Important Formulas – Percentage

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• 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 consumptionso as not to increase the expenditure = [R/(100+R)×100]%
• If the price of a commodity decreases by R%, the increase in consumptionso 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, thenPopulation 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, thenPopulation 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,

ThenValue 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,

ThenValue 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%,

Percentages are a way of expressing a part of a whole as a number out of 100. They are used in many different fields, including mathematics, science, business, and economics.

The percentage formula is:

Percentage = (Part / Whole) x 100

For example, if you have a pizza that is cut into 8 slices and you eat 3 slices, you have eaten 37.5% of the pizza.

The percent change formula is:

Percent Change = (New Value – Old Value) / Old Value x 100

For example, if the price of a gallon of milk goes from $3 to $3.50, the percent change is 16.7%.

The DISCOUNT formula is:

Discount = Original Price – Sale Price

For example, if the original price of a shirt is $50 and it is on sale for $40, the discount is $10.

The markup formula is:

Markup = Selling Price – Cost Price

For example, if the cost of a shirt is $40 and it is sold for $50, the markup is $10.

The commission formula is:

Commission = (Sales x Commission Rate) / 100

For example, if a salesperson sells $1000 worth of products and earns a commission rate of 5%, the commission is $50.

The profit and loss formula is:

Profit = Selling Price – Cost Price

Loss = Cost Price – Selling Price

For example, if the selling price of a shirt is $50 and the cost price is $40, the profit is $10. If the selling price is $40 and the cost price is $50, the loss is $10.

The Simple Interest formula is:

Simple Interest = Principal x Interest Rate x Time

For example, if you borrow $100 at an interest rate of 5% for 1 year, the simple interest is $5.

The Compound Interest formula is:

Compound Interest = Principal x (1 + Interest Rate)^Time

For example, if you invest $100 at an interest rate of 5% compounded annually for 1 year, the compound interest is $5.06.

Percentages are a powerful tool that can be used to solve many different types of problems. By understanding the basic percentage formulas, you can use percentages to calculate Discounts, markups, commissions, profits, losses, simple interest, and compound interest.

Here are some frequently asked questions and short answers about percentages:

• 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 sales tax, discounts, and tips.

• What are some important formulas involving percentages?
  Some important formulas involving percentages include the following:

• Percentage change = (new value – old value) / old value * 100%

• Discount = original price – sale price
• Tip = (service charge) * (total bill)

• Sales tax = (sales tax rate) * (total purchase price)

• What are some common mistakes people make when calculating percentages?
  Some common mistakes people make when calculating percentages include the following:

• Forgetting to multiply by 100.

• Using the wrong denominator.

• Not accounting for rounding errors.

• What are some tips for calculating percentages accurately?
  Some tips for calculating percentages accurately include the following:

• Make sure you understand the problem.

• Use a calculator.

• Double-check your work.

• What are some Resources for Learning more about percentages?
  Some resources for learning more about percentages include the following:

• Online calculators

• Textbooks
• Websites
• Videos

1. A store sells a shirt for $20. The store marks up the shirt by 50%. What is the selling price of the shirt?
   (A) $10
   (B) $15
   (C) $20
   (D) $30

2. A store sells a pair of jeans for $50. The store marks down the jeans by 25%. What is the sale price of the jeans?
   (A) $37.50
   (B) $43.75
   (C) $50
   (D) $62.50

3. A store sells a television for $300. The store offers a 10% discount on the television. What is the sale price of the television?
   (A) $270
   (B) $290
   (C) $330
   (D) $360

4. A store sells a computer for $1,000. The store offers a 20% discount on the computer. What is the sale price of the computer?
   (A) $800
   (B) $900
   (C) $1,200
   (D) $1,400

5. A store sells a car for $20,000. The store offers a 5% discount on the car. What is the sale price of the car?
   (A) $19,000
   (B) $19,500
   (C) $20,500
   (D) $21,000

6. A store sells a house for $300,000. The store offers a 2% discount on the house. What is the sale price of the house?
   (A) $294,000
   (B) $296,000
   (C) $298,000
   (D) $302,000

7. A store sells a boat for $50,000. The store offers a 10% discount on the boat. What is the sale price of the boat?
   (A) $45,000
   (B) $47,500
   (C) $52,500
   (D) $55,000

8. A store sells a car for $20,000. The store offers a 5% discount on the car. The customer also uses a coupon for an additional 10% discount. What is the sale price of the car?
   (A) $17,000
   (B) $17,500
   (C) $18,500
   (D) $19,000

9. A store sells a house for $300,000. The store offers a 2% discount on the house. The customer also uses a coupon for an additional 3% discount. What is the sale price of the house?
   (A) $284,600
   (B) $287,200
   (C) $290,800
   (D) $294,400

10. A store sells a boat for $50,000. The store offers a 10% discount on the boat. The customer also uses a coupon for an additional 15% discount. What is the sale price of the boat?
    (A) $38,500
    (B) $39,250
    (C) $40,000
    (D) $40,750