Percentages: Complete Guide, Formulas & Practice Questions

Now that you have gone through the fractions topic, you might become a little familiar with the percentages. A percent is a fraction. A special fraction that always have 100 in the denominator. For instance

x100\frac{x}{100}

regardless of what the value of x is here.

However, instead of always using 100 in the denominator to identify a fraction as a percentage, we use a handy little symbol that denotes a percentage:

“%““\%”

Whenever you see this symbol after a number, it means the number is a percent. For instance:

x%→where,x=numberx\%→where , x=number

So we can say:

15%=15100,15\%=\frac{15}{100},

or

25%=2510025\%=\frac{25}{100}

Special case, zero %. Is there something as 0%?
Yes, there is. Which might simply mean \(\frac{0}{100}\). As it is okay to have a zero in the numerator rather than in the denominator.

Improper Fraction Percentage:

A percentage greater than a 100% is known as an improper fraction percentage. Question is, can there be one? Yes there can be as follow:

125%=125100125\%=\frac{125}{100}

Converting a percentage into decimals is easier than converting any other fraction. As the bottom number is always a hundred, you are just have to move the decimal. For instance:

65%65\%

or

65100=0.65\frac{65}{100}=0.65

0.65⏟←2 places65.000.65\underbrace{\hspace{1.2cm}}_{\leftarrow\;2\text{ places}}65.00

vice versa, when to convert a decimal to the percent:

0.650.65

First multiply decimal with 100

0.65×1000.65\times100

Then move the decimal two places to the right

0.65⏟→2 places65.00 0.65\underbrace{\hspace{1.2cm}}_{\rightarrow\;2\text{ places}}65.00

you get

=65%=65\%

There are a few you should remember:

Percentage rules
  1. Percent/Percentage Of:

The word “of” usually means multiplication.
If a question asks for p% of a number x, convert the percentage to a decimal or fraction and multiply:

p%ofx=p100×xp\%of x=\frac{p}{100}\times{x}

Lets do an example:

A school has 640 students. Of these students, 62.5% participate in at least one extracurricular activity. Of the students who participate, 35% participate in a sports program. How many students participate in a sports program?

First find 62.5% of 640:

0.625(640)=4000.625(640)=400

Now find 35% of those 400 students:

0.35(400)=1400.35(400)=140

As a result we get:

=140=140

Be careful about what quantity the percentage is “of.” Here, 35% is not 35% of all 640 students; it is 35% of the 400 participating students.

  • 2. Percentage Change

Percentage change measures how large an increase or decrease is relative to the original value.
The fundamental formula is:

PercentageChange=NewValue−OriginalValueOriginalValue×100%PercentageChange = \frac{NewValue-OriginalValue}{OriginalValue}\times100\%

Where,
For an increase:

PercentIncrease=IncreaseOriginalValue×100%Percent Increase=\frac{Increase}{OriginalValue}\times100\%

For a decrease:

PercentDecrease=DecreaseOriginalValue×100%PercentDecrease=\frac{Decrease}{OriginalValue}\times100\%

Lets do an example:

The number of subscribers to an educational website increased from 2,400 to 3,060. By what percentage did the number of subscribers increase?

First calculate the increase:

3060−2400=6603060-2400=660

The original value is 2,400, so:

6602400×100%\frac{660}{2400}\times100\%

0.275×100%0.275\times100\%

=27.5%=27.5\%

The denominator is normally the original value, not the new value. This is one of the easiest places to make a percentage mistake in a time constraint aptitude test.

  • 3. Adding a Percent — Percentage Increase

When a quantity is increased by a percentage, you are keeping 100% of the original quantity and adding another percentage to it.

For example, increasing something by 18% means:

100%+18%=118%100\%+18\%=118\%

or,

(1+p100)(1+\frac{p}{100})

where,

p=percentage=18p=percentage=18
(1+18100)(1+\frac{18}{100})
=1.18=1.18

Therefore, when a certain value is given to be increased by a percentage, following is the formula you can use to solve:

NewValue=OriginalValue(1+p100)NewValue=OriginalValue(1+\frac{p}{100})

Lets do another example:

A company’s annual revenue was $480,000. It increased by 15% in the first year and then increased by another 20% in the second year. What was the company’s revenue after the second increase?

A 15% increase:

(1+15100)=1.15(1+\frac{15}{100})=1.15

After the first increase:

NewValue=480000(1+15100)NewValue=480000(1+\frac{15}{100})
480000(1.15)=552000480000(1.15)=552000

Then, a 20% increase:

(1+20100)=1.20(1+\frac{20}{100})=1.20


Now the second 20% increase applies to $552,000, not the original $480,000:

NewValue=552000(1+20100)NewValue=552000(1+\frac{20}{100})
552000(1.20)=662400552000(1.20)=662400

Therefore,

$662,400\$662,400

Avoid the most common mistake by saying:

15%+20%=35%15\%+20\%=35\%


and increase $480,000 by 35%. That would be incorrect because the second increase applies to an already increased amount. Successive percentage increases compound. They generally cannot simply be added together.

  • 4. Subtracting a Percent — Percentage Decrease

When a quantity decreases by a percentage, subtract that percentage from 100%.

If something decreases by 25%:

100%−25%=75%100\%-25\%=75\%

So the new value is 75% of the original:

NewValue=OriginalValue(1−p100)NewValue=OriginalValue(1-\frac{p}{100})

A 25% decrease therefore means multiplying by:

0.750.75

Lets do an example:

A jacket originally costs $320. During a sale, its price is reduced by 30%. After the discount, a 7.5% sales tax is applied. What is the final price?

First apply the 30% decrease:

320(1−0.30)320(1−0.30)
320(0.70)=224320(0.70)=224

Now apply the 7.5% tax to the discounted price:

224(1.075)=240.80224(1.075)=240.80

Therefore, the final price is:

=$240.80=\$240.80

note that you should not simply calculate:

30%−7.5%=22.5%30\%-7.5\%=22.5\%

because the discount and tax are being calculated from different base values.

One important distinction mandatory to be paid attention to in the aptitude test is the vocabulary used in the question.

“What is 20% of 500?”

500(0.20)500(0.20)

Is different than,

“500 is increased by 20%.”

500(1.20)500(1.20)
Percentage problems

In percent problems, the whole generally will be associated with the word of, and the part will be associated with the word is. The percent can be represented as the ratio of the part to the whole, or the is to the of.

For example: What is 25 percent of 36?

Here you are provided with the percent and the whole. To find the part, change the percent to the fraction, and then multiply. Following formula can be used:

x%ofy=x100×yx\%ofy=\frac{x}{100}\times y

As 25% = \(\frac{1}{4}\), you are really being asked what one-fourth of 36 is.

14×36=9\frac{1}{4}\times36=9

Similarly,
18 is what percent of 3?

m×3=18m\times3=18

m=6m=6

6×100%=600%6\times100\%=600\%

Practice questions

1. In the first quarter of the year, a certain entrepreneur sold 36 percent of the 75 crafts that she put for sale online. In the second quarter of that year, she sold one-fourth of the remaining crafts. What was the percent decrease from the first quarter to the second in the number of crafts the entrepreneur sold?

  • 12%
  • 25%
  • 55\(\frac{5}{9}\)%
  • 61%

2. Alma bought a laptop computer at a store that gave a 20% discount off its original price. The total amount she paid to the cashier was p dollars, including an 8% sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p?

  • 0.88p
  • \(\frac{p}{0.88}\)
  • (0.8)(1.08)p
  • \(\frac{p}{(0.8)(1.08)}\)

3. How many liters of a 40% saline solution must be added to 6 liters of a 20% saline solution to obtain a 25% saline solution?

  • 1.5 Liters
  • 2 Liters
  • 2.5 Liters
  • 3 Liters

4. The result of increasing the quantity x by 400% is 60. What is the value of x?

  • 12
  • 15
  • 240
  • 340

5. 13 is 33\(\frac{1}{3}\)% of what number?

  • 26
  • 39
  • 36
  • 43
Click to reveal answers

  1. C
  2. D
  3. B
  4. A
  5. B
Explanations to the above answers: