**What is the 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%

**Important Points to Remember:**

**a) **If the price of a commodity increases 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]%**

**d)** If A is R% less than B, then B is more than A by

**[(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:

**1.Population after n years = P [1+(R/100)] ^{n}**

**2.Population n years ago = P / [1+(R/100)] ^{n}**

**Results on Depreciation:** Let the present value of a machine be P. Suppose it depreciates at the rate R% per annum. Then,

**1.Value of the machine after n years = P [1-(R/100)] ^{n}**

**2.Value of the machine n years ago = P / [1-(R/100)] ^{n}**

**To solve the percentages problems we have to memorize some basic percentage values to do fast calculations in ****competitive**** exams.**

- 1/2 = 50%
- 1/3 = 33.33%
- ¼ = 25%
- 1/5 = 20%
- 1/6 =16.66%
- 1/7 = 14.28%
- 1/8 = 12.5%
- 1/9 = 11.11%
- 1/10 = 10%
- 1/11 =9.09 %
- 1/12 = 8.33%
- 1/13 = 7.69%
- 1/14 = 7%
- 1/15 = 6.66%
- 1/16 = 6.25%
- 1/17 = 5.88%
- 1/18 =5.55%
- 1/19 = 5.26%
- 1/20= 5%

**5%= 1/20 = 0.05****10% =1/10 =0.1****15% =3/20****20% = 1/5****25% = ¼****30% =3/10****40% =2/5****50% =1/2****55%=11/20****60% = 3/5****70% = 7/10****75% = 3/4****80% = 4/5****90% = 9/10****100%=1****6 ¼ % = 1/6****12 ½ % = 1/8****16 2/3 % = 1/6****33 1/3 % = 1/3****66 2/3 % =2/3****125% =5/4****150% = 3/2**

**Questions**

**Q.1 The marks obtained by Anuj in Physics, Chemistry and Mathematics are 70, 80 and 90 respectively out of 120. What is the percentage of marks obtained by Anuj in Chemistry?**

1. 66.66%

2. 33.33%

3. 50%

4. 45%

5. 75%

**Solution:**

Marks obtained by Anuj in Chemistry = 80

Total marks in Chemistry = 120

Therefore, required percentage = × 100 = 66.66%

**Q. 2 If p% of q is 25 then, find q% of p.**

1. 20

2. 15

3. 25

4. 30

5. 10

**Solution: **

p% of q = 25

Or, = 25

Therefore, q% of p = 25

**Q.3 The score of Ajit is 10% of the score of Akash and the score of Akash is twice the score of Paul. The score of Paul is 40, find the score of Ajit.**

1. 10

2. 8

3. 16

4. 12

5. None of these

**Solution**

The score of Paul = 40

Then, the score of Akash = 2 × 40 = 80

Therefore, the score of Ajit = (10/100) × 80 = 8

Q**.4 The ratio of male and female in a party is 3 : 2. Find the percentage of female in the party.**

1. 60%

2. 30%

3. 50%

4. 40%

5. 25%

**Solution:**

Let the number of male and female in the party be 3x and 2x respectively.

Total number of persons in a party = 3x + 2x = 5x

Therefore, percentage of female in the party = × 100 = 40%

**Q.5 Ajit buys 30 mangoes from the daily market. Ragini buys twice that of Ajit from the daily market. Suresh buys thrice that of Ajit from the daily market’ What is the percentage of mangoes bought by Ragini among the three?**

1. 33.33%

2. 20%

3. 30%

4. 50%

5. 66.66%

**Solution:**

Number of mangoes bought by Ajit = 30

Number of mangoes bought by Ragini = 2 × 30 = 60

Number of mangoes bought by Suresh = 3 × 30 = 90

Total number of mangoes bought by them = 30 + 60 + 90 = 180

Therefore, required percentage = × 100 = 33.33%

**Q.6 In a meeting the number of persons who ordered coffee is 50 and the number of persons who ordered tea is 40. Total number of persons who ordered none is half the number of persons who ordered tea. Find the percentage of persons who does not ordered anything in the meeting. Assuming only these three types of persons in the meeting.**

1. 36.36%

2. 18.18%

3. 20%

4. 15%

5. None of these

**Solution:**

Number of persons who ordered coffee = 50

Number of persons who ordered tea = 40

And, Number of persons who ordered none of the two = × 40 = 20

Therefore, required percentage = × 100 = 18.18%

**Q.7 A’s salary is 25% more than B’s salary. By what percent is A’s salary less than B’s salary?**

1. 25%

2. 10%

3. 20%

4. 15%

5. None of these

**Solution:**

Let A’s salary be Rs.100

Then B’s salary = 100 + 100 × = 125

Percentage by which A’s salary is less than B’s salary

= × 100 = 20%

**Q.8 If the ratio x is 10/20 and ratio y is 10/25. Find the percentage change in ratio 2 with respect to ratio 1.**

1. 10%

2. 30%

3. 15%

4. 20%

5. 5%

**Solution:**

Ratio x = 0.5

And, Ratio y = 0.4

Percentage change in ratio = × 100 = 20%

**Q.9 The length and breadth of a rectangle are changed by 10% and -20% respectively. Then, what is the change in the area of the rectangle.**

1. 11%

2. 10%

3. 9%

4. 8%

5. 12%

**Solution:**

Let the length and breadth of a rectangle be l and b respectively.

Then, the area of the rectangle (A) = l × b

Since, length is changed by 10% and breadth is changed by – 20%.

Therefore, new length and breadth be l’ and b’ respectively.

l’ = l + l × =

b’ = b – b × =

New area of rectangle (A’) = l’ × b’

Or, A’ = × =

Therefore, percentage change in area of the rectangle

= × 100 = 12%

**Q.10 The ratio of the height of A and B is 1 : 2. What percentage more/less is the height of A with respect to height of B?**

1. 50%

2. 30%

3. 40%

4. 25%

5. 100%

**Solution:**

Let the height of A and B be x cm and 2x cm respectively.

The height of A be lesser than the height of B by 2x – x = x cm

Therefore, height of A be lesser than the height of B by × 100 = 50%

**Q.11 The salary of Ajit is 25% of the salary of Ragini. The salary of Ragini is 40% of the Asha and the salary of Ajit is 50 LPA. Find the salary of Asha.**

1. 100 LPA

2. 500 LPA

3. 200 LPA

4. 300 LPA

5. 400 LPA

**Solution:**

The salary of Ajit = 50 LPA

Or, 25% of Ragini = 50 LPA

Therefore, salary of Asha = 200 LPA

Salary of Ragini = 200 LPA

Or, 40% of Asha= 200 LPA

Salary of Asha= 500 LPA

**Q.12 The marks obtained by X, Y and Z are 30, 50 and 40 respectively. What percentage of the marks is obtained by Z with respect to Y?**

1. 70%

2. 50%

3. 80%

4. 90%

5. None of these

**Solution:**

Marks obtained by Z = 40

And, marks obtained by Y = 50

Therefore, required percentage = × 100 = 80%

**Q.13 The number of prize distributed among the 4 students in the ratio 1 : 2 : 3 : 4. What percentage of prize is given to one with the highest number of prize?**

1. 20%

2. 50%

3. 45%

4. 40%

5. 30%

**Solution:**

Let the number of prizes distributed among the students be x, 2x, 3x and 4x.

Total number of prizes distributed = x + 2x + 3x + 4x = 10x

Therefore, required percentage = × 100 = 40%

**Q.14 If 30% of a number is 300, then 50% of that number is?**

1. 500

2. 400

3. 300

4. 200

5. None of these

**Solution:**

Let the number be x.

Then, 30% of x = 300

Therefore, x = 1000

So, 50% of 1000 = 500

**Q.15 20 % of a number when subtracted from 108, gives the number itself. Find the number.**

1. 80

2. 100

3. 90

4. 110

5. None of these

**Solution:**

Let the number be x

Then, according to the question –

108 – 0.2x = x

a = 90

**Q.16 Which of the following is the largest number?**

1. 30% of 300

2. 10% of 500

3. 19% of 100

4. 20% of 400

5. 15% of 150

**Solution:**

Option (1) –

× 300 = 90

Option (2) –

× 500 = 50

Option (3) –

× 100 = 19

Option (4) –

× 400 = 80

Option (5) –

× 150 = 22.5

Therefore, option (1) is the correct answer.

**Q.17 What is the 10% of 20% of 25% of 100?**

1. 0.5

2. 0.2

3. 0.75

4. 1.00

5. None of these

Solution:

Ans = 0.5

**Q.18 Harish bought some kg of rice from the store and distributed among the two families in the ratio of 1 : 4. Find the percentage of rice distributed among the two families.**

1. 25% and 75%

2. 30% and 70%

3. 20% and 80%

4. 40% and 60%

5. None of these

**Solution:**

Let the amount of rice distributed among the two families be x kg and 4x kg.

Total amount of rice distributed = x + 4 x = 5 x

Percentage of rice distributed to 1st family = (x /5 x) × 100 = 20%

Percentage of rice distributed to 2nd family = (4 x /5 x) × 100 = 80%

**Q.19 A fraction is given as 1/5. By how much percent, denominator is greater than the numerator?**

1. 300%

2. 400%

3. 200%

4. 500%

5. None of these

**Solution:**

Numerator of the fraction = 1

And, denominator of the fraction = 5

Therefore, percentage by which denominator is greater than the numerator = × 100 = 400%

**Q.20 Rajdhani express has a capacity of 500 seats of which 10% are in the executive class and the rest chair cars. During one journey, the train was booked to 85% of its capacity. If executive class was booked to 96% of its capacity, then how many chair car seats were empty during the journey?**

1. 73

2. 89

3. 87

4. 90

5. None of these

**Solution:**

Total executive class seats = 50

Total chair cars seats = 450

Total booked seats = 500 × 85% = 425

Total executive class seats booked = 50 -50 × 4% = 48

empty chair car seats = 450 – (425 – 48) = 73

**Q.21 A person has to multiply a number by 5 but by mistake he divides it by 5 then the percent change in this due to the mistake.**

1. 96%

2. 45%

3. 100%

4. 120%

5. 50%

**Solution:**

Let the number is x.

If we multiply this number by 5 then the result is 5 x

→if divide it by 5 then the result be 0.2 x

So the percentage change is 5-0.2=4.8 x

And % Change→ 4.8 x *100/5 x = 96%

**Q.22 In an election between four candidates A, B, C and D, A got 20% of the total valid votes, B got 100% more votes than A and D got 50% votes of B. 5% of votes are invalid from 7500 votes. The number of valid votes that C got, was:**

1. 2630

2. 2235

3. 1860

4. 1425

5. None of these

**Solution:**

Option (4) is the answer.

**Q.23 A person distributes his wealth between 3 sons. He gave 50 % of his wealth to his first son and 50 % of the rest to second, 50 % rest to its third son. If the combined wealth of his all 3 son is 78750. Find the quantity of the wealth the person was initially?**

1. 90000

2. 95000

3. 25000

4. 45000

5. 63012

**Solution:**

Let the wealth be -100

Now , he gave 50 % of this 100 → Remaining Is 50

50 % of 50 and After this 50 % of 25 →50+25+12.5=87.5

87.5 =78750

1=900

100=90000

**Q.24 The sum of 15% of a positive number and 10% of the same number is 70. What is twice of that number?**

1. 440

2. 280

3. 560

4. 140

5. None of these

**Solution:**

Option C is correct.

**Q.25 Nalini decided of spend 45% of her salary on shopping. On completion of her shopping, she realised that she had, spent only Rs.11475, which was 60% of what she had decided to spent. How much is Nalini’s salary?**

1. 29600

2. 38800

3. 42500

4. Cannot be determined

5. None of the above

**Solution:**

Correct Answer is (4)