### PERCENTAGES

A fraction with denominator 100 is called percentage. Numerator of that fraction is called rate percent.
The term percent means per every 100.
50 percent means 50 out of 100.
The symbol  % is used to denote percent. Ex: 20%, 35%, 80% etc.
Percent to fraction: To convert percent into fraction, divide the percent by 100. Fraction to percent: To convert fraction into percent, multiply the fraction by 100. Increment percent = Ex: increment % in 5 and 8  is  3/5  × 100 = 60%.
Decrement percent  = Ex. Decrement % in 5 and 8 is 5/8  × 100 = 37.5%.

Some important percent /fractions. Following rules should be kept in mind for solving the questions on percentage —
1. For converting a given percentage into a fraction, divided it by 100 or multiply by 2. For converting a given fraction into a percentage multiply it by 100.
3. For converting one given quantity, say, x as a percentage of another given quantity, say, y find Note - percentage is never expressed in any unit like rupees or kilos.
4. When there are two articles ‘a’ and ‘b’ in a group and in the ratio of a : b then to express any one article as a percentage of the group, divide the article by the total articles of the group and multiplies the whole by 100.
Percentage of ‘a’ = And, Percentage of ‘b’ = EXAMPLES Example 4. How many percent is 15 cm of 1 metre? Note: Both quantities are converted into same unit.

Example 5.  A person spent 85% of his monthly income and thereafter saves Rs.360 per month. Find out his total income.
Sol: Suppose his monthly income is Rs. 100
Expenditure = 85% = Rs. 85
Saving = Rs. 100 – 85 = Rs. 15
∴  If saving is Rs. 15 income is Rs. 100
∴  If saving is Rs. 360 income is = Rs. 2400

Example 6. If a student gets 38 marks out of 50, what is the percentage of his obtained marks?
Sol:  Percentage of obtained marks = 76%

Example 7.  If two numbers are respectively 30% and 40% more then a third number, what percentage is the first of the second?
Sol: Let the third number be 100
∴ First number  = 100 + 30% of 100
= 130
and Second number   = 100 + 40% of 100
= 140
∴ Percentage of the first number to the second number to the second number Example 8. A student scores 20% marks and fails by 30 marks while another student who scores 32% marks, gets 42 marks more than the minimum required pass marks. What are the maximum marks for the examination?
Sol:   32% - 20% = 12%
And 42–(-30) = 72
∴ 12% marks = 72
∴ 100% marks = Example 9.  An engineering student has to secure 50% marks to pas. He gets 163 marks and fails by 37 marks. Find the maximum marks
Sol: Minimum marks required = 163 + 37  = 200
If 50 are the minimum marks then maximum marks  = 100
∴  200 are the minimum marks then maximum marks = CONVERSION

Percent to Fraction: To convert percent into fraction, divide the percent by 100. Fraction to percent: To convert fraction into percent, multiply the fraction by 100. Ex: A man’s salary increased from Rs.15,000 to Rs.18,000. What is the increment percent? When two variables are inversely proportional, If one is ‘x’ percent more/ less than the second one then the second one will be Note: If first one is more use ‘+’ sign and if it is less use ‘−’ sign

Ex: Petrol rates are increased by 25% . How much percent a consumer has to reduce his consumption to maintain the same expenditure on petrol? Model Questions
1. The difference between 91% of a number and 46% of the same number is 315. What is 37% of that number?
A) 333        B) 185        C) 259        D) 270

2. If the difference between 62% and 80% of a number is 198. Then the difference between 92% and 56% of the number will be... (SSC - CGL - 2019)
A) 396       B) 3564       C) 1100       D) 360

3. 32% of a number exceeds 17% of the same number by 120. What is the value of the number?   (SSC - CGL - 2017)
A) 600       B) 800        C) 500        D) 1200

4. A number was reduced from 5482 to 4385.6. What is the percentage decrement?
A) 15%       B) 25%       C) 18%       D) 20%

5. The number of students in a class is increased by 20% and the number now becomes 66. Initially the number was (SSC - CGL - 2016)
A) 45        B) 50           C) 55            D) 60

6. If the price of an article is reduced by 20%, the number of articles sold is increased by 30%. What will be the effect on the revenue?
A) 4%           B) 10%        C) 6%             D) 15%

7. If the length of a rectangle is increased by 40% and the breadth is decreased by 20% then the area of rectangle is increased by x%. Find the value of x? (SSC - CGL - 2019)
A) 8%        B)12%        C) 15%          D) 9%

8. If the price of sugar increases by 25%, b what percentage should a family reduce its consumption so that there is no change in the expenditure on sugar? 9. If the salary of Manoj is 40% less than that of Subhash, then by how much percentage is the salary of Subhash more than that of Manoj?  (SSC - CGL - 2016) 10. If 120 is reduced by x%, the same result will be obtained if 40 is increased by x%. Then x% of 210 will be what percentage less than (x + 20)% of 180?   (SSC - CGL - 2018) 11. The present population of a town is 26,010. It increases annually at the rate of 2%. What is the population of the town 2 years ago?    (SSC - CGL - 2017)
A) 24500    B) 25000    C) 26000    D) 19500

12. Population of a town is 12,420. If it increases 15% in the first year and decreases 10% in the second year. What was the population of the town two years ago?
A) 12000     B) 12800     C) 11600     D) 13100

13. The income of A is 60% less than that of B and the expenditure of A is equal to 60% of B’s expenditure. If A’s income is equal to 70% of B’s expenditure, then what is the ratio of the savings of A and B? (SSC - CGL - 2019)
A) 2 : 15      B) 5 : 7      C) 1 : 2      D) 12 : 19

14. Raju's income is 20% more than his expenditure. If his income increases by 60% and his expenditure increase by 70%, then what percentage of the savings will increase/ decrease? (SSC - CGL - 2018)
A) will decrease 2%    B) will decrease 10% C) will increase 10%   D) will increase 2%

15. Six − eleventh of a number is equal to twenty two percent of the second number. The second number is equal to one-fourth of the third number. The value of the third number is 2400. What is 45% of the first number?
A) 109.8     B) 111.7     C) 108.9     D) 123.4

Explanations
1.
Difference between 91% and 46% is 45%. If 45% value is 315, then 37% value is 2. Difference between 62% and 80% is 18%. Difference between 92% and 56% is 36%. If 18% value is 198, then 36% value is
36/18 × 198 = 396

3. Difference between 32% and 17% is 15%. If 15% value is 120, then 100% value is 6. Let the price of article be Rs.10 and number of articles sold is 10 then the revenue is
10 × 10 = Rs.100 New price is 80% of 10 = Rs.8 and the articles sold is 13 (30% increased)
New revenue is Rs.8 × 13 = Rs.104
∴ 4% increment in the revenue
Shortcut: For 2 successive changes use the formula (use + sign for increment and − sign for decrement) (Use + sign if the 1st variable increases and − sign if it decreases)
Required percentage reduction is Required percentage increment is = 12420
⇒ x = 12000

13. Let the Expenditure of B be Rs.100
Savings = Income - Expenditure Savings ratio = 10 : 75 = 2 : 15

14. Let the Expenditure of B be Rs.100
Income will be Rs.120
Savings = Income − Expenditure  Key: 1-C; 2-A; 3-B; 4-D; 5-C; 6-A; 7-B; 8-C; 9-C;  10-B; 11-B; 12-A; 13-A; 14-C; 15-C.

Posted Date : 17-02-2021