**Percentages****Percent (%)**

Percent means for every hundred or per hundred. The numerator of “per hundred” is called the rate percent. Example 12/100 can be called as 12%, and 12% is the rate percent, and 12 is the rate. The other way to look at it is if some makes a profit of 20%, then one has gained 20/100 of the value invested.

**Important notes**

**1.**To express percent as a fraction divide it by 100,

50 % = 50/100 = 1/2

**2.**To express a fraction as a percent multiply it by 100,

½ = (1/2)x100 = 50%

**3.**Increase/Decrease Percent= Here Increase/Decrease = (Final value – initial value)

**Important relations in percentage**

**1.**If the price of a commodity increases by r%, then percentage reduction in consumption, so as not to increase expenditure is :-

Expenditure = Price x Consumption Decreases = (r/100+r) x 100

**Example:**If the cost of petrol increases by 40%, by what percent the person should reduce his consumption considering expenditure on petrol remains the same.

Increase in price = 40%, by the formula, decrease in consumption is = = 28.57%

**2.**If the price of a commodity decreases by r%, then increase in consumption, so as not to decrease expenditure is

**Example:**If the cost of petrol decreases by 10 %, by what percent can a person increase his consumption considering expenditure on petrol remains the same?

Decrease in price = 10%, by the formula, increase in consumption is = 11.11%

**3.**If A’s income is r% more than B’s then B’s income is how much % less than A’s.

**Difference =**(r/100+r) x 100

Example: If A’s income is 20% more than B’s, then what percent is B’s income lesser than A?

A’s income more than B = 20%, by the formula, B’s income is = × 100 = 16.66 % less of A

**4.**If A’s income is r % less than B’s then B’s income is by what % more than A’s income

Difference = (r/100-r) x 100

**Example:**If A’s income is 20 % less than B’s, then what percent is B’s income more than A?

A’s income less than B = 20%, by the formula, B’s income is = × 100 = 25% more of A

**CALCULATIONS IN PERCENTAGES**

Let’s start with a number A (= 1A)

**1.**A increased by 10% would become A + 0.1A = 1.1A

**2.**A decreased by 10% would become A – 0.1A = 0.9 A

**3.**A increased by 200% would become A + 2A = 3A

**4.**A decreased by 50 % would become = 0.5A

**5.**Use decimal fractions while adding and subtracting and normal fractions while multiplying.

**SOLVED EXAMPLES**

**Q1.**Express 1/3 as percentages: .

**Ans.**To convert to percent, multiply each by 100

1/3 x 100 = 33.33%

**Q2.**If a pipe, A is 30 meters and 45% longer than another pipe, B find the length of the pipe B.

Ans. A = 30m, A = B + B => A = B => A = 1.45B Therefore B = = = 20.68.

**Q3.**On my sister’s 15th birthday, she was 159 cm in height, having grown 6% since the year before. How tall was she the previous year?

**Ans.**Height this year = 159 cm, growth = 6 %, let last year height be A

Now A = 159 A = = 150

Last year height = 150

**Q4.**Arun spent 25 % of his pocket money, and has Rs 125 left. How much had he at first?

**Ans.**Pocket money spent = 25%, left = 75%, Let original pocket money be A

Therefore A = 125 A = 125 ×

(By now students should be able to do this in single step), A = 166.66

**Q5.**If the cost of electricity increases by 30%, by what percent one should reduce his spend in order that spent on electricity stays the same?

**Ans.**As per the formula, × 100 The reduced percentage spent = × 100 = 23.07%

**Q6.**If the price of petrol increases by 25% and Rajesh intends to spend only 15% more on petrol, by how much % should he reduce the quantity of petrol that he buys?

**Ans.**Let the initial cost of 1 litre be A

Cost after increase = 1.25A (25% increase)

Let Rajesh’s initially buy be ‘B’ litres of petrol

Initial spent = AB

Increase in spent = 15%,

Current spent = 1.15 AB (15% increase)

Let the number of litres he is buying now is C.

Therefore 1.25AC = 1.15 AB, 1.25 C = 1.15 B, C = 0.92B, which means that current consumption is 92% of earlier consumption, therefore Rajesh has reduced his consumption by 8 %

**Q7.**In an election, Congress secured 10% of the total votes more than BJP (consider only two parties in the election and everyone voting). If BJP got 126000 votes, by how many votes did it lose the election?

**Ans.**Let congress secured X % of the total votes, therefore BJP had secured (X – 10) % of votes, being a two party election:

X + X – 10 = 100

2X = 110

X = 55

Therefore Congress has 55% of vote and BJP has 45%, since BJP got 126000 votes

of total votes = 126000

Total votes = 280000

Congress votes = 55/100 × 280000 = 154000

Difference = 28000, which is the victory margin.

**Q8.**If the population is1500000 and the expected birth rate is 50%, while the expected death rate is 31%, What will be the net change in the in the population at the end of the one year.

**Ans.**The current population is 1500000

Number of births will be 50/100 × 1500000 = 750000

Number of deaths was 31/100 × 1500000 = 465000

Net change = 750000 – 465000 = 285000

**Q9.**What is the % change in the area of a square (which will become rectangle) if its length side is increased by 10% and its width side is decreased by 10%?

**Ans.**In these types of problems, assume a percent (100) base and then move forward.

Let the side of square be 100

So length side will become

1.1(10% increase) × 100 = 110

And the width side will become

0.9(10% decrease) × 100 = 90

Old Area = 100 × 100 = 10000

New Area = 110 × 90 = 9900

Difference = 10000 – 9900 = 100

Difference percent = 100/10000 × 100

= 1% decrease in the area

**Q10.**Ram obtains 40 % of the marks in a paper of 200 marks. Shyam is ahead of Ram by 25 % of Ram’s marks, while Bhuvan is ahead of Shyam by one ninth of his own marks. How many marks does Bhuvan get?

**Ans.**Ram’s marks = 40/100 × 200 = 80

Shyam’s Marks = 1.25 (25% ahead) × 80 = 100

Let Bhuvan’s Marks be A, therefore

A = 100 (Shyam’s marks) + 1/9A

8/9 A = 100, A = 112.5

**After grasping the concept of percentages, let us move to its biggest application, problems of profit and losses.**

**Profit & Loss**

Basic concepts in profit and loss

**1.**Cost price (C.P): The price at which an article is bought

**2.**Selling price (S.P): The price at which an article is sold

**3.**Marked Price: The price listed on the label

**4.**Discount: The reduction offered on the list prize, it may be a value or a percent

**5.**Mark-up: The increment over the cost price

**6.**Profit or loss = SP – CP (the negative value indicates the loss)

**7.**Profit or loss percentage = (SP-CP)/CP x 100 % (the negative value indicates the loss percentage). This formula is the most important formula, if you use only this formula in the entire profit and loss chapter, you will never make an error, so try following it as much as possible.

**8.**To calculate gain/loss percentage, it is not required to have all the values of cost price and selling price, you can assume the values to be x, or even 10 or 100.

**Example:**A shopkeeper puts the marked price on his goods 25 % above cost price and gives discount of 12.5 % on marked price. What is his profit %?

Let the CP be 10, then marked price is 1.25*10 = 12.5

Now discount to the cash purchaser = 12.5 %, which is 0.125*12.5 = 1.56

Net gain = 2.5(excess on CP) – 1.56 = 0.94

gain % = 0.94/10*100 = 9.4 %

Important Relations

**1.**If two items are sold each at rupees R, one at a gain of X % and other at a loss of X %, there is always an overall loss given by ( X2 / 100 ) % and the value of loss is given by ( 2X2S )/( 1002 – X2 ). In case of the cost price of both the items is the same and percentage loss and gain are equal, then net loss or profit is zero. The difference between the two cases is the cost price, in first case it is not same, in the second it is same.

**Example:**Amar sells two watches for Rs 1200, one at a profit of 10 % and other at a loss of 10 %. Find his gain or loss percentage and the actual gain or loss.

Using the formula, Loss % = (X2/100) % = 10 x 10/ 100% = 1 %

The value of loss = (2 x X2 x S) / (1002 – X2)

= (2 x 10 x 10 x 1200) / (1002 – 102) = Rs 24.24.

Alternatively, SP of both watches = 1200, Total SP = 2400 (2 watches)

One watch was sold at a profit of 10%, using our basic formula:

Profit or loss percentage = (SP-CP)/CP x 100 %

10 = (1200 –CP)/CP x 100

110CP =120000

CP = 1090.90

One watch was sold at a loss of 10%, so its CP is

-10 = (1200 –CP)/CP x 100

90CP =120000

CP = 1333.33

Total CP = 1333.3333 + 1090.9090 = 2424.24

Total SP = 2400

Loss = 2424.24 – 2400 = 24.24

Loss % = 24.24/2424.24*100 = 1 %

So it is important to understand both formula and basic logic for the question, as explained earlier, if the basic formula of profit and loss is used, you will never go wrong.

Students should learn to do the above calculations in one step, using the formula and concept of percentage, if an item is sold at 10% profit, then:

CP = SP

CP = SP/1.1

**2.**A dishonest shopkeeper claims to sell goods at cost price, but uses a lighter weight , then his Gain % = [ 100 x excess / ( original value – excess ) ]

**Example:**A shopkeeper sells rice to a customer, using false weight and gains 100/8 % on his cost. What weight has he substituted for a kilogram?

Using the formula, Gain % = [100 x excess / (original value – excess)]

100/8 = [100 x excess/(1 – excess)]

From here, Excess = 0.111.. Kg, which is 111.11 grams

Weight used by shopkeeper = 1000 – 111.11 = 888.89 grams

Alternatively, if he is selling 1 gram for Re. 1, then 1000 grams are for Rs. 1000, so once a shopkeeper sells 1 kg, he makes Rs. 1000, using the basic profit and loss formula, Profit or loss percentage = (SP-CP)/CP x 100

100/8 = (1000 – CP)/CP x 100

100 CP = 800000 – 800CP

CP = 8000/9 = 888.89

Since here grams is equal to rupees, the weight used is 888.89 grams

Again we establish that the basic formula works well everywhere.

Miscellaneous Examples:

**Q11.**Ravi sells two tables at same price, one at a profit of 10 % and other at a loss of 10 %. Find his gain or loss percent

**Ans.**As per the formula, there will be loss and it will be given by (X2/100) %

= 10 x 10/ 100% = 1 %

**Q12.**A bookshop sells an old book for Rs. 49.35, making a 6% loss on the cost. What was the cost price of the book, and what is the cash value of the loss?

**Ans.**Since loss is 6%, and assuming cost of the book is ‘A’

0.94 A = 49.35

A = Rs. 52.50

Value of the loss = 52.50 – 49.35 = Rs. 3.15

**Q13.**If the cost price of 20 articles is equal to the selling price of 25 articles, what is the % profit or loss made by the merchant?

**Ans.**CP of 20 Articles = SP of 25 articles

Which is CP of 20 Articles = SP of 20 articles - SP of 5 articles

Now SP of 5 articles = SP of 20 articles - CP of 20 Articles

SP of 5 articles = Loss (Since CP is greater than SP in this case)

Also SP of 5 articles = CP of 4 articles (as per our first equation)

Loss Percent = (SP of 5 articles)/( CP of 20 Articles) x 100

= (CP of 4 articles)/( CP of 20 Articles) x 100

= 1/5 x 100 = 20%

Alternatively, assume the CP = 100, CP of 20 articles will be 2000

And SP of 25 articles will also be 2000, therefore SP of one article is 80, since CP is 100, loss is 20%

**Q14.**If SP of 10 articles is equal to CP of 12 articles. What is the gain percent?

**Ans.**Assume the CP = 100, CP of 12 articles will be 1200

And SP of 10 articles will also be 1200, therefore SP of one article is 120, since CP is 100, gain is 20%

**Q15.**If Kapil had sold a stereo for Rs. 6000, he would have made a 20% profit. Instead, he sold it for a 40% loss. At what price was the stereo sold?

**Ans.**Here at 6000 he was making 20% profit, therefore 1.20 CP = 6000, CP = 5000

And since he sold at a 40 percent loss, it is 0.60 CP

Which is 0.60 x 5000 = Rs. 3000

**Q16**. Rajan sold an article at a profit of 30 %; and had he sold it for Rs 3 more, the profit would have been 40 %. Find the cost price.

**Ans.**Suppose of the CP = 100. To Sell at gain of 30%, SP = 1.3 x 100 = 130

Now, To Sell at gain of 40%, SP = 1.4 x 100 = 140

Here when the difference in 30% and 40% is 10 (140-130) then CP is 100

When the difference is 1 the CP would be 100/10 = 10

When the difference is 3 the CP would be 10 x 3 = 30

Alternatively, it is given 40 % - 30 % of CP = Rs 3

Therefore 10% of CP = Rs. 3

CP = 3 x 100/10 = Rs. 30

**Q17.**A Vegetable vendor buys 240 kilograms of potato for Rs. 380. If 20 percent of the potato is unusable, at what average price per kilogram must he sell the rest of the potato in order to make a profit of 25 percent?

**Ans.**Cost price = Rs. 380, Total potato bought = 240 kg

Potato unusable = 20% = .2 x 240 = 48 kg

Potato left = 192 kg

To gain 25% on CP (380), the total SP should be 1.25 x 380 = 475

Selling Price per kg = 475/192 = Rs. 2.47 per kg.

**Q18.**By selling an article at 80% of its marked price, a merchant makes a loss of 12%. What will be the % profit made by the merchant if he sells the article at 95% of its marked price?

**Ans.**Let the marked price be 100. When the SP is 80%, which is 80, he is at a loss of 12 %, therefore by the basic formula (80 – CP)/CP x 100 = -12

8000 – 100CP = -12CP

CP = 8000/88 = 90.90

If the article is sold at 95%, which is 95 here, profit = 95 – 90.9 = 4.1

Profit % = 4.1/90.9 x 100 = 4.51 %

**Q19.**A shopkeeper marks his goods at 30 % above cost price and allows discount of 15 % for cash payment. What profit % does he make?

**Ans.**Assume the cost price = 100, so the marked price (30% more) = 1.3 x 100 = 130

A discount of 15% on the same is 0.15 x 130 = 19.50

Actual SP = 130 – 19.50 = 111.50

Gain percent = 111.50 – 100 = 11.50 %( since base is 100)

**Q20.**Krishna sold his pen for Rs 24 and got a % of profit equal to the cost price; find the cost price.

**Ans.**By the basic formula, Gain % = (SP-CP)/CP x 100 %

Here gain % = CP and SP = 24

Therefore, CP = (SP-CP)/CP x 100

(CP)2 = 2400 - 100CP

(CP)2 +100CP - 2400 = 0

Solving CP = 20

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