Ratios and Percentages

Ratios and Percentages: Comparing quantities is easy, each of the quantities is defined to a specific standard and then the comparison between them takes place after that. Comparing quantities can be effectively done by bringing them to a certain standard and then comparing them related to that specified standard. For instance, if we quantify a particular thing with respect to 100 portions of the same thing, this is referred to as a percentage comparison. Or else if we compare the two things with respect to each other, then we calculate the ratios of these two quantities.

Ratio

In simple terms, comparing Quantity1 with Quantity2 is Ratio. The first quantity(say p) is mentioned in the numerator and the second quantity(say q) is mentioned in the denominator. The way to depict the ratio is p/q.

Ratio Formula: Quantity 1/Quantity 2

Example: 

Quantity1 has 5 portions and Quantity2 has 8 portions. The ratio of both the quantities will be 5:8.

Ratio To Percentage Formula

The formula for converting a ratio to a percentage is:

Percentage = Ratio x 100

How to Convert Ratio to Percentage

Converting Ratios to Percentages

To convert a ratio to a percentage, you compare one part to the whole. Here's an example using a student's performance in semester one:

Given Ratio: 586 : 600

Step 1: Form a Fraction

Start by converting the given ratio to a fraction, comparing the student's marks against the total. In this case, the fraction is:

586 (marks obtained) / 600 (total marks), which can also be written as 586 : 600.

Step 2: Perform the Division

Divide the fraction:

586 ÷ 600 = 0.9766666

Rounding to two decimal places: 0.98

Step 3: Convert to Percentage

Multiply the result by 100 to convert it into a percentage:

0.98 × 100 = 98%

Therefore, the student scored 98% in semester one.

Percentage

Comparing Quantity with 100 portions of the same quantity in the denominator is Percentage. The first quantity(say p) is mentioned in the numerator and the second quantity(say q) is mentioned in the denominator. This quantity when multiplied by 100. The way to depict the ratio is (p/q)*100.

Percentage Formula: (Quantity/Total Quantity) * 100

Example: 

Quantity1 has 5 portions and Quantity2 has 8 portions. 

Percentage of Quantity1: (5/13)*100

Percentage of Quantity2: (8/13)*100

Both are comparing techniques, and are used accordingly in different situations depending on the need.

Percentage as Fraction:

80% expressed as fraction will be 80/100

30% expressed as fraction will be 30/100

Percentage as Ratio:

80% expressed as ratio will be 80/100 = 4 : 5

30% expressed as ratio will be 30/100 = 3 : 10

Percentage as Decimal:

80% expressed as decimal will be 80/100 = 0.8

30% expressed as decimal will be 30/100 = 0.3

Calculating Percentage of a Number:

To calculate the percentage, simply multiply the number by the percent.

Example: 

90% of 10 can be calculated as follows:

(90/100)*10 = 9

So, 90% of 10 is 9.

Some Quick Formulas

• If a/b = c/d, then ad = bc
• If a/b = c/d, then b/a = d/c
• If a/b = c/d, then a/c = b/d
• If a/b = c/d, then (a - b)/b = (c - d)/d
• If a/b = c/d, then (a + b)/b = (c + d)/d
• If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d), Componendo- Dividendo Rule
• If (a/b)*100 then it's a/b %

Practice Problems on Ratios and Percentages

Question 1: A quantity is 25% of 200. Find the quantity?

Solution: 

Quantity = (25/100) * 200 = 50

Question 2: If 60% of a quantity(x) is 300, then find the quantity x?

Solution:

60x/100 = 300

=> x = 30000/60 = 500

The quantity x is 500.

Question 3: A cricket team won 3 matches out of a total of 5 matches in a series. Depict the winning ratio and percentage?

Solution: 

Winning Ratio: 3/5

Winning Percentage: (3/5) * 100 = 60%

Question 4: Winning percentage of a cricket match series is 60%. Find the total matches won out of 5 in the cricket series?

Solution: 

Matches Won = (60/100) * 5 = 3 

Question 5: A nursery has 10000 plants. 10% of the plants are oranges and 3% are mangos plants. What is the total number of other plants?

Solution: 

Total Plants = 10000

Orange plants = 10% of total

 Mango Plants = 3% of total

 Other plants = 100% - (10% + 3%) = 87% of total

                      = (87/100) * 10000 

                      = 8700

The total number of other plants are 8700.

Question 6: GeeksforGeeks has 90 engineer employees. In this company, engineers constitute 50% of its workforce. How many people are employed in the company?

Solution:

50% of total workforce = 90 engineers

Let total workforce be x

Then, (50/100) * x = 90

                           x = (90 * 100) / 50

                           x = 180

 So, the total workforce of GeeksforGeeks is 180 employees.

Ratios and Percentages Worksheet

1. A class has 30 students where the ratio of boys versus girls is 3:2. How many boys are there in this class?

2. Mixing apples and grapes for a fruit salad in the ratio 2:3, how many grapes will you put if you have 8 apples?

3. A shirt originally sold for $40 was on sale for 20% off. How much is the sale price of the shirt?

4. If 25% of a number is 50, what is 40% of the same number?

5. The ratio of apples to oranges in a fruit basket is 4 : 5. If the fruits are 30% apples, how many oranges does the basket contain?

6. If a school's ratio of students who play football to students playing basketball is 5 : 3, and if 60 percent of the students are playing football, how many play basketball?