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 percentage comparison. Or else if we compare the two things with respect to each other, then we calculate the ratios of these two quantities.

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.

The formula for converting a ratio to a percentage is:

Percentage = Ratio x 100

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.

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 Formulae

• 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

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: w3wiki 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 w3wiki is 180 employees.