PERCENTAGE
What is Percentage: A fraction with its denominator as ‘100’ is called a percentage. Percentage means per hundred. So it is a fraction of the form 6/100, 37/100, 151/100, and these fractions can be expressed as 6%, 37%, and 151% respectively. By a certain percent, we mean that many hundredths.
Thus x percent means x hundredths, written as x%.
To express x% as a fraction:We have, x% = x/100.
Thus, 20% =20/100 =1/5; 48% =48/100 =12/25, etc.
To express a/b as a percent:We have, a/b = ((a/b)*100)%
Thus, ¼ =[(1/4)*100] = 25%; 0.6 =6/10 =3/5 =[(3/5)*100]% =60%
Why Percentage: Percentage is a concept evolved so that there can be a uniform platform for comparison of various things. (Since each value is taken to a common platform of 100)
Example: To compare three different students depending on the marks they scored we cannot directly compare their marks until we know the maximum marks for which they took the test. But by calculating percentages they can directly be compared with one another.
Important Points to Remember:
a)If the price of a commodity increase by R%, then the reduction in consumption so as not to increase the expenditure is
[R / (100+R))*100] %
b)If the price of the commodity decreases by R%, then the increase in consumption so as to decrease the expenditure is
[(R / (100-R)*100] %
c)If A is R% more than B, then B is less than A by
[(R/(100+R))*100]%
[(R/(100-R))*100]%
Results on Population: Let the population of the town be P now and suppose it increases at the rate of R% per annum, then:
Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then,
Solved Examples:
1. A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?
Solution:
Number of runs made by running = 110 – (3 × 4 + 8 × 6) = 110 – 60 = 50
Therefore, required percentage = 50/110 × 100 = 500/11 = 45 5/11%
2. 8 is 4% of a, and 4 is 8% of b. c equals to b/a. What is the value of c?
Solution:
4% of a = 8 => a = 8 x 100/4 = 200; 8% of b = 4 => b = 4 x 100/8 = 50
c = b/a = 50/200 = ¼
3. Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are:
Solution:
Let the marks be (x + 9) and x.
Then, (x + 9) = (56/100) x (x + 9 + x)
=> 25(x + 9) = 14(2x + 9) => x = 33
So there marks are 42 and 33
4. In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was:
Solution:
Number of valid votes = 80% of 7500 = 6000
Valid votes polled by other candidate = 45% of 6000 = 2700
5. Teacher took exam for English, average for the entire class was 80 marks. If we say that 10% of the students scored 95 marks and 20% scored 90 marks then calculate average marks of the remaining students of the class
Solution:
Let’s assume that total number of students in class is 100 and required average be x. Then from the given statement we can calculate = (10 * 95) + (20 * 90) + (70 * x) = (100 * 80)
=> 70x = 8000 – (950 + 1800) = 5250 => x = 75.
6. In a hotel, 60% had vegetarian lunch while 30% had non-vegetarian lunch and 15% had both types of lunch. If 96 people were present, how many did not eat either type of lunch?
Solution:
So People who do not have either lunch were = 96 – 72 = 24
7. Due to a 25% increase in the price of rice per kilogram, a person is able to purchase 20 kg less for Rs. 400. What is the increased price of rice per kilogram?
Solution:
20kg = 25% of 400 = Rs.100 => 1kg = Rs.5
8. Wheat is now being sold at Rs. 27 per kg. During last month, its cost was Rs. 24 per kg. Find by how much percent a family reduces its consumption, so as to keep the expenditure fixed.
Solution:
Percentage increase in cost = (27 – 24) x 100/24 = 12.5
Required decrease = (12.5%) x 100/(100 + 12.5) = 11.11
9. In a competitive examination in State A, 6% of candidates got selected from the total appeared candidates. State B had an equal number of candidates appear and 7% of candidates got selected with 80 more candidates got selected than A. What was the number of candidates appearing from each State?
Solution:
State A and State B had an equal number of candidates appear.
In-state A, 6% of candidates got selected from the total appeared candidates
In-state B, 7% candidates got selected from the total appeared candidates
But in State B, 80 more candidates got selected than State A
From there, it is clear that 1% of the total appeared candidates in State B = 80
=> total appeared candidates in State B = 80 x 100 = 8000
=> total appeared candidates in State A = total appeared candidates in State B = 8000
10. In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?
Solution:
Let the total number of students = x
20% of students are below 8 years of age
=> number of students whose age ≥ 8 years = 80% of x ______(i)
number of students whose age is 8 years =48 _________(ii)
number of students whose age is greater than 8 years = 48 x 2/3 = 32 _______ (iii) From (i), (ii), (iii)
80% of x = 42 + 32 = 80
=> 100% of x = 80 x 100/80 = 100 => x = 100
Solution:
Price of the car = Rs.3,25,000
Car was insured to 85% of its price
Insured price = 325000 x 85/100
Insurance company paid 90% of the insurance.
Amount paid by insurance company = 325000 x 85/100 x 90/100 = 325 x 85 x 9 = 248625 Difference between the price of the car and the amount received = 325000 – 248625 = Rs.76375
12. 30% of the men are more than 25 years old and 80% of the men are less than or equal to 50 years old. 20% of all men play football. If 20% of the men above the age of 50 play football, what percentage of the football players are less than or equal to 50 years?
Solution:
Let total number of men = 100
Then, 20 men play football. 80 men are less than or equal to 50 years old. The remaining 20 men are above 50 years old.
Number of football players above 50 years old = 20 x 20/100 = 4
Number of football players less than or equal to 50 years old = 20 – 4 = 16
Required percentage = 16/20 x 100 = 80%
13. A candidate who gets 20% marks fails by 10 marks but another candidate who gets 42% marks gets 12% more than the passing marks. Find the maximum marks.
Solution:
From the given statement pass percentage = 42% – 12% = 30%
By this 30% of x – 20% of x = 10 marks => 10% of x = 10marks => 100% = 100 marks
14. A mixture of 20 liters of milk and water contains 20% of water. The new mixture is formed by adding 5 lit of water. What is the percentage of milk in the new mixture?
Solution:
20 li = 4 li water 16 li milk
By adding 5 li water
Total = 9 li water 16 li milk (new)
16/25×100=64%
15. From the salary of Roja, 20% is deducted as house rent, 10% of the rest she spends on children’s education, and 20% of the balance she spends on watching movies. If her savings are Rs.5760/- then her total salary is:
Solution:
First value = last value× [100/(100-p1)] × [100/(100-p2)]
= 5760× [100/(100 – 20)] × [100/(100 – 10)] × [100/(100 – 20)] = 5760 × 100/80× 100/90× 100/80 = 10000/-
Practice Problems:
This post was last modified on October 23, 2021 3:45 pm