RATIO AND PROPORTION

CONCEPT 1:
RATIOS:
1. Numbers can be used to make comparisons in day-to-day situations. When comparing any two numbers, sometimes it is necessary to find out how many times one number is greater than the other. In other words, we often need to express one number as a fraction of the other.
2. For example, Akshay scored 76 runs and Harsh scored 19 runs in the finals of the CB series. Then we can say that Akshay scored 4 times as many runs as Harsh. In simple way, by ratio of 4:1
3. Ratios are useful in making comparisons. They represent a relationship between the two quantities of the same unit. One of the values is divided by the other to find the value of one quantity in terms of the other.
4. Ratio of two terms is denoted by a:b which is equal to a/b, where a is the antecedent and b is the consequent.
For example, 5000g of type A rice is mixed with 3 kg of type B rice. The ratio of type A to type B rice can be calculated by dividing the amount of type A rice by the amount of type B rice. To calculate a ratio, the two quantities should be of same unit=> Convert 3 kg to 3000g and then find the ratio.
=> Ratio of type A to type B rice = 5000/3000 = 5/3
The ratio can be represented as 5:3 The fraction 5/3 amounts to the amount of type A rice to the amount of type B rice.
However, the fraction 5/8 represents the amount of type A rice in the mixture.


CONCEPT 2:
SCALING RATIOS:
1. The order of the terms in a ratio is important. => a:b is not the same as b:a
2. The two quantities should be of same unit. For example, 30 marks can be compared with 45 marks but not with Rs.45
3. Ratios are usually reduced to the lowest form for simplicity. Multiplying or dividing the terms in ratio by the same number does not change it
For example, if there are 5000 students in college A, 4000 students in college B and 3500 students in college C, the ratio of students in the three colleges will be 5000:4000:3500 => On simplifying, we get 10:8:7 Both the ratios are equivalent.
4. Ratios can also be expressed in percentages. To express the value of a ratio as a percentage, multiply the ratio by 100.
5. There are many situations where there are more than two quantities and they are not in the same ratio. The ratios can be scaled to find a common ratio.
For example, of the ratio of red marbles to blue marbles is 2:5 and the ratio of blue marbles to yellow marbles is 6:7, then we can find the common ratio using the scaling ratio method.
Red            Blue           Yellow
2                  5                
                    6                 7
Blue is common in both the ratios, so find the LCM of 5 and 6 => 30.
The value of 5 corresponds to 30. So, any other value in the same ratio should be multiplied by 6 => The value of 2 corresponds to 12. (2*6=12) So, 2:5 equals to 12:30
Similarly, the value of 6 corresponds to 30. So, any other value in the same ratio should be multiplied by 5=> The value of 7 corresponds to 35 (7*5=35) So, 6:7 equals to 30:35
Therefore, the ratio of red marbles to blue marbles to yellow marbles be 12:30:35


CONCEPT 3:
FINDING THE COMMON RATIO WHEN MORE THAN TWO RATIOS ARE INVOLVED:
1. The formula to find the common ratio when more than two ratios are given is as follows
Let a/b=n1/d1   b/c=n2/d2   c/d=n3/d3   d/e=n4/d4
Then a:b:c:d:e =(n1n2n3n4) : (d1n2n3n4) : (d1d2n3n4) : (d1d2d3n4) : (d1d2d3d4)
Example,
a/b=3/7  b/c=2/11  c/d=4/5  d/e=9/13
a=3*2*4*9 =216  b=7*2*4*9  c=7*11*4*9  d=7*11*5*9   e=7*11*5*13
=> a:b:c:d:e = 216:504:2772:3465:5005


CONCEPT 4:
COMPARISON OF RATIOS:
1. Let us consider two ratios a:b and c:d. Now, a:b is greater than c:d.if a/b > c/d
Multiplying both sides by bd we get ad>bc => a:b is greater than c:d is ad>bc and vice versa.
2. Thus to determine which of the two given ratios a:b and c:d is greater, we compare a*d and b*c where b>0 and d>0
For example, to compare 4:5 and 3:4, we compare 4*4 and 3*5
Since 16>15, 4:5 is greater than 3:4


CONCEPT 5:
PROPERTIES OF RATIOS:
1. When a ratio, say a:b is multiplied with itself, the new ratio formed is a^2:b^2, is known as duplicate ratio.  Also a^3:b^3 is called triplicate ratio, a^(1/3) : b^(1/3) => Sub-triplicate ratio
a^(1/2): b^(1/2) => Sub-duplicate ratio. Moreover, b:a is called reciprocal ratio of a:b
2. Consider that we are given two simultaneous equations with three unknown variables (say x, y and z). Although we required a third equation to find all three unknowns, two equations are enough to determine the ratio of the variables, i.e., x:y:z This can be done as follows
p1x+q1y+r1z = 0 and p2x+q2y+r2z = 0
Then x:y:z = (q1r2-q2r1) :  (r1p2-r2p1) : (p1q2-p2q1)
3. Multiplying or dividing the same number (say x) to both the numerator and the denominator of a ratio (say a:b) will not change the value of the ratio.
=> a/b = (a*x):(b*x) = (a/x):(b/x)
4. Effect of either adding or subtracting a number (say x) from the numerator and denominator of a ratio a:b
=> If a<b or (a/b)<1 , then for a positive quantity x,
(a+x)/(b+x) > a/b    and    (a-x)/(b-x) < a/b
=> If a>b or (a/b)>1, then for a positive quantity x
(a+x)/(b+x) < a/b   and     (a-x)/(b-x) > a/b
5. If a:b = c:d, then a:b = (a+c) : (b+d)


CONCEPT 6:
PROPORTION:
1. The equality of two ratios is called proportion. A proportion is an equation that has two equivalent ratios on either side. In other words, if a/b = c/d then a, b, c and d are in proportion.
This equality of ratios is denoted as a:b :: c:d
2. When a, b, c and d are in proportion, they are called the first, second, third and fourth proportional respectively. a and d are called the extremes and b and c are called the means. When four members are in proportion, the product of extremes is equal to the product of means.
=> if a/b = c/d, then (a*d) =(b*c)
3. The concept of proportion is not restricted to only two ratios. It can be extended to more than two equal ratios. => is a/b = c/d = e/f = g/h then a, b, c, d, e, f, g and h are said to be in proportion.


CONCEPT 7:
CONTINUED PROPORTION:
1. If a/b = b/c, then a, b and c are said to be in continued proportion. In this case, b is called mean proportional and it is also the geometric mean of a and c, b^2= ac
2. Also, in the case of a continued proportion, the ratio of the first and the third proportional is equal to the duplicate ratio of the first and second proportion.
If a/b = b/c, then a/c= a^2/b^2


CONCEPT 8:
PROPERTIES OF PROPORTIONS:
1. If a:b :: c:d or a/b = c/d, then
a/c = b/d                          [Alternendo law]
d/c = b/a                          [Invertendo law]
(a+b)/b = (c+d)/d             [Componendo law]
(a-b)/b = (c-d)/d               [Dividendo law]
(a-b)/(a+b) = (c-d)/(c+d)  [Componendo and Dividendo law]


2. If a/b = c/d = e/f = k,  then (a+c+e)/(b+d+f) = k
3. Also, if a/b = c/d = e/f = k and p,q and r are real numbers,
then (p*a^n+ q*c^n + r*e^n) / (p*b^n+ q*d^n+ r*f^n) = k^n
4. However, if a/b, c/d, e/f,.... are not equal, then (a+c+e+...) / (b+d+f+...) lies between the highest and lowest of the given fractions.


CONCEPT 9:
RATIO IN INVESTMENT:
1. If P and Q invest amounts a1 and a2 respectively for time period t1 and t2 respectively then the ratio in which the total profit earned will be divided amongst P and Q will be (a1*t1) : (a2*t2)
The unit of measurement of the time period t1 and t2 should be the same.
2. If n people invest amounts a1, a2, a3 and a4 for the time period t1, t2, t3 and t4
then the profit earned will be divided in the ratio (a1*t1) : (a2*t2) : (a3*t3) : (a4*t4)

PROFIT, LOSS AND DISCOUNT

CONCEPT 1:
PROFIT AND LOSS FORMULA:
1. Profit and loss are a part of every day transactions. The term 'cost price' and 'selling price' are used at every stage of goods exchanging hands.
2. The price at which a person buys (or produces) a product is called the 'cost price' (CP) of the product w.r.t to that person and the price at which a person sells a product is the sales price or the 'selling price' (SP) of the product again w.r.t that person.
3. At each stage, the cost price for a person becomes the selling price for another. For example, if Amit buys apples at Rs. 75/kg form the wholesaler and sells them to Sumit at Rs. 80/kg, then for one kg of apples that exchanged hands between the wholesaler, Amit  and Sumit,
The wholesaler's selling price = Rs.75 which becomes the cost price for Amit
Amit's selling price = Rs. 80 becomes the cost price of Sumit.
4. When a person is able to sell a product at a price higher than its cost price for him, then we can say that he has earned profit.
Profit = Selling price - Cost price 
5. Similarly, if  a person is able to sell an item for a price lower than its cost price for hime, we say that it is a loss (L)
Loss = Cost price - Selling price


CONCEPT 2:
PERCENTAGE PROFIT OR LOSS:
1. We often need to compare gains and losses of business transactions.
2. Percentage profit = (Selling price - Cost price)/Cost price *100
3. Percentage loss = (Cost price - Selling price)/ Cost price *100
We must always determine the percentage profit on the cost price of an item, unless it is stated otherwise in the question.
4. Selling price = [(Percentage profit/100) +1 ] *100
5. Cost price = (Selling price *100 )/ (Percentage profit+100) => When there is profit
6. Cost price = (Selling price *100)/ (100- Percentage loss) => When there is loss
7. Percentage profit = [ (SP/CP) - 1)]*100


CONCEPT 3:
TYPES OF COSTS:
1. If two items are sold for the same SP, one at a gain of a% and the other at a loss at a%, then there is an overall loss and the loss percentage = a^2/100
2. Direct costs/ variable costs: These are costs that apply to each produced commodity. They are called variable costs because these costs vary depending on the number of units of goods produced/sold. 
For example, costs such as the amount spent in buying raw materials for one unit of the product, or the amount paid to a salesman who is paid on a piece wise basis are direct costs.
3. Indirect costs/Fixed costs (Overhead costs): These are costs that remain constant. For example, costs such as rent paid for office space, monthly wages paid to employees etc.


CONCEPT 4:
BREAK-EVEN POINT:
1. When the total sales revenue earned from the selling of certain number of products equal to the total cost incurred in producing/purchasing that number of units (includes fixed cost and variable cost), then there is neither profit nor a loss.
2. In this situation, the entity involved in the transaction is said to have 'broken even', and the sales value in terms of the number of units sold is called the break-even point or break-even sales.
3. In other words, the break-even point is the number of units of a product to be sold to recover the costs.
4. Hence, if n is the number of units at which the entity breaks even, then,
Selling price of one unit * n = (Variable cost of one unit * n) + Total fixed costs
5. n (Break-even point) = Total fixed costs / (Selling price - Variable cost) for one unit.
6. Selling price - Variable cost for one unit = unit contribution margin.
7. At the break-even point, the fixed costs = Unit contribution margin * n (Total contribution)
8. Usually, the selling price is more than the variable cost associated with each unit. Hence, every unit sold beyond the break-even point contributes to the profit of the company.
Profit = (Total units sold - Break-even point) * unit contribution margin.
And if the total units sold is less than the break-even point, then a loss has been incurred.


CONCEPT 5:
FALSE WEIGHTS:
1. If an item is claimed to be sold at cost price, using false weights, then the overall percentage profit is given by,
Percentage profit = [(Claimed weight of the item/ Actual weight of the item)-1]*100


CONCEPT 6:
MARKED PRICE AND DISCOUNT:
1. The discount between selling price of a good and its cost price is known as markup.
2. Manufacturers add a markup to the cost price of an item in order to make profits. The price that is written on an article or written on the label attached to it is the sum of the cost price and the markup, and is called as the marked price => Marked price = Cost price + Markup
or Marked price = Cost price + (Markup, as a percentage/100) * Cost price
Generally, MP = SP.
3. However, sometimes, in order to increase sales or to sell of the old stock, the retailers reduce the marked price of an article by a certain amount called Discount. In this case, selling price will be the reduced price 
=> Selling price = Marked price - Discount
4. Similar to markup, discount can also be represented both as an amount and as a percentage
Selling price = Marked price - (Discount, as a percentage/100) * Marked price
5. Discount percentage = (Discount/ Marked price) *100
6. SP/MP = 1- (Discount percentage/100)


CONCEPT 7:
BUY X AND GET Y FREE:
1. If articles worth Rs.x are bought and articles worth Rs. y are obtained free along with the Rs.x article, then the discount is equal to Rs.y and the discount percentage is given by
Discount percentage = [y/ (x+y)] *100
2. Successive discounts => When a discount of a% is followed by another discount of b%, then the total discount is given by (a + b + (ab/100)) %

INTEREST AND GROWTH RATES

CONCEPT 1:
1. Money borrowed today is repaid with a higher amount tomorrow. This gives rise to the concept of interest.
2. Creditor is a person who lends money to someone who wishes to borrow it.
3. The person who receives the money from him is termed as debtor.
4. The amount of money which the creditor lends initially is known as the Principal (P) or Capital and the time frame for which he lends the money is known as the Time or Period (T or n).
5. The difference between the amount of money which the debtor borrows today (principal) and the amount of money which he needs to repay at the end of time period is called the Interest (I) over the principal amount. Also, the total money which he repays is termed as Amount (A).
Amount (A) = Principal (P)  + Interest (I)
6. The interest is calculated based on the Rate of Interest (R), which is specified in terms of percent per annum (p.c.p.a) unless otherwise stated.
7. There are two ways in which the Interests are calculated => Simple Interest and Compound Interest.


CONCEPT 2:
SIMPLE INTEREST:
1. The interest which is calculated for the given time duration only on the original principal, is called simple interest.
2. Simple interest = (Principal * Rate * Time)/100
3. Amount = Principal + Interest = P+I = P + (P*r*T)/100 
4. An example: Suppose Mukesh borrows Rs.100 from Rajesh for a period of 2 years and agrees to pay simple interest at the rate of 10% per annum. Now, the amount of interest due to Rajesh at the end of 1 year will be (100*1*10)/100 => Rs.10
So, Mukesh owes Rajesh Rs.110 at the end of the year.
5. What is the concept here? Since he has agreed to pay simple interest, he pays the interest only on the original Rs.100 for the second year too. That is, for the second year also the interest is calculated based on the original principal Rs.100 and not on Rs.110
This implies that he pays the interest of Rs.10 for the second year too. So, the cumulative interest will be Rs.20 for the time period of two years
6. This can also be calculated together for two years using the formula,
SI = (100*10*2)/100 = Rs.20 => Total amount = 100+20 = Rs.120.
7. Thus, when we calculate the simple interest, the accumulated interest is not taken into account for the purpose of calculating interest for later years. The interest remains constant every year.
Example 1:
Find the simple interest on a principal of Rs.3500 at the rate of 4% per annum for a period of 8 years.
Solution:
SI = (3500*4*8)/100 = Rs.1120
Example 2:
Calculate the simple interest on Rs. 75000 for 2.5 years at 47/3% per annum. Find the amount to be paid after 2.5 years.
Solution:
SI = (75000*2.5* (47/3))/100 = Rs. 29375.
Amount = SI + P = 29375+75000 = Rs. 104375
Example 3:
In 4 years, Rs. 6000 amounts to Rs. 8000. In what time at the same rate will Rs. 525  amount to Rs. 700?
(SNAP 2009)
Solution:
We can see that (8000/6000) = (700/525) = (4/3)
Since the ratio is same between the two, the time required for 525 to grow up to 700 will be the same as the time required for 6000 tp grow upto 8000. So, the required time period is 4 years.


CONCEPT 3:
COMPOUND INTEREST:
1. Now consider the example of Mukesh and Rajesh again. There, Mukesh owes to Rajesh Rs.110 at the end of year 1. If Mukesh pays to Rajesh an interest of 10% on Rs.110 at the end of second year, we can say that he pays compound interest. So, in this case the interest for second year is (10/100)*110 = Rs.11
So, in this case, Mukesh pays a cumulative interest of Rs.10+ Rs.11 = Rs. 21 to Rajesh at the end of 2 years. 
2. Note that the interest to be paid is more when it is compounded. In the above illustration, we consider the frequency of compounding to be yearly. This frequency of compounding is specified in questions and can be quarterly, half-yearly, 2 yearly etc.
3. What is the concept of Compound Interest? When the money is lent at compound interest, at the end of a fixed period, the interest for that fixed period is added to the principal and this amount is considered to be the principal for the next year or period. This is repeated until the amount for the last period  has been calculated.
Compound interest = Final amount - Original principal
Final amount = P * ( 1 + (r/100) )^n     => r = rate of interest and n = number of years
In the case of Rajesh and Mukesh,
Amount = 100 * (1.1)^2 = Rs.121    => CI = 121-100 = Rs. 21
Example 1:
If Rs.12000 has been lent out at 10% rate of interest, the interest being compounded annually, then what is the interest for the third year?
Solution:
Note that the compound interest at the end of 3 years is not asked, But the interest for the third year alone is asked. This can be solved by calculating the amount which is compounded for 2 years and then finding out the interest for third year (By SI method) on the compounded amount for 2 years.
Amount at the end of second year = 12000*(1+ (10/100))^2 = Rs. 14520
The interest for the third year is calculated on Rs. 14520 (In case the interest is not compounded, then it will be calculated on Rs.1200 which is the original principal)
Therefore, Interest for the third year = 14520 * (10/100) = Rs.1452
Example 2:
If the compound interest on a certain sum for 3 years at 4% is Rs.1500, then what would be the simple interest on the same sum at the same rate and the same time period?
Solution:
Let the sum be P.
CI = P*(1+ (4/100))^3 - P = 1500
=> (0.124864)P = 1500  => P= (1500/0.124864)
SI = 0.12 * (15000/0.124864) = (1500/1.04) = Rs.1442 (app)
Example 3:
Mungeri Lal has two investment plans- A and B, to choose from. Plan A offers interest of 10% compounded annually while plan B offers simple interest at 12% per annum. Till how many years is plan B a better investment? (XAT 2009)
Solution:
Let he invested Rs.100. In plan B, SI is calculated 
Plan A, Interest is compunded. For first year: (100*10)/100 = 10 => Amount =110
Second year: (110*10)/100= 11 => 121 Third year : (121*10)/100 = 12.1 => 133.1
Fourth year: (133.1*10)/100 = 13.31 => 146.41 Fifth year: (146.41*10)/100 => 14.641 => 161.051
Now, we calculate plan B for four years
(100*4*12)/100 => 48 => Amount = Rs.148 which is greater than the fourth years'value of plan A
For 5 years (100*5*12)/100 =60 => Amount = Rs.160 which is less than the fifth year's value of plan A
So, plan B is a better investment for four years.
Example 4:
Mr.Jeevan wanted to give some amount of money to his two children, so that although today they may not be using it, in the future the money would be of use to them. He divides a sum of Rs.18750 between his two sons of age 10 years and 13 years respectively in such a way that each would receive same amount at 3% p.a compound interest when he attains the age of 30 years. What would be the original share of the younger son? (IIFT 2007)
Solution:
Let P be the share of younger son and 18750-P be the share of the elder son
It is given that the amount received by the two at the end is equal.
Younger son's no. of years for maturity = 20 years and Elder son's no. of years = 17
P* (1+(3/100))^20 = (18750-P) * (1+(3/100))^17
By solving the equation, P = Rs.8959.60


CONCEPT 4:
COMPOUNDING MORE THAN ONCE A YEAR:
1. As mentioned earlier, the frequency of compounding may vary. It can be done half yearly, quarterly, monthly etc.
2. When compounding is done more than once a year, the rate of interest for the time period will be less than the effective rate of interest for the entire year.
3. For example, if the rate of interest for the whole year is 10%, then the rate of interest when the amount is compounded half-yearly will be 5% and the time period of compounding becomes doubled.
4. For half yearly rate, A = P * (1+(r/2/100))^2n
    For quarterly rate , A = P * (1+ (r/4/100))^4n
    For monthly rate, A = P * (1+ (r/12/100))^12n
NOTE:
The difference between the simple interest and compound interest (calculated on the same principal and with the same rate of interest) for the second year is equal to the interest calculated for one year on one year's simple interest.
In mathematical terms, CI-SI = P*(r/100)   => when the P and R are same for both SI and CI
Example 1:
The compound interest on a certain sum for 2 years is Rs.360 and the simple interest on the same sum for 2 years if Rs.300. Find the principal and the rate percent.
Solution:
Since SI remains the same for all years, SI for the first year= 300/2 = Rs.150
The SI and CI remain same for the first year. So, CI for the first year = Rs.150
Now CI is more because it has an additional component of interest on the SI of the first year. So, CI for the first year  = Rs.60
So, Rate of interest = (60*100)/150 = 40%   => Principal =150/0.4 = Rs.375


CONCEPT 5:
POPULATION FORMULA:
1. If the original population of a town is P and the annual increase is r%, then the population is n years (P')
P' = P* (1+ (r/100))^n
For example, if the rate of growth of population of rabbits in a warren is 100% per year or if the population doubles every year, 2 rabbits will become 16 rabbits in a matter of 3 years.
2. If the annual decrease is r%, then the population in n years is given by a change of sign in the formula:
P' = P* (1- (r/100))^n
3. Depreciation of a value: The value of an asset decreases with time and this decrease is depreciation. If the rate of depreciation is r%, then the value will be P*(1-(r/100))^n
Example 1:
The population of a city is currently 30 million. The number has been increasing at a steady rate for the past 10 years. If it is observed that the rate of increase is 15% every year, then what will be the population of the city 3 years from now?
Solution:
P= 30 million r=15 and n=3 years Formula => P' = 30* (1+(15/100))^3 = 45.6 million.


CONCEPT 6:
GROWTH RATE:
1. The absolute growth rate =  Final value - Initial value
2. Growth rate of a year = [(Final value - Initial value)/initial value ]*100
3. Simple annual growth rate or Average annual growth rate 
 => AAGR =(Sum of the growth rates (a%) of all the years)/ Number of years
4. Compounded Annual Growth Rate (CAGR) = [(Final value/Initial value)^(1/Number  of years)] -1
Example 1:
Bennett distribution company, a subsidiary of a major cosmetics manufacturer Bavlon, is forecasting the zonal sales for the next year. Zone I with current yearly sales of Rs.193.8 lakh is expected to achieve a sales growth of 7.25% Zone II with current sales of Rs.79.3 lakh is expected to grow by 8.2% and zone III with sales of 57.5 lakh is expected to increase sales of Rs. 57.5 lakh is expected to increase sales 7.15%. What is the Bennett's expected sales growth for the next year? (IIFT 2009)
Solution:
Total sales this year = 193.8+79.3+57.5 = Rs. 330.6 lakhs
Expected sales next year = (193.8*1.0725) + (79.3*1.082) + (57.5*1.0715) = Rs.355.26 lakhs.
Sales growth = ((355.26-330.6)/330.6)*100 = 7.46%


CONCEPT 7:
OFFERING LOANS ON A DISCOUNT BASIS:
1. If a bank or a company offers loan on a discount basis at a% then if the original value of the loan is Rs. 100 then the bank or the company loans out Rs. [100 - (a% of 100)] or Rs. (100-a) (In this case) and get Rs.100 back at the time of maturity.
In general, if d is the discount rate, then the effective interest rate is (d/1-d) *100
Example 1:
ICICI bank offers a 1-year loan to a company at a interest rate of 20 percent payable at maturity, while Citibank offers on a discount basis at a 19% interest rate for the same period. How much should the ICICI bank decrease/ increase the interest rate to match up the effective interest rate of Citibank? (FMS 2009)
Solution:
Assume that the ICICI bank gives a loan of Rs.100 at 20% interest=> The amount that it ll receive will be Rs.120. To calculate ICICI's effective rate such that it is competitive with Citibank's, we need to find out how much the Citibank earns if it lends out Rs.100. Citibank lends at at 19% on a discount basis.
When Citibank gives 19% interest rate on discount basis and if the original value of the loan is Rs.100, it means that the bank gets only Rs.81 right now and pays Rs.100 at the end of  the year.
So, the effective rate of interest = (19/81)*100 = 23.45%
Thus for the interest rate of ICICI bank to match up the effective interest rate of Citibank. it has to be increased by (23.45-20) = 3.45%

CONCEPTS ON PERCENTAGES FOR CAT- II

CONCEPT 3
CHANGES IN THE NUMERATOR:
The changes in the numerator and the denominator will change (increase or decrease)the overall value of the ratio.
1. The changes in the numerator is directly proportional to the value of the ratio.
=> If the value of the numerator increases, then the value of the ratio will increase.
=> If the value of the numerator decreases, then the value of the ratio will decrease.          
2. In fact, the percentage change in the value of the numerator is equal to the percentage change in the value of the ratio. An example is, if a value of an item increases from 20/10 to 30/10, the percentage change in the value of the numerator is (30-20/20 = 50%. Also, the percentage change in the value of the ratio is (3-2)/2 =50%
Example 1:
If there is threefold increase in all the sides of a cyclic quadrilateral, then the percentage increase in its areawill be? (IIFT 2009)
Solution:
A square is a cyclic quadrilateral. So, we can find the area as x^2 when the side is x
If the sides increase three folds, it means 3x then the area will be 9x^2
So, the percentage increase in area will be (9x^2 -x^2)/x^2 = 8x^2/x^2 = 800%


CONCEPT 4:
CHANGES IN THE DENOMINATOR:
1. The changes in the denominator is inversely proportional to the value of the ratio.
=> If the value of the denominator increases, the value of the ratio will decrease.
=> If the value of the denominator decreases, the value of the ratio will increase.
2. For example, consider the value of an item decreased from 20/10 to 20/12 (that is, denominator has increased by 20%). The value of the ratio decreased by 16.67%. It means that, a 20% increase in the denominator has resulted in 16.67% decrease in the value of the ratio.
3. A real-life example could be that if the price of a commodity increases by 20%, then in order for the expenditure( which is the product of the price and consumption) to remain constant, the percentage reduction in the consumption should be 16.67%
NOTE:
1. If the numerator increases by a%, then the value of the ratio will increase by a% and vice versa. But this is not the case in concern with the changes in denominator. Instead for the changes in denominator, the change in the value of the ratio is calculated as below.
2. If the price of the commodity increases by a%, then the percentage reduction in consumption, so that the expenditure remains same is (a/a+100)*100.
3. If the price of the commodity decreases by b%, then the percentage increase in consumption, so that the expenditure remains same is (b/b+100)*100.
Example 1:
Rice is now being sold at Rs.27 per kg. During last month it was sold at Rs.24 per kg. Find how much percent that a family should reduce its consumption to keep the expenditure fixed. (SNAP 2009)
Solution:
Assume the family consumes 1kg.
The reduction in consumption is 24/27 => 8/9
So, the percentage of reduction reduced is [1-(8/9)] /1* 100 = 11.11%
Example 2: 
The price of a commodity decreases by 20%. By what percentage should the quantity increase so as to keep revenue constant?
Solution:
The formula for percentage increase is b/100-b => 20/100-20 => 25%
Example 3:
Sowmiya generally spent Rs.800 for buying a month's provision of potatoes. However, lower yield this year caused the cost of one kilogram of potatoes to be increased by 60%. Owing to this, Sowmiya had to buy 30 kg less potatoes than usual. What was the cost of potatoes this year?
Solution:
If the cost of one kilogram of potatoes increases, the decrease in the consumption so as to keep the expenditure constant should be (60/(100+60))*100 = 37.5%
Decrease in the consumption is what which is given (the decrease of 30 kg potatoes in whole consumption than usual) So, 37.5% decrease = 30 kg 
Actual consumption after price increase= (30/37.5)*100 =  80kg
So, the usual consumption is 80 kg. This year's consumption = 80-30 = 50 kg
=> Cost of potatoes this year = 800/50 = Rs.16 per kg.


CONCEPT 5:
CHANGES IN BOTH NUMERATOR AND THE DENOMINATOR:
1. If the numerator increases and the denominator decreases, then the value of the ratio will surely increase.
2. If the numerator decreases and the denominator increases, then the value of the ratio will surely decrease.
3. However, if the numerator and the denominator simultaneously increase/decrease then it is not quote so apparent how the value of the ratio will change.     
4. We already know how the individual changes in the numerator and the denominator affect the ratio. Now, we ll try to combine these concepts. 
5. Consider that if the numerator increases by a% and the denominator increases by b%. Since the numerator increases by a%, the value of the ratio will also increase by a%. And since the denominator increase by b%, the value of the ratio will decrease by b/100+b% Thus by applying successive changes, we can find out the net percentage increase.
Example 1:
A student took a certain entrance exam and scored 180 out of 250 marks in the first attempt. However, during the second attempt, the total marks were increased to 330, out of which the student scored 250. Find the percentage change in the percentage marks of the student in the first two attempts.
Solution:
The numerator increases by (250-180)*180 = 38.89% (a%)
The denominator increases by (330-250)*250 =32%
So, b% is calculated as 32/(100+32) = 24.24% (b%) (value of ratio will decrease by 24.24%)
Net percentage change= a%+(-b%)+ (a*-b)/100 = 5.22% (approximately)
Example 2:
The numerator of the ratio M/N is increased by 20% and its denominator is increased by 25%. Then the numerator of the new ratio formed is increased 10% to get the fraction P/Q. What is net percentage change between the ratios M/N and P/Q?
Solution:
An increase of 25% in denominator results in 25/100+25 = 20% decrease in the value of the ratio.
So, consider the initial value of the ratio to be 100
100 => increase if 20% as per increase in num => 120 => decrease of 20% as per increase in den => 96
The net percent change will be |96-100| = 4% decrease
The numerator of the new ratio is increased by 10%
100=> decrease of overall 4% => 96 => increase of 10% as per increase in num => 105.6
Net percentage change between the two ratios = 105.6 - 100 = 5.6% increase


CONCEPT 6:
PERCENTAGE CHANGE IN A QUANTITY (A*B) WHEN BOTH A AND B CHANGE:
The difference between this method and the above mentioned is, earlier it is stated how to find the percentage change in one of the variables (say b), when the other changes (say a) and the product remains constant (a*b) 1. Now, we see how to calculate the percentage change in (a*b) when both a and b change.
2. An illustration : Let the original value of an item be A=a*b. This changes to B= x*y in the next year. So, to find the percentage change between A and B, we will find the percentage changes between a and x (p%) and b and y (q%). So,
p= (x-a)/a % and q = (y-b)/b % (Here p and q will be negative if there is a percentage decrease)
3. An example: Let A = 10*12 and B= 9* 16.=> p=-10% and q=33.33%
So, the net percentage change = 120-100 = 20%
Example 1:
Due to erosion of the soil from some parts of his field, a farmer considered increasing the length of his rectangular field by 25% and reducing the breadth by 12%. What will be the percentage change in the area of his plot?
Solution:
Let the area be 100. => 25% increase => 125 => 12% decrease => 110 
The percentage change will be 110-100 = 10%


CONCEPT 7:
SUCCESSIVE PERCENTAGE CHANGES:
1. Two successive increases on a particular value of a% and b% would be equal to a net increase of
[a+b+(ab/100)] %
In case of decline in growth or a discount, the value of a,b or both is negative.
2. Three successive increases on a particular value of a%, b% and c% would be equal to a net increase of
([(100+a)/100]*[(100+b)/100]*[(100+c)/100] -1) %
Example 1:
Ravi's income has increased by 10% over the last year and will be 20% higher next year. If last year his salary was Rs.15000, what will it be next year?
Solution:
Successive increases of 20% and 10% = 20+10+ (20*10)/100 = 32%
Ravi's salary next year = 15000*1.32 = Rs.19800
Example 2:
At the end of year 1998, Shepherd bought nine dozen goats. Henceforth, every year he added p% of the goats at the beginning of the year and sold q% at the end of the year where p>0 and q>0. If Shepherd had nine goats at the end of the year 2002, after making the sales for that year, which of the following true?
1. p=q     2.p<q     3. p>q    4.p=q/2    (CAT 2003)
Solution:
Shepherd bought 9 dozen goats at the end of year 1998. Since p% and q% are same every year, and that at the end of year they should negate which means the number of goats added and sold are same every year,
So, consider he bought 1 dozen of goats to it, Percentage added = 1/9*100 = 11.11% (p)
He sold 1 dozen of goats to get back the same, Percentage sold = 1/10* 100 = 10%(q)
So, always P>Q
Example 3:
A student took five papers in an examination where the full marks were the same for each paper. His marks in these papers were in proportion of 6:7:8:9:10 In all papers together, the candidate obtained 60% of marks. Then the number of papers in which he got more than 50% marks are?
Solution:
Average scored in all the papers = (6+7+8+9+10)/5 = 40/5 = 8
Because he scored a total of 60% of marks, 50% of marks will be (8/60)*50 = 6.67
So, number of subjects he scored more than 6.67 is 4
Example 4:
A college raised 75% of the amount needed for a new building by receiving an average donation of Rs.600  from the people already solicited. The people already solicited represents 60% of the people the college will ask for the donations. If the college is to raise exactly the amount needed for the new building, what should be the average donation from the remaining people solicited? (CAT 2001)
Solution:
Let the population be p. Then the amount already received = 600* 0.6p
Remaining amount = 25% => (3/4) of the total amount = 360p
(1/4) of the total amount = 25 percent = 120p
Remaining people = 0.4p So, Amount per head (Average donation) = 120p/0/4p = Rs.300
Example 5:
Fresh grapes contain 90% of water by weight while dry grapes contain 20% of water by weight.
What will be the weight of dry grapes available from 20 kg of fresh grapes? (CAT 2001)
Solution:
Total weight of fresh grapes = 20 kg
As water has 90% by weight in this total weight, the weight of grape mass will be 10%
So, the actual weight of grape mass in total weight of fresh grapes = (10/100)*20 = 2 kg
(Note- In this question it is evident that both fresh and dry grapes has same grape mass, only the water content percentage is different)
Therefore, the weight of dry grapes available from 20 kg of fresh grapes is calculated by taking into account the percentage of grape mass that can be contained (which is 100 -20% of weight of water) => 80%
For 80% of grape mass, the actual weight can be 2 kg.
So, for 100% of grape mass, what can be the weight that can be made available from 20 kg of fresh grapes?
=> (2/80)*100 = 2.5 kg
Example 6:
The present value of an optical instrument is Rs.20000. If its value will depreciate 5% in the first year, 4% in second year and 2% in the third year, what will be its value after three years? (FMS 2009)
Solution:
Value after 3 years = 0.95*0.96*0.98*20000 = Rs. 17875.2
Example 7:
BSNL offers its share at a premium of Rs.40, whereas its par value is Rs.160. Parul Mehra invested Rs.50000 in this stock. After one year, BSNL declared a dividend of 19%. What rate of interest did Ms.Mehra receive on her investment? (FMS 2009)
Solution:
Premium is the excess of money that the company receives over its par value
So, cost of buying one share = 160+40 = Rs.200
No. of shares she bought = 50000/200 = 250 shares
Dividend = 0.19*250*160 = Rs. 7600 (Dividend is always calculated on par value)
Rate of interest of dividend on investment = (7600/50000)*100 = 15.2 %

CONCEPTS ON PERCENTAGES FOR CAT- I

METHODS TO CALCULATE PERCENTAGES:
1. Percentage refers part of or a fraction of the whole.
2. It is a way to describe a number as a fraction with denominator 100.
3. Percent implies 'for every hundred' and is denoted by % This concept is developed to facilitate easier comparison of fractions. Hence, it is very useful in both data interpretation and solving quantitative problems.
Example:
If a student receives 20 marks out of 50, then his percentage is (20/50)*100 = 40%
More examples:
1/5 = 20%, 4/25 = 16%, 0.36 = 36 %, 0.52 = 52%
Likewise, percentages are used to understand many situations in our daily lives. Some situations are percentage of students scoring first class in an exam and the percentage increase in the cost of a product.
Case 1:
If in a class, 15 out of 25 students are girls, then what is the equivalent saying the proportion of girls?
(15/25)*100 = 60% In ratios, we can say that the ratio of girls to the boys is 3:2
Case 2:
x is 15% of 70. Find x => x= (15/100)*70 => x=1050/100 = 10.50%
Case 3:
484 is 40% of x. Find x => 484= (40/100)*x = 484 *(100/40) = 1210
Case 4:
x% of 85 is 15. Find x => (x/100)*85= 15 => 15*(100/85) = 17.647%
Example 1:
An agent sells goods of value of Rs.15000. The commission which he receives at the ratio of 12.5% is? (SNAP 2009)
Value of goods = 15000 => Commission = (12.5/100) * 15000 = Rs.1875.
Example 2:
Sonali invests 15% of her monthly salary in insurance policies. She spends 55% of her monthly salary in shopping and household expenses. She saves the remaining amount of Rs.12750. What is sonali's monthly income? (SNAP 2009)
Solution:
Total percentage of monthly salary spent = 15+55 = 70%
Remaining 30% which is saved = 12750 => (30/100)*Monthly income = Rs.12750
=> Monthly income = 12750*(100/30) = Rs.42500.
ALTERNATIVE METHOD OF CALCULATING PERCENTAGES:
There is an alternative method of finding the percentages when a ratio is given.
The basic idea of this method is to remove certain percentages of the denominator from the numerator preferably multiples of 10.
Consider an illustration- To find the percentage for 7526/4626.
Step 1: Remove 100% of 4646 from the numerator since 7526>4626. => 7526-4626 = 2900.
Step 2: Now, the ratio can be written as 7526/4626 = 100% + (2900/4626)
Step 3: We can see that 10% of 4626 is 462.6 and 400*6 = 2400 So, we can try by calculating 60% of 4626= 2775.6 Now remove 60% from the numerator.
(The concept is we need to find by trial and error that how much percent of the denominator is approximately the numerator.For that first, calculate 10% of it and then find out through the multiples of that value)
Step 4: Removing 60% => 2900-2775.6 = 124.4 So, the step will be
 7526/4626 = 100% + 60% + (124.4/4626)
As we know that the 10% of 4626 is 462.6 and now the numerator is far below this, the actual value will lie between 160% and 170%.
Depending on the accuracy required, we could either stop at this point or move on further, (Applicable mostly in DI problem sets)
Step 5: Still if we need to move on , calculate 5% of 4626 => 232.3 which is greater than the numerator 124.4 
So, the value will be between 160% and 165%
Step 6: Again moving on further accurate, calculate 2.5% = 116.15 which is less than the numerator 124.4 
So, the answer will be appoximately 163% ( more than 162.5%)
The greatest advantage of this approach us is that it gives the liberty of stopping calculation as soon as the required accuracy is arrived at depending on the available answer options.
Example 3:
If Roshini scored 71 marks out of 93 marks in Mathematics, what was her percentage score?
Solution:
We need to find (71/93)*100
Applying the method stated above, calculate 50% of 93= 42.5
The numerator, 71 is far greater than 42.5
So, calculating 20% of 93= 2*10% of 93 = 18.6=> 50%+20% = 42.5+18.6 = 61.1
Removing 70% from the numerator (71/93) = 70% +(5.9/100)
We know that 10% is 9.3 and 5% is 4.15 So, the value should lie between 75% and 80%
5% is 4.15 and 1% is 0.93 => Adding 5% and 1% => 4.15+0.93 = 5.88 (approximately equal to 5.9 which is the numerator) So, the accurate percentage score will be 76% 


CONCEPT 2
PERCENTAGE INCREASE OR DECREASE:
Percentages are used to often indicate the changes in the quantity.
Percentage change = (Final value- Initial value)/Initial value
For example, if the cost of a product changes from 6 to 78, the percentage change is given by, (8-6)/6 
1. If a quantity increases by a% , then its value is multiplied by (100+a)/100. For example, if there is 25% increase in a product worth Rs.464, its new price will be 1.25*464 = Rs.580.
2. If a quantity decreases by b%, then its value is multiplied by (100-a)/100. For example, if there is 25% decrease in a product worth Rs.464, its new price will be 0.75*464 = Rs.348.
Example 1:
A's salary is 20% more than B's salary. By what percentage is B's salary less than A's salary?
Solution:
Let B's salary be Rs.100. A's salary will be Rs.120. 
B's salary is (120-100)/120 =>16.67% less than A's salary (Because here the initial value is 120 and the final value is 100)
ABSOLUTE VALUE CHANGE AND PERCENTAGE CHANGE:
1. Absolute value change is the actual change that occurs in the measure of a quantity 
2. Percentage change is the absolute change with respect to the measure of the original quantity.
Example 2:
If the cost of a product increases from Rs. 500 in 2000 to Rs.750 in 2001, then calculate the absolute value change and the percentage change between the two years.
Solution:
The absolute value change = |Final vale - Original value| = |750 - 500| = Rs.250
The percentage change = (750-500)/500 = 50%
Example 3:
The interest rate of a bank is increased from 11% in 2003 to 12.5% in 2004.  Calculate the percentage point change and percentage change.
Solution:
The percentage point change = 12.5% - 11% = 1.5%
The percentage change = 1.5/11 = 13.63%