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Topic#1 : AVERAGE

Introduction

• If a player has scored 3, 112, 42 and 56 runs in his first four matches, how many runs can be expected to score when he goes out to bat in the next match? The answer is around 53 and is called his average score.
• By definition, an average is the most likely middle value of the data set. It is the value that each element of the set would take if all the elements of the set were to be the same. The average of the elements of a set can be calculated as the sum of all the values in the set divided by the total number of the values.
• Average = (Sum of all the values in the data set)/(Number of values in the data set)

Example:

A group of 5 friends scored 10, 12, 16, 20 and 18 in a class test. Find the average score of the students.

Solution:

Average = (10 + 12 + 16 + 20 + 18) / 5 = 15.2

Hence, the average score is 15.2

This implies that if any student from this class now takes the exam, s/he can be expected to obtain approximately 15 marks in the test.

Points to Remember

• If the value of each item group is increased/decreased by the same value x, then the average of the group also increases/decreases by x. For instance, if the income of each person in a group increases by Rs. 10, the average income of the group also increases by Rs. 10. This is valid only when the value of each item increases/decreases by the same amount.
• If the average age of the group of people is x years, then their average age after n years will be (x + n) years and their average age n years ago would be (x – n) years. This is because with each passing year, each person’s age increases by 1 and vice versa.
• If the value of each item in a group is multiplied/divided by the same value x (where x is not equal to 0, in case of division), then the average of the group also gets multiplied/divided by x.
• The average of a group always lies between the smallest value and the largest value in that group.
• If the average of a set is x, and an element having a value n is added, such that n > x, then the average of the new set is greater than x. On the other hand, if n < x, then the average of the new set is less than x.
• Conversely, if the average of a set is x, and an element having a value n is removed from the set, such that n > x, then the average of the new set is less than x. On the other hand, if n < x, then the average of the new set is greater than x.

Weighted Averages

• If a person invests 30% of his money in Gold, 25% of his money in the stock market, 40% of his money in fixed deposits and the remaining 5% in a savings account; and wants to know the return obtained in his entire investment, he needs to make his calculations keeping in mind the different proportions allocated to various categories. This can be done using the concept of “Weighted Averages”. The term ‘weight’ stands for the relative importance that is attached to the values.
• The average in such a case is called the weighted average and is given by the following formula.
• Weighted Average = (w1x1 + w2x2 + w3x3 + ————– + wnxn) / (w1 + w2 + w3 + ————– + wn), where w1, w2, w3.. are the weights of the respective values.
• If all the weights are equal, then the weighted average is same as simple average or arithmetic mean.

Topic#2 : LOGARITHM

Introduction

• If N = a^p, where a is positive and ≠ 1, then

p = log_aN = log₁₀ N / log₁₀ a

• In words, p is referred to as the logarithm of N to the base a. Here a can not be negative, 0 or 1 because logarithm of negative numbers and 0 is not defined and the logarithm of 1 (to the base 10) is equal to 0 (⸪ a⁰ = 1).
• The reverse is also true. That is, if log_a N = p, then we can write N = a^p.
• For example,

Since 81 = 3⁴,

Therefore, log₃ 81 = 4.

• Logarithms are generally expressed to the base 10. These are common logarithms.

COMMON LOGARITHMS

• Common logarithms are expressed to the base of 10. If no base is mentioned, it is assumed that the base is 10.

[Note: You might encounter logarithm with base ‘e’. They are out of scope for discussion. They are known as natural logarithms.]

• For example, the common logarithm of 100 would be expressed as log₁₀100 and is equal to 2. (⸪ 100 = 10²).
• Consider the below table for values of some common logarithms

N	log10N
0.001	-3
0.01	-2
0.1	-1
1	0
10	1
100	2
1000	3
10000	4

LAWS OF LOGARITHMS

Most of the problems related to logarithms can be solved by the following laws and formulae:

1. The logarithm of 1 to any base is always 0. i.e. log_b1 = 0. (where b is positive and ≠ 1).
2. The logarithm of a number to a base which is equal to that number is 1. i.e. log_bb = 1. (where b is positive and ≠ 1).
3. log_ab X log_ba = 1 OR log_ab = 1/(log_ba).
4. The logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.

i.e. log_b(m x n) = log_bm + log_bn

Please note : log_b(m x n) ≠ log_bm x log_bn

5. The logarithm of the ratio of numbers is equal to the difference obtained when the logarithm of the denominator is subtracted from that of numerator.

i.e. log_b(m / n) = log_bm – log_bn

Please note : log_b(m / n) ≠ log_bm / log_bn

6. The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

i.e. log_b (mⁿ) = nlog_bm

7. Logarithms expressed in one base can be converted to logarithms expressed in any base.

i.e. log_b m = log_a m / log_a b = log_a m x log_b a

8. b^{logb n} = n
9. Given an equation, log_aM = log_bN,

(i) If M = N, then a will be equal to b.

(ii) If a = b, then M will be equal to N.

10. In order to compare two numbers, when the comparison between their respective logarithms is given:

(i) If b is greater than 1 and log_bm > log_bn, then m > n

(ii) Also, If b is less than 1 and log_bm > log_bn, then m < n

Topic#3 : SURDS AND INDICES

INTRODUCTION

• The entire domain of numbers also includes numbers like √2, √3 and 4√5. Numbers such as √2, √3, √5… are also called irrational numbers. In this topic, the focus will be on learning the laws of indices to manipulate the irrational numbers.

SURDS

• A surd is defined as an irrational number which is represented as the n^th root of a rational number.
• Hence, an irrational number which represents the n^th root of a positive rational number a is called a surd and is represented as ⁿ√a.
• This representation uses indices (discussed later). Therefore, √5, ³√6, ⁴√7 are surds that have the order 2, 3 and 4 respectively, which is read as the third root (or cube root) of 6, the fourth root of 7 and so on. If the index of the radical is not given, it is assumed to be 2.
• Conditions for a number to be a surd are:

1. 1. It is a positive irrational number.
   2. It is of the type ⁿ√a, where a is a positive rational number.

Hence, rational numbers like √4 or ³√27 are not surds, because these are not unresolved i.e. these can be resolved in rational numbers as √4 = 2 and ³√27 = 3.

PURE SURDS

• Surds with unit coefficients are known as pure surds.
• For example, √2, ³√7, ⁴√5 are pure surds.

MIXED SURDS

• Surds with coefficients other than unity are known as mixed surds.
• For example, 3√3, 10⁴√7 and 7√6 are mixed surds.

COMPOUND SURDS

• Any surd which is the sum or difference of two or more surds is called a compound surd. A compound surd can be a combination of two or more pure surds, two or more mixed surds or a combination of pure and mixed surds.
• For example, 7 + √2 + 3√6 + 5√6 is a compound surd.

SIMILAR AND DISSIMILAR SURDS

• Surds which have the same irrational part are known as similar surds. Surds with different irrational parts are known as dissimilar surds. Hence, 3√5 and 4√5 are similar surds, whereas 4√3 and 4√5 are dissimilar surds.

MULTIPLYING SURDS

• Surds which have the same index can be multiplied without changing their index. The result of multiplying or dividing surds having the same index will also have the same index as the original surd.
• For example, √3 and √5 can be multiplied and the result will also have an index of 2. However, √2 and 3√5 cannot be multiplied directly as their indices are different i.e. 2 and 3 respectively.

Example 1: Simplify the following

(i) 2√3 + √75 – √27

(ii) (3√15 x 2√5) / 5√3

Solution:

To add surds, make them similar surds,

(i) 2√3 + √75 – √27 = 2√3 + 5√3 – 3√3

                                  = √3( 2 + 5 – 3)

                                  = 4√3

                                  = √48

(ii) (3√15 x 2√5) / 5√3 = ((3 x 2) / 5) √((15 x 5)/3)

 = (6/5) x √25

= 6/5 x 5

= 6

RATIONALISATION OF SURDS

• The process of converting a surd to a rational number by multiplying it with a suitable number is called rationalisation. To rationalise, we multiply the surd with a rationalising factor. When the rationalising factor is multiplied with the surd, we get a rational number.
• For example, To rationalise √5 , we multiply it multiply it √5 (rationalising factor) to get √5 x √5 = 5, and to rationalise √27 we multiply it with √3 to get √27 x √3 = √81 = 9.

RATIONALISING FACTORS

• Consider the term √a + √b which is the sum of two surds. The rationalising factor for this is √a – √b. Multiply these terms to get (√a ₊ √b) x (√a – √b) = a – b.

USE OF RATIONALISATION

• Rationalisation is mostly used to rationalise the denominator of an expression, if the denominator features a sum of surds.

Example: Rationalise : c/(√a + √b)

Solution: c/(√a + √b) = (c(√a – √b))/(( √a + √b)( √a – √b))

= (c(√a – √b)) / (a – b)

COMPARISON OF SURDS

• It is difficult to compare two surds having different indices. The key to solve this problem is to change both surds to the same order. You can then compare them by the value of their radicands i.e. the value under the root sign. The new index of the two surds is the LCM of the original indices of the surds.

Example : Compare √2 and ³√3

Solution:

The order of the surds is 2 and 3, hence convert both the surds to surds of an order which is the LCM of the original orders i.e 6 (LCM of 2 and 3 is 6).

√2 = 2^1/2 = (2³)^1/6 = 8^1/6

And,³√3 = 3^1/3 = (3²)^1/6 = 9^1/6

Now, it can be clearly seen that 8^1/6 < 9^1/6

Hence, √2 < ³√3.

RULES OF INDICES

If a and b are non-zero rational numbers and m and n are rational numbers, then

1. a⁰ = 1
2. a^-m = 1/a^m
3. ^m√a = a^1/m
4. a^m/n = ⁿ√a^m
5. a^m x aⁿ = a^{(m + n)}
6. a^m / aⁿ = a^{(m – n)}
7. (a^m)ⁿ = a^mn
8. (ab)^m = a^mb^m
9. If a^m = aⁿ, then m = n ; only if a ≠ 1, -1
10. If a^m = b^m and m ≠ 0, then a = b if m is odd and a = ± b, if m is even.

Topic#4 : PROFIT, LOSS AND DISCOUNT

PROFIT AND LOSS

• The price at which a person buys(or produces) a product is the Cost Price (CP) of the product with respect to that person and the price at which a person sells a product is called the sales price or the Selling Price (SP) of the product with respect to that person.
• When a person is able to sell the product at a price higher than its cost price for him, then he can be said to have earned a Profit(P).

Profit = Selling Price – Cost Price

P = SP – CP

• Similarly, if a person sells an item for a price lower than its cost price for him, then a Loss(L) has been incurred.

Loss = Cost Price – Selling Price

L = CP – SP

• If a person sells a product at the same price at which he bought it i.e. at the cost price, the transaction is said to have been conducted on a “no profit no loss” basis. Thus, there is neither profit nor loss in such a transaction.

Percentage Profit    = (Actual Profit / Investment) X 100

                          = ((Selling Price – Cost Price)/Cost Price) X100

Percentage Loss      = (Actual Loss / Investment) X 100

                = ((Cost Price – Selling Price)/Cost Price) X 100

• Please note : Profit percentage and loss percentage are determined on the basis of Cost Price
• Percentage Profit can only be calculated when the number of goods sold and the number of goods bought is equal.
• When the amount of money spent and earned are equated, then

Percentage Profit = (Remaining Goods / Sold Goods) X 100

MARKED PRICE AND DISCOUNT

• The difference between the Selling Price of a good and its Cost Price is known as markup. The price that is printed on an article or written on the label attached to it is the sum of the Cost Price and the markup, and is called the Marked Price (MP) or List Price of the item.

i.e. Marked Price = Cost Price + Markup

• Markup can also be expressed as percentage of cost price,

Marked Price = Cost Price + ( (Markup(as percentage) X CP) / 100 )

Markup (as percentage) = ( (Marked Price – Cost Price) / Cost Price ) X 100

• Generally, MP = SP. However, sometimes discount is given on the marked price. So, selling price becomes less than the marked price.
• In this scenario, Selling Price = Marked Price – Discount
• When discount is expressed as percentage, SP = MP – ( ( Discount(as percentage) / 100 ) X MP )Discount (as percentage) = ( (Marked Price – Selling Price) / Marked Price ) X 100Percentage Discount = ( Discount / Marked Price ) X 100

  SP/MP = 1 – (Discount Percentage / 100)

•  To solve problems with successive discounts, use the below formula,
• When a discount of a% is followed by another discount of b%, then the total discount is given by ( a + b – (ab)/100) %
• In general, if there are successive discounts of p%, q% and r% in 3 stages, then,
• Total discount = [ 1 – {((100 – p) / 100) x ((100 – q) / 100) x ((100 – r) / 100)}] x 100
        

Topic#5 : SIMPLE and COMPOUND INTEREST

CONCEPT

• Money borrowed today is repaid with a higher amount tomorrow. This difference leads to the concept of interest.
• 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 Time or Period (T or n).
• The difference between the Principal and the amount of money which the borrower needs to repay at the end of the time period is called the Interest(I) over the Principal Amount. The final amount being paid at the end is known as Amount.
• Amount = Principal + Interest
• The interest is calculated based on the rate of the interest(R).
• There are two ways of calculating the rate of interest: Simple and Compound.

SIMPLE INTEREST

• The interest calculated only on the original principal, for the given time duration, is called Simple Interest.
• Simple Interest = ( P X R X T ) / 100
• Amount = Principal + Interest

COMPOUND INTEREST

• Compound interest is interest accrued on principal as well as the previous year’s interest. When 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. The difference between the final amount and the original principal is the Compound Interest (CI).
• Amount = P x (1 + R/100)ⁿ
• CI = Amount – Principal

COMPOUNDING MORE THAN ONCE A YEAR

• The frequency of compounding can vary. It can be done half-yearly, quarterly or monthly etc. When compounding is done more than once a year, the rate of interest for that period will be less than the effective rate of interest for the entire year.
• For half yearly rate,  A = P X (  1 + (r/2)/100)²ⁿ
• For quarterly rate, A = P X (  1 + (r/4)/100)⁴ⁿ
• For monthly rate, A = P X (  1 + (r/12)/100)¹²ⁿ

POPULATION FORMULA

• If the original population of a town is P and the annual increase is r%, then the population is n years(P1) is :

P1 = P X ( 1 + r/100 )ⁿ

• If the annual decrease is r%, then the population in n years is given by a change of sign in the formula:

P1 = P X ( 1 – r/100 )ⁿ

DEPRECATION OF VALUE

• The value of any asset decreases with time due to any of a number of factors including wear and tear, outdated technology, etc. This decrease is called its deprecation.
• If P is the original value and r is the rate of deprecation per year, then the final value (F) after n number of year, then the final value (F) after n number of years is given by the formula,

F = P X ( 1 – r/100 )ⁿ

Topic#6 : PERCENTAGES

CONCEPT

• A percentage is used to convert a part of or fraction of the whole. It is a way to describe a number as a fraction with the denominator 100. “Percent” implies “for every hundred” and is denoted by the symbol “%”.
• To write a fraction or decimal as a percentage, convert it to an equivalent fraction with a denominator of 100.
• For example: ½ = 50%, 1/3 = 33.33%, 1/5 = 20% , 40% = 40/100, 50% = 50/100 = ½ and so on.

PERCENTAGE INCREASE AND DECREASE

• Percentages are often used to indicate changes in a quantity. A percentage is a good measure to compare the change in two different quantities depending on the initial (or base) value of the quantity. For instance, if two persons A and B have salary Rs. 100 and Rs 20 respectively. Both get salary hike of Rs . 20, the increment in terms of percentage will be higher for B, even though the absolute hike is same in both the cases.
• Percentage Change = (Final Quantity – Initial Quantity) / Initial Quantity , where final quantity can be less than, greater than or equal to initial quantity.
• When the final value > initial value, the percentage change is positive. Such a percentage change is also called percentage increase.
• When a final value = initial value, the percentage change is zero.
• When the final value < initial value, the percentage change is negative. This kind of percentage change is also called percentage decrease.

PERCENTAGE POINT

• A percentage point is defined as the difference between the two percentage values.
• For example, the GDP growth of a country grows from 11% in 2019 to 12.5% in 2020. So, the percentage point change is 12.5 – 11 = 1.5 and the percentage change is ( 1.5/11 ) x 100 = 13.63%
• 1% is also considered as 100 basis points.

IMPACT OF CHANGES IN THE VALUE OF NUMERATOR AND DENOMINATOR ON THE OVERALL VALUE OF THE RATIO

1. CHANGE IN THE NUMERATOR

• The numerator is directly proportional to the value of the ratio. In fact, the percentage change in the numerator is equal to the percentage change in the value of the ratio.

2. CHANGES IN THE DENOMINATOR

• The denominator is inversely proportional to the value of the ratio; i.e. if the value of the denominator increases, then value of the ratio decreases and vice-versa.

Two important concepts for solving numerical problems:

(a) If the price of a commodity increases by a%, then the percentage reduction in the consumption, so that the expenditure remains the same is:

                        ( a / (a + 100 ) ) x 100

(b) If the price of the commodity decreases by b%, then the percentage increase in consumption, so that the expenditure remains the same is:

   ( b / ( b – 100 ) )  x 100

3. CHANGES IN BOTH NUMERATOR AND DENOMINATOR

• If the numerator increases and the denominator decreases, it is clear that the ratio will increase. If the numerator decreases and the denominator increases, then it is apparent that the ratio will decrease. On the other hand, if both the numerator and denominator simultaneously increase/decrease, then it is not quite so apparent how the ratio will change.

SUCCESSIVE PERCENT CHANGES

• 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.
• In general, if there are successive increases of p%, q% and r% in 3 stages, then:
• Total percentage increase

= [{((100 + p) / 100 ) X ( (100 + q)/ 100) X ( (100 + r)/100)} – 1] X 100

 

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