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### Business Mathematics

Percentage Formulas

Lesson Objectives

After studying this module, you should be able to:

- identify percentage, rate and base
- find the percentage when the rate and the base are given
- find the rate when the percentage and the base are given
- find the base when the percentage and the rate are given
- apply the concept in solving business-related word problems

Lesson Content

Definition of Terms

- Rate ( R ) – the rate is the percent
- Base ( B ) – the result obtained when a number is divided by a percent
- Percentage ( P ) – is the product of the rate (in decimal) and the base.

In symbols,

P = R x B

How do you identify percentage, rate and base?

Study the statement below:

Forty percent of one hundred fifty is sixty.

Forty percent of one hundred fifty is sixty.

40% x 150 = 60

40% in decimal is 0.40

Therefore, the above statement written in mathematical symbols is

0.40 x 150 = 60

Remember

The quantities to the left and right of the word “of” are the rate and the base, respectively. These two are separated from the percentage by the word “is.”

Example 1

Sixty percent of one thousand eighty is what number?

Now, let’s identify the parts.

R B P

Sixty percent of one thousand eighty is what number?

60% x 1080 = Percentage (P)

Therefore, the statement written in mathematical symbols is,

0.60 x 1080 = P

648 = P

Example 2

Twenty percent of three thousand fifty is what number?

25% x 3,050 = P

0.25 x 3,050 = P

762.5 = P

Example 3

What number is twenty percent of five hundred seventy?

P = 20% x 570

P = 0.20 x 570

P = 114

Sometimes, the rate or the base is the one missing. Consider the formula P = R x B, again.

P = R x B

To find rate (R) when P and B are given, divide both sides by B.

Then,

(Note: R here is in decimals)

Example 1 the formula P = R x B, again.

R B P

What percent of sixty is forty – two?

R x 60 = 42

Since the rate is unknown, use the formula given to solve R.

Substitute all the given values to the formula. the formula P = R x B, again.

Change 0.7 to percent, that is, 70%.

Recall the formula P = R x B, again.

In changing a decimal number to percent form, simply multiply the given decimal number by 100 and affix percent sign (%).

Example 2 the formula P = R x B, again.

What part of eight hundred ten is one hundred sixty – two?

R x 810 = 162

(simplify )

Note! the formula P = R x B, again.

If asked “what part” in solving the rate, the final answer is express in decimal or fraction form.

Example 3 the formula P = R x B, again.

Thirty is what percent of eighty?

30 = R x 80

R = 0.375

R = 37.5%

Similarly, to find base (B) when P and R are given, divide both sides of the original formula by R.

That is,

Then,

Example 1 both sides of the original formula by R.

R B P

Five percent of what number is nine hundred?

5% x B = 900

Since the base is unknown, use the formula given to solve B.

Substitute all the given values to the formula. both sides of the original formula by R.

(5% is 0.05 in decimal)

B = 18,000

Example 2 both sides of the original formula by R.

Eighty percent of what number is three hundred?

80% x B = 300

B = 375

Example 3 both sides of the original formula by R.

Forty – six is two percent of what number?

46 = 2% x B

B = 2,300

Note! both sides of the original formula by R.

It should be noted that the base is usually preceded by the preposition “of”. Sometimes, however, “as great as”, “as many as”, and “as much as” are used instead of “of”.

Key Concept both sides of the original formula by R.

- To find the percentage when the base and the rate are known, multiply the base by the rate.
- To find the base when the percentage and the rate are given, divide the percentage by the rate expressed in decimal or in fraction form.
- To find the rate when the percentage and the base are given, divide the percentage by the base.

Assessment both sides of the original formula by R.

I. Solve for the value of the unknown.

1. Find 12% of 843.

2. Php 42 is 14% of what amount?

3. What is 35% of 80?

4. 6% of 1,080 is what number?

5. 5% of Php1.50 is what part of of Php30.00?

II. Solve the following problems: both sides of the original formula by R.

6. Bamhad 96 pairs of shoes on hand. He sold 50% of them at P85.00 each, 25% of the remainder at P82.50 each and the rest at P80.00 each. How much did he receive for the entire lot?

7. Charm, a government employee was given an increase in salary of Php 880. If this was a raise of 5% of the salary which she had been receiving, what was her previous monthly salary?

8. Miranda bought a second-hand laptop for Php12,000 and resold it for Php Php9,000. What part of the resale price was the buying price?

9. After investing 60% of her capital in construction. Engineer Ysabel had Php70,000 left. What was her capital?

10. Xayvion, a financial analyst of a certain firm receives a monthly salary of Php45,000 which is

as much as vice-president’s salary. What is the vp’s salary?

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