Life Mathematics

Percentage

      It is used to find the amount or share of something in terms of 100. In its simplest form, percent means per hundred. To express a number between zero and one, percentage formula is used.

      It is defined as a number represented as a fraction of 100. Denoted by the symbol = %, the percentage is majorly used to compare and find out ratios.

Q1.Express  as a percentage.

Solution:

       )

            

           

 

Q2. There are 150 students in a class. Out of them, 75 are girls. Find the percentage of girls in the class?

Solution:

      Total students in the class=150

      Girls in the class = 75

      % of girls in the class = (75 ⁄ 150) × 100 = (7500 ⁄ 150) = 50%

 

Percentage Increase

      It is the ratio of value increased to the original value and multiplied by 100. It is expressed in percentage. If there is an increase in the value of anything, then there is an increase in percentage. 

 

                                         

 

Q. What is the percentage change in the rent of the house if on the month of January it was Rs. 10,000 and in the month of March it is Rs. 15,000.?

Solution:

      We can clearly see that there is an increase in the value of rent, so the percentage increase will be given as:

      Percentage Increase = (Increased Value ⁄ Original Value) × 100

      Here, Increased value = 15000 – 10000 = 5000

      Original Value = 10000

      5000 ⁄ 10000 = 33.33%

      Hence, there is a 33.33% increase in the value of rent.

 

Percentage Decrease:

When there is a decrease in the value of something, then, to know the percentage decrease, that is, to know the decrease per hundred, – this formula is used. It is the ratio of the decrease of the decreased value to the original value multiplied with 100.

 

                             

 

1.     Few days ago, the cost of mangoes per 100 kg was Rs. 10000. Now is it being sold at rs. 8000. What is the percentage decrease in the value?

Solution:

      Percentage decrease in the land value

                              =

                              =

                              = 0.2 × 100

                              = 20%

      So there is a 20% decrease in value of 100 kg mangoes.

 

Some Solved Examples:

In an election, candidate A got 75% of the total valid votes. If 15% of the total votes were declared invalid and the total numbers of votes is 560000, find the number of valid vote polled in favour of candidate.

Solution:

      Total number of invalid votes = 15 % of 560000

                                                                 = 15/100 × 560000

                                                                 = 8400000/100

                                                                 = 84000

      Total number of valid votes 560000 – 84000 = 476000

      Percentage of votes polled in favour of candidate A = 75 %

      Therefore, the number of valid votes polled in favour of candidate A = 75 % of 476000

                                          = 75/100 × 476000

                                          = 35700000/100

                                          = 357000

 

A shopkeeper bought 600 oranges and 400 bananas. He found 15% of oranges and 8% of bananas were rotten. Find the percentage of fruits in good condition.

Solution: Total number of fruits shopkeeper bought = 600 + 400 = 1000

      Number of rotten oranges = 15% of 600

                                         = 15/100 × 600

                                         = 9000/100

                                         = 90

 

      Number of rotten bananas = 8% of 400

                                   = 8/100 × 400

                                   = 3200/100

                                   = 32

      Therefore, total number of rotten fruits = 90 + 32 = 122

      Therefore Number of fruits in good condition = 1000 - 122 = 878

      Therefore Percentage of fruits in good condition = (878/1000 × 100)%

                                                                 = (87800/1000)%

                                                                 = 87.8%

2.   Profit & Loss

 

      Cost Price (C.P)- The amount for which an article is bought is called its Cost Price (C.P)

      Selling Price (S.P)- The amount for which an article is sold is called its Selling Price (S.P)

      Profit/Gain-When the S.P is more than the C.P, then there is a profit or gain.

                                                                 

 

      Loss- When the S.P is less than the C.P, then there is a loss.

                                                                 

 

      Some important Formulae:

                 

                 

                 

                 

 

 

Q1. Henry sold a bicycle at 8% gain. Had it been sold for $ 75 more, the gain would have been 14%. Find the cost price of the bicycle.

Solution:

      Let the cost price of the bicycle be $ x. 

      SP of the bicycle at 8% gain = $ [{(100 + gain %) /100} × CP] 

                                                      = $ [{(100 + 8)/100} × x] 

                                                      = $ {(108/100) × x} 

                                                      = $ (27x/25)  

      SP of the bicycle at 14% gain = $ [{(100 + 14)/100} × x] 

                                                = $ {(114/100) × x} 

                                                = $ (57 x/50) 

      Therefore, (57 x /50) - (27 x/25) = 75 

                  ⇔ (57 x – 54 x)/50 = 75

                  ⇔ 3 x = (50 × 75) 

                  ⇔ x = (50 × 25) 

                  ⇔ x = 1250

      Hence the CP of the bicycle is $ 1250. 

 

Q2. Suppose a shopkeeper has bought 1 kg of apples for 100 rs. And sold it for 120 kg. How much is the profit he got?

Solution:

      Cost Price for apple is 100 rs.

      Selling Price for apple is 120 rs.

      Then profit gained by shopkeeper is; P = SP – CP

                              P = 120 – 100 = Rs. 20/-

What is a Discount?

Discount means reducing the price or value of an object or item. In other words, In order to increase the sale and also to clear the old stock, a certain percentage of rebates on the marked price of the articles. This rebate is known as discount.

 

Marked Price-

      The price on the label of an article/product is called the marked price or list price. This is the price at which product is intended to be sold. However, there can be some discount given on this price and the actual selling price of the product may be less than the marked price. It is generally denoted by MP.

 

      When Discount is offered, M.P. > S.P.

      When Discount is not offered, M.P. < S.P.

      Discount = Marked Price – Selling Price

      Discount percentage = [(Discount)/(Marked price)]× 100. 

      Discount = Discount % of Marked Price

Additional expenses made after buying an article are included in the cost price and are known as overhead expenses.

·       cost price = buying price + overhead expenses

      Sales Tax is charged on the sale of an item by the government and is added to the Bill Amount.

·       Sales tax = Tax% of sale amount.

 

Q1. Akshay offers a discount of 20% on all the items at her shop and still makes a profit of 12%. What is the cost price of an article marked at Rs 280?

Solution:

      We have the Marked Price = Rs 280.

      Also we have the Discount = 20% of Rs 280.

      Thus we can write it as = (20/100) × 280 = Rs 56. 

      So the selling price = Rs (280 – 56) = Rs 224.

      Let the cost price be Rs 100. 

      Profit = 12% of Rs 100 which is = Rs 12.

      So selling price = Rs (100 + 12) = Rs 112.

Now let us see further:

      If the selling price is Rs 112, cost price = Rs 100.

      If the selling price is Rs 224, cost price = Rs (100/224) × 112 which is = Rs 200. 

                  Ans=Rs 200

Q2. A bicycle marked at Rs 1,500 is sold for Rs 1,350. What is the percentage of the discount?

Solution:

Marked Price = Rs 1500, and Selling Price = Rs 1350.

Amount of discount is = Marked Price – Selling Price.

 In other words we can say that = (1500 – 1350) = Rs 150.

Discount for Rs. 1500 =Rs 150
Therefore, the Discount for Rs 100 = (150/1500) × 100 = 10%
Thus, the Percentage of discount = 10%

Q3.  An almirah is sold at Rs 5,225 after allowing a discount of 5%. Find its marked price.

Solution:

      The discount is given in percentage. Hence, the M.P. is taken as Rs 100. Rate of discount = 5%.

                  Amount of discount = (5/100) ×100 = Rs 5.

                  Selling Price = M.P. – Discount = 100 – 5 = Rs. 95.

                  If S.P. is Rs 95, then M.P. is Rs. 100.

                  When S.P. is Rs. 5225, M.P. = Rs. (100/95) × 5225.

                  Therefore the M.P. of the almirah = Rs. 5,500 

 

3. Simple & Compound Interest:

Simple Interest is the rate at which we lend or borrow money. 

Let the principal amount be equal to P. Let the rate at which the interest is levied is equal to R% per annum (per year). let the time for which the amount is lent = T years. Then we can write:

Simple Interest = [{P×R×T}/100]

Compound interest (or compounding interest) is interest calculated on the initial principal, which also includes all of the accumulated interest of previous periods of a deposit or loan. 

Compound Interest = Total amount of Principal and Interest in future (or Future Value) less Principal amount at present (or Present Value)

                              = [P (1 + i)n] – P

                              = P [(1 + i)n – 1]

(Where P = Principal, i = nominal annual interest rate in percentage terms, and n = number of compounding periods.)

Q. A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

Solution:

      Amount after 1 year on Rs. 1600 (deposited on 1st Jan) at 5% when interest calculated half-yearly

                              =P (1+(R/2)100)^2T

                              =1600(1+ (5/2)100)^2×1

                              =1600(1+140) ^2

                              =P (1+(R/2)100) ^2T

                              =1600(1+ (5/2)100) ^2×1

                              =1600(1+140) ^2

      Amount after 1/2 year on Rs. 1600 (deposited on 1st Jul) at 5% when interest calculated half-yearly

                              =P (1+(R/2)100)^2T

                              =1600(1+ (5/2)100) ^2×12

                              =1600(1+140)

                              =P (1+(R/2)100) ^ 2T

                              =1600(1+ (5/2)100) ^2×12

                              =1600(1+140)

      Total Amount after 1 year

                              =1600(1+140)2+1600(1+140)

                              =1600(4140)2+1600(4140)

                              =1600(4140) [1+4140]

                              =1600(4140) (8140)

                              =41×81=Rs. 3321

                              =1600(1+140)2+1600(1+140)

                              =1600(4140)2+1600(4140)

                              =1600(4140) [1+4140]

                              =1600(4140) (8140)

                              =41×81

                              =Rs. 3321

Compound Interest = Rs.3321 - Rs.3200 = Rs.121