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 Math Help > Basic Math > Word Problems and Basic Arithmetic > Percentage Change

Previously, I gave you some tips on figuring percentages.  This page tells you how to work problems involving percentage change -- percentage increase or percentage decrease.

A percentage increase or decrease is a way of comparing two quantities -- an earlier quantity and a later one.  The percentage change is always the difference divided by the earlier quantity.  Let me illustrate with an example:

Question: If Joe had 6 apples yesterday, and he has 8 apples today, what is the percentage increase of his apples?

Answer: It's the difference, 2, divided by the earlier quantity, 6, expressed as a percentage.

2/6 is what percent?  My previous page helps you answer that question -- it tells you to write the question this way:

2/6 = x/100

To solve for x, just multiply both sides by 100, so you get 200/6 = x, so x = 33 1/3.

The final answer, then, is that Joe has had an increase of 33 1/3 percent in his number of apples.

Now, fast forward another day, after Joe has eaten two of his apples.  Here's the question:

Question: If Joe had 8 apples yesterday, but then ate two of them (leaving him 6 apples), what is the percentage decrease of his apples?

Answer: Again, it's the difference, 2, divided by the earlier quantity.  In this question, the earlier quantity is 8, not 6, do you understand that?  It's important!

So the answer is the difference, 2, divided by the earlier quantity, 8.

2/8 is what percent?

2/8 = x/100

Solve it for x, and you'll get x = 25, so the final answer is that Joe has had a decrease of 25 percent in his number of apples.

Something odd has happened.  Did you catch it?  Joe's apple ownership started at 6, then increased by 33 1/3 %, then decreased by 25 %, and now he owns 6 apples again.  Unless you've been dealing with percentages a lot in your life, this fact comes as a bit of a surprise, doesn't it?  The moral of the story is that a percentage increase can be offset by a relatively smaller percentage decrease.  This happens because both the increase and the decrease are measured as a fraction of the earlier number -- remember?  It's the difference divided by the earlier number.  When the change is an increase, the earlier number is smaller, so a difference translates to a larger percentage change.  When the change is a decrease, the earlier number is larger, so the same difference (an oxymoron, I know!) translates to a relatively smaller percentage change.

Another example:

Question: Mary receives an allowance of \$6, which is a 20% increase over last week's allowance.  What was last week's allowance?

Answer: Let "x" be the amount of last week's allowance, which was smaller than \$6.

The difference between this week's allowance and last week's allowance is 6-x.

The percentage change is the difference divided by the earlier amount, so it is

(6-x) / x, which we are told is 20%, and 20% means 20/100.  So we have the equation,

(6-x) / x = 20 / 100

Now it's just a matter of solving for x.  First multiply by 100, then multiply by x, to get rid of all the denominators, so you get

100(6-x) = 20x

600 - 100x = 20x

600 = 120x

x = 600/120 = 60/12 = 5

So x=5.  That is, Mary's allowance last week was \$5.

Maybe you were hoping not do have to do all that algebra just to answer a "percentage question" but let me tell you: you'll need to learn algebra sooner or later, so you might as well start now.  Then you'll have two useful skills -- figuring percentages and doing algebra.  And you'll never forget either one, I swear!

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The webmaster and author of this Math Help site is Graeme McRae.