In this article, we will discuss some common problems students encounter with percent problems, which can come in a variety of formats. Here are some quick pointers:

Percents MUST be APPLIED to something

A percent means nothing on it’s own.

Example: 16% of men, or 30% off the sales price

Percents are basically fractions with a denominator of 100

Learn your common percents, and convert to fractions whenever possible.

Example: 20% = 1/5, 62.5% = 5/8

The word “of” means multiply

Example: 80% of men = 4/5 * (total # of men)

Percents higher than 100 are numbers higher than 1

Example: 125% = 100% + 25% = 1 + 0.25 = 1.25

Recognize the difference between percent MORE/LESS THAN and percent OF

Example:
What is 25% less than 8?
¼* 8 = 2, so 8 – 2 = 6

Example:
What is 25% of 8?
¼*8 = 2

Use shortcuts

20% less than means 80% of. So instead of taking 20%, then subtracting from the original, just take 80% and be done. Conversely, 50% more than 10 should be calculated by multiplying 10*3/2 [10*(1 + 0.5)] in one neat step, versus two tougher ones.

See the previous example:
What is 25% less than 8?
¾*8 = 6, and we’re done! On easy numbers like this, it might not seem necessary, but as numbers get larger, it will save lots of time.

The higher the number, the higher the resulting percent

Applying the same percent to a higher number will yield a higher number.

Example:
A certain positive integer x is increased by 10%, and then decreased by 10%. Which is bigger, x or the resulting number?
The 10% increase of x in the first round increases x by a certain amount. The 10% decrease in the 2nd round is applied to a higher number, so will yield a larger change. The original x will be bigger.

Percent change = Total Change/Original Value

Example:
Before trading began, James’ investment portfolio was worth $10,000.  At the end of market close, James’ investment portfolio grew by $2,000.  What was the percent change in James’ portfolio?
Percent change = $2,000/$10,000 = 0.2, or 20%

Don’t add constants and percents

You should never find yourself trying to figure out what 5 + 6% equals. In this case, you are probably missing what to apply the percent to.

Let’s take a look at two examples!

Example 1:

A tour group of 25 people paid a total of $630 for entrance to a museum. If this price included a 5% sales tax, and all the tickets cost the same amount, what was the face value of each ticket price without the sales tax?

A. $22

B. $23.94

C. $24

D. $25.20

E. $30

Without a calculator, fractions are always easier. They cancel well, and are typically neater.

5% = 1/20 since 5*20 = 100.

Now we set up the equation, setting x = ticket price before tax.

25 people * x dollars/person * 1.05 (with tax) = $630

Note we can convert to fractions, cancel and simplify. Look how easy it gets?

25*(21/20)*x = 630

5*(21/4)*x = 630

x = 630*4 / 5*21

x = $24

Choice C

Example 2:

During an auction, Jerome sold 75% of the first 1,000 items he offered for sale, and 30% of his remaining items. If he sold 40% of the total number of items he offered for sale, how many items did Jerome offer for sale?

A. 750

B. 1,050

C. 1,800

D. 3,500

E. 4,500

Again, we want to set up the equation – this will make things a lot easier. And again, switching to fractions is always best.

3/4*1000 + 3/10*R = 4/10*T

We have 2 equations, and 1 unknown. This is a good hint that there may be a hidden 2nd equation.

1000 + R = T

Now, we have 2 equations and 2 unknowns. We can solve!

750 + 3R/10 = 400 + 4R/10

350 = R/10

R = 3,500

We always look back to the original question to see exactly what we are looking for. In this case, T. Not R.

T = R + 1,000 = 3,500 + 1,000 = 4,500

Choice E

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