Percentage

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         Important Formulas – Percentage

• Percentage

  Percent means for every 100

  So, when percent is calculated for any value, it means that we calculate the value for every 100 of the reference value.

  percent is denoted by the symbol %. For example, x percent is denoted by x%

• x%=x/100

  Example : 25%=25/100=1/4

• To express x/y as a percent, we have x/y=(x/y×100)%

  Example : 1/4=(1/4×100)%=25%

• If the price of a commodity increases by R%, the reduction in consumption so as not to increase the expenditure = [R/(100+R)×100]%
• If the price of a commodity decreases by R%, the increase in consumptions o as not to decrease the expenditure = [R/(100−R)×100]%
• If the Population of a town = P and it increases at the rate of R% per annum, then Population after n years = P((1+R)/100))ⁿ
• If the population of a town = P and it increases at the rate of R% per annum, then Population before n years = P((1+R)/100))ⁿ
• If the present value of a machine = P and it depreciates at the rate of R% per annum,

Then Value of the machine after n years = P((1-R)/100))ⁿ

• If the present value of a machine = P and it depreciates at the rate of R% per annum,

Then Value of the machine before n years = P((1-R)/100))ⁿ

Solved Examples

Level 1

Answer : Option A

Explanation :

A = xy/100 ………….(Equation 1)

B = yx/100……………..(Equation 2)

From these equations, it is clear that A = B

Answer : Option C

Explanation :

20% of a = b

=> b = 20a/100

b% of 20 = 20b/100=(20a/100) × 20/100

=(20×20×a)/(100×100)=4a/100 = 4% of a

Answer : Option D

Explanation :

5% of A + 4% of B = 2/3(6% of A + 8% of B)

5A/100+4B/100=2/3(6A/100+8B/100)

⇒5A+4B=2/3(6A+8B)

⇒15A+12B=12A+16B

⇒3A=4B

⇒AB=43⇒A:B=4:3

Answer : Option C

Explanation :

Increase in the population in 10 years = 2,62,500 – 1,75,000 = 87500

% increase in the population in 10 years = (87500/175000)×100=8750/175=50%

Average % increase of population per year = 50%/10=5%

Answer : Option A

Explanation :

Votes received by the winning candidate = 11628

Total votes = 1136 + 7636 + 11628 = 20400

Required percentage = (11628/20400)×100=11628/204=2907/51=969/17=57%

Answer :Option B

Explanation :

He sells 40% of oranges and still there are 420 oranges remaining

=> 60% of oranges = 420

⇒(60×Total Oranges)/100=420

⇒Total Oranges/100=7

⇒ Total Oranges = 7×100=700

Answer : Option C

Explanation :

Total runs scored = 110

Total runs scored from boundaries and sixes = 3 x 4 + 8 x 6 = 60

Total runs scored by running between the wickets = 110 – 60 = 50

Required % = (50/110)×100=500/11%

Answer :Option B

Explanation :

Total numbers = 70

Total numbers in 1 to 70 which has 1 in the unit digit = 7

Total numbers in 1 to 70 which has 9 in the unit digit = 7

Total numbers in 1 to 70 which has 1 or 9 in the unit digit = 7 + 7 = 14

Required percentage = (14/70)×100=140/7=20%

                    Level 2

Answer :Option B

Explanation :

Total number of votes = 7500

Given that 20% of Percentage votes were invalid

=> Valid votes = 80%

Total valid votes = (7500×80)/100

1^st candidate got 55% of the total valid votes.

Hence the 2^nd candidate should have got 45% of the total valid votes

=> Valid votes that 2nd candidate got = (total valid votes ×45)/100

=7500×(80/100)×(45/100)=75×(4/5)×45=75×4×9=300×9=2700

Answer : Option D

Explanation :

State A and State B had an equal number of candidates appeared.

In state A, 6% 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 these, 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

Answer : Option A

Explanation :

Let the total number of students = x

Given that 20% of students are below 8 years of age

then The number of students above or equal to 8 years of age = 80% of x —–(Equation 1)

Given that number of students of 8 years of age = 48 —–(Equation 2)

Given that number of students above 8 years of age = 2/3 of number of students of 8 years of age

=>number of students above 8 years of age = (2/3)×48=32—–(Equation 3)

From Equation 1,Equation 2 and Equation 3,

 80% of x = 48 + 32 = 80

⇒80x/100=80

⇒x100=1⇒x=100

Answer : Option B

Explanation :

Let the number of candidates applied for the examination = x

Given that 5% of the applicants were found ineligible.

It means that 95% of the applicants were eligible (∴ 100% – 5% = 95%)

Hence total eligible candidates = 95x/100

Given that 85% of the eligible candidates belonged to the general category

It means 15% of the eligible candidates belonged to other categories(∴ 100% – 85% = 15%)

Hence Total eligible candidates belonged to other categories

=(total eligible candidates×15)/100=(95x/100)×(15/100)

=(95x×15)/(100×100)

Given that Total eligible candidates belonged to other categories = 4275

⇒(95x×15)/(100×100)=4275

⇒(19x×15)/(100×100)=855

⇒(19x×3)/(100×100)=171

⇒(x×3)/(100×100)=9

⇒x/(100×100)=3

⇒x=3×100×100=30000

Answer :Option A

Explanation :

Let the number = 1

Then, ideally he should have multiplied 1 by 5/3.

Hence the correct result was 1 x (5/3) = (5/3)

By mistake, he multiplied 1 by 3/5.

Hence the result with the error = 1 x (3/5) = (3/5)

Error = 5/3−3/5=(25−9)/15=16/15

percentage error = (Error/True Value)×100={(16/15)/(5/3)}×100

=(16×3×100)/(15×5)=(16×100)/(5×5)=16×4=64%

Answer :Option A

Explanation :

Price of the car = Rs.3,25,000

Car insured to 85% of its price

=>Insured price=(325000×85)/100

Insurance company paid 90% of the insurance

⇒Amount paid by Insurance company =(Insured price×90)/100

=325000×(85/100)×(90/100)=325×85×9=Rs.248625

Difference between the price of the car and the amount received

= Rs.325000 – Rs.248625 = Rs.76375

Answer : Option A

Explanation :

Assume that the initial price of 1 Litre petrol = Rs.100 ,Benson spends Rs.100 for petrol,

such that Benson buys 1 litre of petrol

After the increase by 25%, price of 1 Litre petrol = (100×(100+25))/100=Rs.125

Since Benson spends additional 15% on petrol,

amount spent by Benson = (100×(100+15))/100=Rs.115

Hence Quantity of petrol that he can purchase = 115/125 Litre

Quantity of petrol reduced = (1−115/125) Litre

Percentage Quantity of reduction = ((1−115/125))/1×100=(10/125)/×100=(10/5)×4=2×4=8%

Answer :Option C

Explanation :

Let total number of men = 100

Then

80 men are less than or equal to 50 years old

(Since 80% of the men are less than or equal to 50 years old)

=> 20 men are above 50 years old (Since we assumed total number of men as 100)

20% of the men above the age of 50 play football

⇒Number of men above the age of 50 who play football = (20×20)/100=4

Number of men who play football = 20 (Since 20% of all men play football)

Percentage of men who play football above the age of 50 = (4/20)×100=20%

=>Percentage of men who play football less than or equal to the age 50 = 100%−20%=80%

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A percentage is a number or ratio that represents a part of a whole. It is expressed as a number out of one hundred, and is often used to compare different quantities. For example, if a store is having a sale and everything is 20% off, that means that the price of each item has been reduced by 20%. To calculate a percentage, you can use the following formula:

Percentage = (Part / Whole) * 100

For example, if a store sells a shirt for $20 and it is on sale for 20% off, the new price of the shirt would be $16. This is because 20% of $20 is $4, and $20 – $4 = $16.

Percentages can also be used to calculate increases and decreases. To calculate an increase, you can use the following formula:

Increase = (New Value – Old Value) / Old Value * 100

For example, if the price of a shirt goes from $20 to $25, the increase would be 25%. This is because (25 – 20) / 20 = 0.25, and 0.25 * 100 = 25.

To calculate a decrease, you can use the following formula:

Decrease = (Old Value – New Value) / Old Value * 100

For example, if the price of a shirt goes from $20 to $16, the decrease would be 20%. This is because (20 – 16) / 20 = 0.2, and 0.2 * 100 = 20.

Percentages can be used in a variety of different fields, including mathematics, statistics, finance, and economics. They are also commonly used in everyday life to compare different quantities. For example, you might use a percentage to compare the price of two different items, or to calculate the amount of tip you should leave at a restaurant.

Here are some additional examples of how percentages are used in everyday life:

• When you buy a product on sale, the percentage off is a way of expressing the DISCOUNT you are getting.
• When you pay your taxes, the percentage of your income that you owe is called your tax rate.
• When you invest in the stock market, the percentage of your Investment that goes up or down is called your return.
• When you take a test, your score is expressed as a percentage of the total possible points.
• When you vote in an election, the percentage of the vote that each candidate receives is called their share of the vote.

Percentages are a powerful tool that can be used to compare different quantities and to make informed decisions. By understanding how to calculate and use percentages, you can better understand the world around you.

Here are some frequently asked questions and short answers about percentages, without mentioning the topic:

• What is a percentage?
  A percentage is a number or ratio that is expressed as a fraction of 100. It is written with a percent sign (%). For example, 50% is equal to 50/100, which is also equal to 0.5.

• How do you calculate a percentage?
  To calculate a percentage, you divide the part by the whole and multiply by 100. For example, to calculate the percentage of students who passed a test, you would divide the number of students who passed by the total number of students and multiply by 100.

• What are some common uses of percentages?
  Percentages are used in a variety of fields, including mathematics, statistics, finance, and business. They are also used in everyday life to express things like Discounts, sales tax, and tips.

• What are some common mistakes people make when using percentages?
  Some common mistakes people make when using percentages include:

• Not understanding the difference between a percentage and a proportion. A percentage is a number or ratio that is expressed as a fraction of 100, while a proportion is a comparison of two quantities.

• Not converting percentages to decimals or FRACTIONS when necessary.
• Not rounding percentages correctly.

• Not using percentages consistently. For example, if you are calculating the percentage of students who passed a test, you should use the same method for all of the students.

• What are some tips for using percentages correctly?
  Here are some tips for using percentages correctly:

• Understand the difference between a percentage and a proportion.

• Convert percentages to decimals or fractions when necessary.
• Round percentages correctly.
• Use percentages consistently.

• Be aware of the limitations of percentages. For example, percentages can be misleading if they are not based on a representative sample.

• What are some Resources for Learning more about percentages?
  There are many resources available for learning more about percentages, including:

• Textbooks on mathematics, statistics, finance, and business.

• Online resources, such as Khan Academy and Wikipedia.
• Calculators that can convert percentages to decimals and fractions.
• Practice problems.

Sure, here are some multiple choice questions without mentioning the topic “Percentage”:

1. A store sells a shirt for $20. If the store offers a 20% discount, what is the new price of the shirt?
   (A) $16
   (B) $18
   (C) $24
   (D) $32

2. A store sells a pair of shoes for $50. If the store offers a 10% discount, what is the new price of the shoes?
   (A) $45
   (B) $55
   (C) $60
   (D) $65

3. A store sells a computer for $1000. If the store offers a 15% discount, what is the new price of the computer?
   (A) $850
   (B) $950
   (C) $1150
   (D) $1250

4. A store sells a TV for $500. If the store offers a 25% discount, what is the new price of the TV?
   (A) $375
   (B) $425
   (C) $575
   (D) $625

5. A store sells a car for $20,000. If the store offers a 10% discount, what is the new price of the car?
   (A) $18,000
   (B) $19,000
   (C) $21,000
   (D) $22,000

6. A store sells a house for $500,000. If the store offers a 5% discount, what is the new price of the house?
   (A) $475,000
   (B) $485,000
   (C) $525,000
   (D) $550,000

7. A store sells a boat for $100,000. If the store offers a 10% discount, what is the new price of the boat?
   (A) $90,000
   (B) $95,000
   (C) $105,000
   (D) $110,000

8. A store sells a piece of jewelry for $5000. If the store offers a 10% discount, what is the new price of the jewelry?
   (A) $4500
   (B) $4750
   (C) $5250
   (D) $5500

9. A store sells a car for $20,000. If the store offers a 20% discount, what is the new price of the car?
   (A) $16,000
   (B) $18,000
   (C) $22,000
   (D) $24,000

10. A store sells a house for $500,000. If the store offers a 20% discount, what is the new price of the house?
    (A) $400,000
    (B) $450,000
    (C) $550,000
    (D) $600,000