METHODS TO CALCULATE PERCENTAGES:
1. Percentage refers part of or a fraction of the whole.
2. It is a way to describe a number as a fraction with denominator 100.
3. Percent implies 'for every hundred' and is denoted by % This concept is developed to facilitate easier comparison of fractions. Hence, it is very useful in both data interpretation and solving quantitative problems.
Example:
If a student receives 20 marks out of 50, then his percentage is (20/50)*100 = 40%
More examples:
1/5 = 20%, 4/25 = 16%, 0.36 = 36 %, 0.52 = 52%
Likewise, percentages are used to understand many situations in our daily lives. Some situations are percentage of students scoring first class in an exam and the percentage increase in the cost of a product.
Case 1:
If in a class, 15 out of 25 students are girls, then what is the equivalent saying the proportion of girls?
(15/25)*100 = 60% In ratios, we can say that the ratio of girls to the boys is 3:2
Case 2:
x is 15% of 70. Find x => x= (15/100)*70 => x=1050/100 = 10.50%
Case 3:
484 is 40% of x. Find x => 484= (40/100)*x = 484 *(100/40) = 1210
Case 4:
x% of 85 is 15. Find x => (x/100)*85= 15 => 15*(100/85) = 17.647%
Example 1:
An agent sells goods of value of Rs.15000. The commission which he receives at the ratio of 12.5% is? (SNAP 2009)
Value of goods = 15000 => Commission = (12.5/100) * 15000 = Rs.1875.
Example 2:
Sonali invests 15% of her monthly salary in insurance policies. She spends 55% of her monthly salary in shopping and household expenses. She saves the remaining amount of Rs.12750. What is sonali's monthly income? (SNAP 2009)
Solution:
Total percentage of monthly salary spent = 15+55 = 70%
Remaining 30% which is saved = 12750 => (30/100)*Monthly income = Rs.12750
=> Monthly income = 12750*(100/30) = Rs.42500.
ALTERNATIVE METHOD OF CALCULATING PERCENTAGES:
There is an alternative method of finding the percentages when a ratio is given.
The basic idea of this method is to remove certain percentages of the denominator from the numerator preferably multiples of 10.
Consider an illustration- To find the percentage for 7526/4626.
Step 1: Remove 100% of 4646 from the numerator since 7526>4626. => 7526-4626 = 2900.
Step 2: Now, the ratio can be written as 7526/4626 = 100% + (2900/4626)
Step 3: We can see that 10% of 4626 is 462.6 and 400*6 = 2400 So, we can try by calculating 60% of 4626= 2775.6 Now remove 60% from the numerator.
(The concept is we need to find by trial and error that how much percent of the denominator is approximately the numerator.For that first, calculate 10% of it and then find out through the multiples of that value)
Step 4: Removing 60% => 2900-2775.6 = 124.4 So, the step will be
7526/4626 = 100% + 60% + (124.4/4626)
As we know that the 10% of 4626 is 462.6 and now the numerator is far below this, the actual value will lie between 160% and 170%.
Depending on the accuracy required, we could either stop at this point or move on further, (Applicable mostly in DI problem sets)
Step 5: Still if we need to move on , calculate 5% of 4626 => 232.3 which is greater than the numerator 124.4
So, the value will be between 160% and 165%
Step 6: Again moving on further accurate, calculate 2.5% = 116.15 which is less than the numerator 124.4
So, the answer will be appoximately 163% ( more than 162.5%)
The greatest advantage of this approach us is that it gives the liberty of stopping calculation as soon as the required accuracy is arrived at depending on the available answer options.
Example 3:
If Roshini scored 71 marks out of 93 marks in Mathematics, what was her percentage score?
Solution:
We need to find (71/93)*100
Applying the method stated above, calculate 50% of 93= 42.5
The numerator, 71 is far greater than 42.5
So, calculating 20% of 93= 2*10% of 93 = 18.6=> 50%+20% = 42.5+18.6 = 61.1
Removing 70% from the numerator (71/93) = 70% +(5.9/100)
We know that 10% is 9.3 and 5% is 4.15 So, the value should lie between 75% and 80%
5% is 4.15 and 1% is 0.93 => Adding 5% and 1% => 4.15+0.93 = 5.88 (approximately equal to 5.9 which is the numerator) So, the accurate percentage score will be 76%
CONCEPT 2
PERCENTAGE INCREASE OR DECREASE:
Percentages are used to often indicate the changes in the quantity.
Percentage change = (Final value- Initial value)/Initial value
For example, if the cost of a product changes from 6 to 78, the percentage change is given by, (8-6)/6
1. If a quantity increases by a% , then its value is multiplied by (100+a)/100. For example, if there is 25% increase in a product worth Rs.464, its new price will be 1.25*464 = Rs.580.
2. If a quantity decreases by b%, then its value is multiplied by (100-a)/100. For example, if there is 25% decrease in a product worth Rs.464, its new price will be 0.75*464 = Rs.348.
Example 1:
A's salary is 20% more than B's salary. By what percentage is B's salary less than A's salary?
Solution:
Let B's salary be Rs.100. A's salary will be Rs.120.
B's salary is (120-100)/120 =>16.67% less than A's salary (Because here the initial value is 120 and the final value is 100)
ABSOLUTE VALUE CHANGE AND PERCENTAGE CHANGE:
1. Absolute value change is the actual change that occurs in the measure of a quantity
2. Percentage change is the absolute change with respect to the measure of the original quantity.
Example 2:
If the cost of a product increases from Rs. 500 in 2000 to Rs.750 in 2001, then calculate the absolute value change and the percentage change between the two years.
Solution:
The absolute value change = |Final vale - Original value| = |750 - 500| = Rs.250
The percentage change = (750-500)/500 = 50%
Example 3:
The interest rate of a bank is increased from 11% in 2003 to 12.5% in 2004. Calculate the percentage point change and percentage change.
Solution:
The percentage point change = 12.5% - 11% = 1.5%
The percentage change = 1.5/11 = 13.63%
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