A fraction whose denominator is 100 is read as percent, for example $ \frac{23}{100} $ is read as twenty three percent or 23%. The symbol % is used for the term percent.
Percentage: Percent means per every hundred and denoted by the symbol %. A fraction with denominator 100 is called a Percent.
Fraction and Percent
To write a fraction as percent, multiply the fraction by 100, simplify it and attach % symbol.
To write a percent as a fraction, drop the % sign, multiply the number by 1/100 (or divide the number by 100) and simplify it.
Decimal and Percent
To write a decimal as a percent, move the decimal point two places to the right and put the % sign.
To write a percent as a decimal, drop the % sign and insert or move the decimal point two places to the left.
Example: She obtained 18 marks in a test of 25 marks. What was her percentage of marks?
Example: One-fourth of the total number of shoes in a shop were on discount sale. What percent of the shoes were there on normal price?
Example: Out of 40 students in a class, 32 opted to go for a picnic. What percent of students opted for picnic?
Example: In the word ARITHMETIC, what percent of the letters are I’s?
Example: A mixture of 80 litres, of acid and water, contains 20 litres of acid. What percent of water is in the mixture?
Calculation of Percent of Quantity or Number
To determine a specified percent of a number or quantity, first change the percent to a fraction or a decimal and then multiply it with the number or the quantity.
Example: In an examination, a student scored 62% marks. If the total marks in the examination are 600, then what are the marks obtained by the student?
Example: Naresh earns Rs. 30800 per month. He keeps 50% for household expenses, 15% for his personal expenses, 20% for expenditure on his children and the rest he saves. What amount does he save per month?
Example: What percent of 360 is 144?
Example: If 120 is reduced to 96, what is the reduction percent?
Example: The cost of an article has increased from Rs. 450 to Rs. 495. By what percent did the cost increased?
Example: 60% of the students in a school are girls. If the total number of girls in the school is 690, find the total number of students in the school. Also, find the number of boys in the school.
Example: A's income is 25% more than that of B. B's income is 8% more than that of C. If A’s income is Rs. 20250, then find the income of C.
Example: A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea for Rs. 22500. What is the reduced price per kg of tea? Also, find the original price per kg.
Example: A student got 45% marks in the first paper and 70% in the second paper. How much percent should he get in the third paper so as to get 60% as overall score?
Example: Find the sum which when increased by 15% becomes Rs. 19320.
Profit and Loss
1. Cost Price (CP): The Price at which an article is purchased, is called its cost price.
2. Selling Price (SP): The Price at which an article is sold, is called its selling price.
3. Profit (Gain): When SP > CP, then there is profit.
Profit = SP - CP
4. Loss: When CP > SP, then there is loss.
Loss = CP - SP
Gain and Loss are always calculated on the CP.
$$ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 $$
$$ \text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 $$
$$ \text{SP} = \text{CP} \times \left( 1 + \frac{\text{Profit Percentage}}{100} \right) $$
$$ \text{SP} = \text{CP} \times \left( 1 - \frac{\text{Loss Percentage}}{100} \right) $$
$$ \text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit Percentage}} $$
$$ \text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss Percentage}} $$
Example: A shopkeeper buys an article for Rs. 360 and sells it for Rs. 270. Find his gain or loss percent.
Example: Sudha purchased a house for Rs. 4,52,000 and spent Rs. 28,000 on its repairs. She had to sell it for Rs. 4,92,000. Find her gain or loss percent.
Example: By selling a book for Rs. 258, a publisher gains 20%. For how much should he sell it to gain 30%?
Example: A man bought oranges at 25 for Rs. 100 and sold them at 20 for Rs. 100. Find his gain or loss percent.
Example: A man sold two horses for Rs. 29700 each. On one he lost 10% while he gained 10% on the other. Find his total gain or loss percent in the transaction.
Example: The cost price of 15 articles is equal to the selling price of 12 articles. Find the gain percent.
Example: A watch was sold at a profit of 12%. Had it been sold for Rs. 33 more, the profit would have been 14%. Find the cost price of the watch.
Discount
A discount is a reduction in the marked (or list) price of an article. For example, 20% discount means a reduction of 20% in the marked price of an article. For example, if the marked price of an article is Rs. 100, it is sold for Rs. 80, i.e. Rs. 20 less than the marked price.
Marked Price or List Price (MP): Price at which the article is listed for sale. Since this price is written (marked) on the article, so it is called the marked price.
Discount: The discount is the reduction from the marked price of the article.
Net Selling Price (SP): SP = MP - Discount
Example: A coat is marked at Rs. 2400. Find its selling price if a discount of 12% is offered.
Example: A machine listed at Rs. 8400 is available for Rs. 6300. Find the rate of discount offered.
Example: A wholesaler’s list price of a fan is Rs. 1250 and is available to a retailer at a discount of 20%. For how much should the retailer sell it, to earn a profit of 15%.
Example: A shopkeeper marks his goods 25% more than their cost price and allows a discount of 10%. Find his gain or loss percent.
Example: An article listed at Rs. 5400 is offered at a discount of 15%. Due to festival season, the shopkeeper allows a further discount of 5%. Find the selling price of the article.
Example: A retailer buys books from a wholesaler at the rate of Rs. 300 per book and marked them at Rs. 400 each. He allows some discount and gets a profit of 30% on the cost price. What percent discount does he allow to his customers?
Simple Interest
When a person has to borrow some money as a loan, he promises to return it after a specified time period along with some extra money for using the money of the lender.
The money borrowed is called the Principal, usually denoted by P, and the extra money paid is called the Interest, usually denoted by I. The total money paid back, that is, the sum of Principal and the Interest is called the Amount, and is usually denoted by A.
A = P + I
The interest is mostly expressed as a rate percent per year.
Interest depends on, how much money (P) has been borrowed and the duration of time (T) for which it is used. Interest is calculated according to a mutually agreed rate percent, per annum (R).
I = P × R × T
Example: Find at what rate of simple interest per annum will Rs. 5000 amount to Rs. 6050 in 3 years.
Example: In how many years will a sum of Rs. 2000 yield an simple interest of Rs. 560 at the rate of 14% per annum?
Example: A certain sum of money at simple interest amounts to Rs. 1300 in 4 years and to Rs. 1525 in 7 years. Find the sum and rate percent.
Example: Out of Rs. 70,000 to invest for one year, a man invests Rs. 30,000 at 4% and Rs. 20,000 at 3% per annum simple interest. At what rate percent, should he lend the remaining money, so that he gets 5% interest on the total amount he has?
Compound Interest
When the interest is calculated on the Principal for the entire period of loan, the interest is called simple interest.
But if this interest is due (not paid) after the decided time period, then it becomes a part of the principal and so is added to the principal for the next time period, and the interest is calculated for the next time period on this new principal. Interest calculated, this way is called compound interest.
The time period after which the interest is added to the principal for the next time period is called the Conversion Period.
The conversion period may be one year, six months or three months and the interest is said to compounded, annually, semi-annually or quarterly, respectively.
$$ \text{CI} = \text{P} \times \left[ \left( 1 + \frac{\text{R}}{100} \right)^{\text{T}} - 1 \right] $$
$$ A = \text{P} \times \left( 1 + \frac{\text{R}}{100} \right)^{\text{T}} $$
Example: Find the compound interest on a sum of Rs. 2000, for two years when the interest is compounded annually at 10% per annum.
Example: Calculate the compound interest on Rs. 20,000 for 3 years at 5% per annum, when the interest is compounded annually.
Example: Calculate the compound interest on Rs. 20,000 for 9 months at the rate of 4% per annum, when the interest is compounded quarterly.
Example: At what rate percent per annum, will a sum of Rs. 15,625 become Rs. 17576 in three years, when the interest is compounded annually?
Example: In how much time will a sum of Rs. 8000 amount to Rs. 9261 at 10% per annum, compounded semi-annually?
Instalment Buying
Instalment purchase scheme enables a person to buy costly goods, on convenient terms of payment. Under this scheme, the customer, after making a partial payment in the beginning, takes away the article for use after signing the agreement to pay the balance amount in instalments. Such a scheme also encourages the buyer to save at regular intervals, so as to pay the instalments.
Cash Price: The cash price of an article is the amount which a customer has to pay in full for the article at the time of purchase.
Cash Down Payment: The amount to be paid (in cash) under an instalment plan at the time of purchase of a commodity, is called the cash down payment. It is the partial payment made by the customer at the time of signing the agreement and taking away the article for use.
Instalments: It is the amount which is paid by the customer at regular intervals towards the remaining part of the selling price of the article.
Interest under the Instalment Plan: In an instalment plan only part payment of the total cost is paid by the customer at the time of purchase. The remaining part of cost is paid on subsequent dates; and therefore the seller charges some extra amount for deferred payments. This extra amount is actually the interest charged on the amount of money which the customer owes to the seller at different times of payment of instalments.
Example: A Television set is sold for Rs. 20000 cash or for Rs. 6000 as cash down payment followed by Rs. 16800 after six months. Find the rate of interest charged under the instalment plan.
Example: A table fan is sold for Rs. 450 cash or Rs. 210 cash down payment followed by two monthly instalments of Rs. 125 each. Find the rate of interest charged under the instalment plan.
Example: A microwave oven is available for Rs. 9600 cash or for Rs. 4000 cash down payment and 3 monthly instalments of Rs. 2000 each. Find the rate of interest charged under the instalment plan.
Example: A computer is sold for Rs. 30,000 cash or Rs. 18000 cash down payment and 6 monthly instalments of Rs. 2150 each. Find the rate of interest charged under the instalment plan.
Example: A ceiling fan is marked at Rs. 1940 cash or for Rs. 420 cash down payment followed by three equal monthly instalments. If the rate of interest charged under the instalment plan is 16% per annum, find the monthly instalment.
Example: A computer is available for Rs. 34000 cash or Rs. 20000 cash down payment together with 5 equal monthly instalments. If the rate of interest charged under the instalment plan is 30% per annum, calculate the amount of each instalment.
Example: The cost of a washing machine is Rs. 12000. The company asks for Rs. 5200 in advance and the rest to be paid in equal monthly instalments. The rate of interest to be charged is 12% per annum. If a customer can pay Rs. 1400 each month, then how many instalments he will have to pay?
Example: A bicycle is sold for Rs. 500 cash down payment and Rs. 610 after one month. If the rate of interest being charged is 20% p.a., find the cash price of the bicycle.
Example: A camera is sold for Rs. 2500 as cash down payment and Rs. 2100 after 3 months. If the rate of interest charged is 20% p.a., find the cash price of the camera.
Example: A mixer was purchased by paying Rs, 360 as cash down payment followed by three equal monthly instalments of Rs. 390 each. If the rate of interest charged under instalment plan is 16% p.a., find the cash price of the mixer.
Example: A refrigerator is available for Rs. 12000 cash or Rs. 3600 cash down payment along with 2 equal half yearly instalments. If the dealer charges an interest of 20% p.a. compounded semi-annually, under the instalment plan, find the amount of each instalment.
Example: A washing machine was available for Rs. 15000 cash but was purchased under an instalment plan after paying Rs. 2250 as cash down payment followed by two equal half yearly instalments. If interest charged was 8% per annum compounded semi-annually, find the value of each instalment.
Example: A juicer is available for Rs. 3500 cash but was sold under instalment plan where the purchaser agreed to pay Rs. 1500 cash down and 3 equal quarterly instalments. If the dealer charges interest at 12% p.a. compounded quarterly, find the amount of each instalment to the nearest rupee.
Example: A television set is sold for Rs. 7110 cash down payment along with 2 equal monthly instalments of Rs. 5581.50 each. If the dealer charges interest at 20% p.a. compounded monthly under the instalment plan, find the cash price of the television set.
Example: A dealer offeres a micro-oven for Rs. 5800 cash. A customer agrees to pay Rs. 1800 cash down and 3 equal annual instalments. If the dealer charges interest at 12% p.a. compounded annually, what is the amount of each instalment.
Example: A flat is available for Rs. 1600000 cash or Rs. 585500 cash down payment and three equal half yearly instalments. If the interest charged is 16% per annum compounded half yearly, calculate the value of each instalment. Find also the total interest charged.