# Solving Percent Problems

Basically, there are 3 types of percent problems:

• What is p% of m?
• p% of what is m?
• What percent of m is n?

Now, we will practice in these percent problems.

These types of problems can be easily solved using proportions.

Example 1. What is 20% of 35?

Let n represents required number.

Percent can be written as ratio 20/100.

From another side the same ratio can be represented as n/35.

We obtained proportion 20/100=n/35.

Solving it, we obtain that n=7.

Therefore, 20% of 35 is 7.

We can generalise this result.

p% of m is p/100*m.

Example 2. 95% of what is 237.5?

Let n represents required number.

Percent can be written as ratio 95/100.

From another side the same ratio can be represented as 237.5/n.

We obtained proportion 95/100=237.5/n.

Solving it, we obtain that n=250.

Therefore, 95% of 250 is 237.5.

We can generalise this result.

p% of what is m? Of m/100*p.

Example 3. What percent of 15 is 27?

Let p represents required percent.

Percent can be written as ratio p/100.

From another side the same ratio can be represented as 27/15.

We obtained proportion p/100=27/15.

Solving it, we obtain that p=180%.

Therefore, 27 is 180% of 15.

We can generalise this result.

n is n/m*100% of m.

Using above 3 types of percent problems, we can solve some real-world problems.

Example 4. Initially population of some town was 200000 people. Recently it has grown by 15%. What is the current population?

First, we need to find by how many people population has grown?

In other words, what is 15% of 200000? Answer is 15/100*200000=30000.

So, the current population is sum of initial population and growth: 200000+30000=230000 people.

Let's see how to solve "backward" problem.

Example 5. Initial price of the dress is $175. Discounted price is$105. What is the discount (in percents)?

First, let's calculate discount in dollars. It is simply $175-$105=$70. Now, we need to find what percent of initial price$175 is $70. Answer is 70/175*100=40%. Therefore, discount is 40%. Now, it is time to exercise. Exercise 1. What percent of 150 is 37.5? Answer: 25%. Exercise 2. What is 12% of 57? Answer: 6.84. Exercise 3. 45% of what is 99? Answer: 220. Exercise 4. Discounted price of the toy is$30. It appears, that discount was 4%. What was the initial price?

Answer: \$31.25. Let n is initial price. Then discount in dollars is n-30. Percent discount is (n-30)/n=4/100.

Exercise 5. 5 years ago population of small town was 800 people. Current population is 1000 people. By how much percents did the population increase?

Answer: 25%. Increase in people is 1000-800=200. Percent increase is 200/800*100=25%.