportions||Percents|
When you express something as a percent, you are really determining how much per every
100. The same way cents are a portion of a dollar,
percents are a portion of 100%. Suppose you answered 24 out of 25 questions
correctly on your test. You just like to wreck the curve for everybody,
don’t you? First we find out how many times 25 goes into
100. So we can multiply 25 by 4 to get our magical number of 100. Then we multiply 24
by 4, which equals 96. Congratulations! You got a 96% on your test.
Percents pop up just about anywhere you can imagine.
For example, money matters are often expressed using percents.
Here’s a hint: the percentages in credit card terms and conditions usually are not
on your side – the best APR’s come in small packages.
Even seemingly small percentage changes in the stock market can have massive real-world
effects. Percents are a key part of the language of
economics, and sometimes just thinking in percents helps people to make good financial
decisions. For example, even if an outfit is “totally
super cute” you probably shouldn’t buy it if it costs more than fifty percent of
your net worth. If you’re a sports fan, you’re used to
hearing percents all the time. <<Announcer voice>> “Jones is only shooting
27 percent from the field tonight.” Field goal and free throw accuracy in basketball,
batting averages in baseball, passing completion rates in football… are all expressed in
percents. We hope your knowledge of percents has now
been improved by at least 62%. In fact, you even look 62% smarter than when
we started… But wait… there’s more!
You know that rare, valuable painting you found in the attic?
The one of all the trees? Well, apparently some collectors really like trees, because
it’s now worth $45,055. That is some fancy foliage.
Thank goodness you didn’t uncover it five years ago and sell it back then.
Now that we mention it… how much could you have sold it for five years ago?
Well, the painting’s value is appreciating annually by 15%.
So… how much would it have been worth half a decade earlier?
A - $10,265 B - $19,990
C - $21,780 D - $22,400
or E - $34,210? First, we’re going to need to pull a variable
out of our bag of tricks. Let’s make “V” the value of the painting
five years ago. Because we have the appreciated value and
want to find the original value, we should set our equation like this:
45,055 equals V times one plus point-one-five to the fifth.
Whoa – how did we get there? Well, the value of the painting increases
by 15% every year… …which is the “one plus point-one-five.”
The value is multiplied by that amount – one-point-one-five – for every year that passes.
So it’s the same as V times 1.15 times 1.15 times 1.15 times 1.15 times 1.15.
Okay, but our V is crowded with a whole mix of junk on the right side of the equation.
We want to get it by itself, so we can reconfigure the equation to look like this:
V equals 45,055 divided by 1.15 to the fifth. Then it’s simply a matter of doing the math…
or politely asking your calculator to do it for you
. Implore your calculator to first take 1.15
to the fifth… ...and then divide 45,055 by that result.
So looks like V equals $22,400.29. After dropping those 29 cents into your math
teacher’s tip jar, it appears our answer is option D.
Wow – it’s already worth nearly $23,000 more than it was five years ago!
At this rate, you’ll be able to retire in a few decades.
Assuming your future kids don’t one day try to add a few more trees.