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