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`%.