### Key Concepts

- Find percent of numbers.
- Use percent’s greater than 100%.
- Use percent’s less than 1%.

**3.1 Analyze percent of numbers**

## Percentages:

The word percent is taken from the Latin word “per centum”, which means “by the hundred”. In other words, percentage is a fraction with 100 as the denominators.

**Example:** There are 52 girls in a class of 100 students.

Percentage of girls =52/100

**What are percentages?**

Percentage is a fraction or ratio in which the value of the whole is always 100. For example, if Bob scores 30% marks in a math test, it means that he scores 30 marks out of a hundred.

**Expression of marks:** 30% (in percentages) or 30/100 (in fractions) or 30:100 (in ratios).

**How do we find percent?**

**Case 1:**

When we have two or more values that add up to 100, then the percentage of the individual value to the total value is the individual number itself. Let’s understand how to write percentages using the example given below.

**Example: **A contractor orders 100 tiles of four different colors. Find the percentage of each color from the table given below.

**Case 2:**

When we have two or more values that do not add up to 100. In such cases, we convert the fractions

to equivalent fractions with denominators as 100. Let us understand using the example given below.

**Example: **In a class of 50 students, 35 were present, and 15 were absent during a day. Find the present and absent percentages of that day.** **

**3.1.1 Find percent of numbers**

**Example 1: **Emily goes on a solo trip. If she travels a total distance of 300 miles, out of which 35% is covered on a bike. Find the distance traveled by Emily on bike.

**Solution: **We observe that 100% of the trip is considered as 300 miles, out of which we are finding 35%. Let us consider the distance traveled on the bike as ‘b’ miles.

**Step 1: **Draw a bar diagram and write equivalent ratios to represent the distance covered on the bike and the percentage covered on the bike.

35/100 = b/300

**Step 2: **Use the equivalent ratios to find 35% of 300 miles.

35/100 =b/300

**Multiply by 300 on both sides of the equation.**

35/100 × 300 = b/300 × 300

35 × 3 = b

b = 105 miles.

35% of 300 is 105 miles.

Therefore, we conclude that 105 miles of the total trip was covered on the bike.

**Example 2: **A student must secure 70% marks in a final examination so that he will be considered for the research assistant post in the college. If the total marks for the examination is 450. Find the number of marks a student must score to gain the post.

**Solution: **We observe that 100% of marks is considered as 450, out of which we are finding 70%. Let us consider the marks to be secured by the student as ‘x’.

**Step 1: **Draw a bar diagram and write equivalent ratios to represent the qualifying marks and the total marks.

70 / 100 =x/450

**Step 2: **Use the equivalent ratios to find 70% of 450 marks.

70/100 =x/450

**Multiply by 450 on both sides of the equation.**

70/100× 450 = x / 450×450

7 × 45 = x

x = 315 marks.

70% of 450 is 315 marks.

Therefore, we conclude a student must secure 315 marks to qualify for the post of a research assistant.

**3.1.2 Use percent’s greater than 100%**

**Example 1: **Bryan currently earns $80000 per annum. He is offered a 120% hike if he joins a new company. Find the expected salary of Bryan in his new company.

**Solution: **We observe that 100% of salary is considered as $80000, out of which we are finding 120%. Let us consider the expected salary of Bryan as ‘$y’.

**Step 1: **Draw a bar diagram and write equivalent ratios to represent the current and expected salaries.

120 / 100 = y/80000

**Step 2: **Use the equivalent ratios to find 120% of $80000.

120/100=y/80000

**Multiply by 80000 on both sides of the equation.**

120/100× 80000=

y/80000 ×80000

120 × 800 = y

y = $96000.

120% of 80000 is $96000.

Expected salary = Current salary + Hike

= 80000+96000

= 176000.

Therefore, we conclude Bryan earns $176000 per annum in his new company.

**Example 2: **With most of the Xerox machines, you can reduce or enlarge your original by entering a percentage for the copy. Reshma wanted to enlarge a 4 cm by 2cm drawing. She set the Xerox machine for 150% and copied her drawing. What will be the dimensions of the copy of the drawing be?

**Solution: **We observe that 100% of the original copy has dimensions 4 cm by 2 cm. The xerox machine is set to enlarge the original copy by 150%. Let the dimensions of the copy after enlarging be ‘x cm’ by ‘y cm’.

**Step 1: **Draw a bar diagram and write equivalent ratios to represent the original and enlarged copy. Firstly, find the dimensions of the length and next breadth.

**Length:**

150/100=x/4

**Step 2: **Use the equivalent ratios to find 150% of 4.

150/100=x/4

**Multiply by 4 on both sides of the equation.**

150/100× 4 =x/4×4

3 × 2= x

x = 6.

150% of 4 is 6 cm.

**Breadth:**

150 / 100=x/2

**Step 2: **Use the equivalent ratios to find 150% of 2.

150 / 100=y/2

**Multiply by 2 on both sides of the equation.**

150/ 100× 2 =y /2×2

3 × 1= y

y = 3.

150% of 2 is 3 cm.

Therefore, we conclude that new dimensions after enlarging the original are 6 cm by 3 cm.

**3.1.3 Use percent’s less than 1%**

**Example 1: **Gilbert classical academy has an enrolment of 400 students. Only 1/2 % of them ride their bicycles to school every day. How many students come on their bicycles?

**Solution: **We observe that 100% of students is considered as 400, out of which we are finding

1/2 %. Let us consider the number of students who come on a bicycle as ‘x’.

**Step 1: **Draw a bar diagram and write equivalent ratios.

0.5 / 100=x/400

**Step 2: **Use the equivalent ratios to find 0.5% of 400.

0.5 / 100= x/400

**Multiply by 400 on both sides of the equation.**

0.5/100× 400 =x/400×400

0.5 × 4 = x

x = 2 students.

0.5% of 400 students is 2 students.

Therefore, we conclude that 2 students come on a bicycle from home to school.

**Example 2: **The registration fee for a used car is 0.8% of the sale price of $4,500. How much is the fee?

**Solution: **We observe that 100% of the cost is considered as $4500, out of which we are finding 0.80%. Let us consider the registration fee as ‘$r’.

**Step 1: **Draw a bar diagram and write equivalent ratios.

0.8 / 100 =r/4500

**Step 2: **Use the equivalent ratios to find 0.8% of 4500.

0.8 / 100 =r/4500

**Multiply by 4500 on both sides of the equation.**

0.8 / 100× 4500 = r/4500 × 4500

0.8 × 45 = r

r = $36.

0.8% of $4500 is $36.

## Exercise:

- In a group of 50 students, 25% are non-swimmers. How many students cannot swim?
- A foster care has 150 kids, of which 68% are above 18 years. How many kids are below 18 years?
- A bike costs $350 but is reduced by 35% in a sale. What does it cost in a sale?
- The city of valley view received 20 inches of rainfall last year. This year the rainfall was noted to be increased by 150%. How many inches of rainfall was received this year?
- A company earned a profit of $35000 in the month of July. If their sales increased by 125% in August, find the profit earned in the month of August.
- Kate has a collection of 200 CDs. Mike has 350% more collection than Kate. How many CDs does Mike have?
- A ship travels 300 miles per day. If it travels continuously for 31 days at the same rate. What is 0.01% of the total distance it covers in 31 days?
- A bacterium is of length 5 cm when observed under the microscope. Find its original length which is 0.02% of the enlarged length.
- 185 is 0.5% of what number?
- Jimmy spends 0.54% of his income on internet bill payments. If his salary is $40000 per annum. What is expenditure on internet bill?

### Concept Map

### What have we learned?

- Finding percent of numbers.
- Using percent’s greater than 100%.
- Using percent’s less than 1%.

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: