# 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 $$$\frac{{20}}{{100}}$$$.

From another side the same ratio can be represented as $$$\frac{{n}}{{35}}$$$.

We obtained proportion $$$\frac{{20}}{{100}}=\frac{{n}}{{35}}$$$.

Solving it, we obtain that $$${n}={7}$$$.

Therefore, 20% of 35 is 7.

We can generalise this result.

$$${p}$$$% of $$${m}$$$ is $$$\frac{{p}}{{100}}\cdot{m}$$$.

**Example 2.** 95% of what is 237.5?

Let $$${n}$$$ represents required number.

Percent can be written as ratio $$$\frac{{95}}{{100}}$$$.

From another side the same ratio can be represented as $$$\frac{{237.5}}{{n}}$$$.

We obtained proportion $$$\frac{{95}}{{100}}=\frac{{237.5}}{{n}}$$$.

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 $$$\frac{{m}}{{100}}\cdot{p}$$$.

**Example 3.** What percent of 15 is 27?

Let $$${p}$$$ represents required percent.

Percent can be written as ratio $$$\frac{{p}}{{100}}$$$.

From another side the same ratio can be represented as $$$\frac{{27}}{{15}}$$$.

We obtained proportion $$$\frac{{p}}{{100}}=\frac{{27}}{{15}}$$$.

Solving it, we obtain that $$${p}={180}$$$%.

Therefore, 27 is 180% of 15.

We can generalise this result.

$$${n}$$$ is $$$\frac{{n}}{{m}}\cdot{100}$$$% 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 $$$\frac{{15}}{{100}}\cdot{200000}={30000}$$$.

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 $$$\frac{{70}}{{175}}\cdot{100}={40}$$$%.

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 $$$\frac{{{n}-{30}}}{{n}}=\frac{{4}}{{100}}$$$.

**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 $$$\frac{{200}}{{800}}\cdot{100}={25}$$$%.