Partnership

What is Partnership?

When a business is run jointly by two or more than two persons then they are called partners and the deal partnership.

Ratio of division of gains/loss

When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.

Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year:

(A's share of profit) : (B's share of profit) = x : y

When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now gain or loss is divided in the ratio of these capitals.

Suppose A invests Rs. x for p months and B invests Rs. y for q months then,

(A's share of profit) : (B's share of profit)= xp : yq.

Working Partner and Sleeping Partner

A partner who manages the business is known as the Working Partner and the person who simply invests the money is known as the Sleeping Partner.

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Ratio and Proportion

Ratio

The ratio of two quantities in the same unit is the fraction a/b and is written as a:b

In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.

Important : If you multiply or divide each term of a ratio with the same non zero number then the ratio does not change.

Proportion

The equality of two ratios is called proportion.

If a:b = c:d then a:b::c:d and we say that a, b, c, d are in proportion.

In this case a and d are called the extremes and b and c are called mean terms

Important : Product of Extremes = Product of Means i.e. (a x d) = (b x c)

Fourth Proportional

In a:b::c:d, d is called fourth proportional to a, b, c.

Third Proportional

In a:b::c:d, c is called third proportional to a and b.

Mean Proportional

Mean Proportional between a and b is √ab

Compounded Ratios

The compounded ratio of the ratios (a:b), (c:d), (e:f) is (ace:bdf)

Duplicate Ratios

Duplicate ratio of (a : b) is (a² : b²).

Sub-duplicate ratio of (a : b) is (√a : √b).

Triplicate ratio of (a : b) is (a³ : b³).

Sub-triplicate ratio of (a : b) is (a^1/3 : b^1/3).

If a/b = c/d then (a + b)/(a - b) = (c + d)/(c - d) [By componendo and dividendo]

Variations

We say that x is directly proportional to y, if x = ky for some constant k and then x y

We say that x is inversely proportional to y, if xy = k for some constant k and then x 1/y

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Profit and Loss

Cost Price

The price at which an article is purchased is known as the Cost Price (C.P.) of the article.

Sell Price

The price at which an article is sold is known as the Sell Price (S.P.) of the article.

Profit

If S.P. > C.P. then the seller makes a profit.

Loss

If C.P. < S.P then the seller incurs a loss.

Important Formula

1. Gain = S.P. - C.P.

2. Loss = C.P. - S.P.

3. Gain % = (Gain x 100/C.P.)

4. Loss % = (Loss x 100/C.P.)

5. Gain or Loss is always reckoned on C.P.

6. Selling Price (S.P.) = [ (100 + Gain %)/100 x C.P.]

7. Selling Price (S.P.) = [ (100 - Loss %)/100 x C.P.]

8. Cost Price (C.P.) = [ (100 + Gain %)/100 x S.P.]

9. Cost Price (C.P.) = [ (100 - Loss %)/100 x S.P.]

10. If an article is sold at a gain of say 25%, then S.P. = 125% of C.P.

11. If an article is sold at a loss of say, 25% then S.P. = 75% of C.P.

12. When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by:

Loss % = (Common Loss and Gain %/10)² = (x/10)²

13. If a trader professes to sell his goods at cost price, but uses false weights, thenGain % = [{ Error/True Value - Error} x 100]

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Percentage

Concept of Percentage

By a certain percent, we mean that many hundredths.
Thus, x percent means x hundredths denoted by x %.

Expressing a percentage as a fraction or decimal number:

i) x % = x/100

ii) 50 % = 50/100 = 1/2 = 0.5

Expressing a fraction or decimal number as percentage:

i) x/y = {(x/y) x 100} %

ii) 1/4 = {(1/4) x 100} % = 25 %

Percentage Increase/Decrease

If the price of a commodity increases R % then the reduction in consumption so as not to increase the expenditure is:

[{R/(100+R)} x 100] %

If the price of a commodity decreases R % then the increase in consumption so as not to decrease the expenditure is:

[{R/(100-R)} x 100] %

Results on Population

Let the population of a town be P now and suppose it increases at the rate of R % per annum, then

1. Population after n years = P(1 + R/100)ⁿ

2. Population n years ago = P/{(1 + R/100)ⁿ}

Results on Depreciation

Let the present value of a machine P. It depreciates at a rate of R% per annum. Then

1. Value of machine after n years = P(1 - R/100)ⁿ

2. Value of the machine n years ago = P/{(1 - R/100)ⁿ}

3. If A is R% more than B, then B is less than A by [{R/(100+R)} x 100] %

4. If A is R% less than B, then B is more than A by [{R/(100-R)} x 100] %

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Average

Average

Average is the measure of the middle or typical value of a data set. In other words it is a measure of central tendency.

Average = Sum of Observations/No. of Observations

Average Speed

In order to understand this let us consider an example.

Suppose a car covers a certain distance at x km/h and an equal distance at y km/h.Then, the average speed of the car is (xy/x+y) km/h.

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Square Root and Cube Root

Square Root

If x² = y then the square root of y is x which is written as √y = x.
For Example, √4 = 2, √16 = 4, √144 = 12

Cube Root

The cube root of a given number x is the number whose cube is x.
The cube root of x is denoted by x^1/3.

Note: √xy = √x x √y

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Simplification

BODMAS Rule

This rule depicts the correct sequence in which the operations are to be performed, so as to find out the value of the given expression.

B - Bracket
O - Of
D - Division
M - Multiplication
A - Addition
S - Subtraction

Thus while simplifying an expression the operations must be executed according to this BODMAS rule.

Modulus of a Real Number

Modulus of a real number a is defined as

If a > 0 then |a| = a. If a < 0 then |a| = -a.

Hence, |7| = 7 and |-7| = -(-7) = 7.

Bar (or Virnaculum)

If an expression contains a bar over it then before applying the BODMAS Rule we simpify the expression under the bar.

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Decimal Fraction

What are Decimal Fractions?

Fractions in which denominators are powers of 10 are known as decimal fractions.

Examples : 1/10 = 1 tenth = 0.1; 9/1000 = 9 thousandths = 0.009

Converting a decimal into vulgar fraction

1 is put in the denominator just below the decimal point. Now as many zeros are annexed to the denominator as is the number of digits after the decimal point. Then the decimal point is removed and the fraction is reduced to its lowest terms.

Example : 0.25 = 25/100 = 1/4

Some Important Results

Annexing zeros to the extreme right of a decimal fraction does not change its value.

If numerator and denominator of a fraction has same number of decimal places, then the decimal point is removed.

Operations on Decimal Fractions

a) Addition and Subtraction

The numbers are placed under each other so that the decimal points lie in one column. Now the numbers are added or subtracted in the usual way.

b) Multiplication

The given numbers are multiplied without taking into account the decimal point. In the product as many decimal places are marked off as is the sum of the number of decimal places in the given numbers

Example : 2.5 x 0.5

First we multiply the numbers without decimal point. 25 x 5 = 125.

Now, sum of decimal places = (1 + 1) = 2. Then product of 2.5 and 0.5 is 1.25

c) Division by a natural number

The given number is divided by the natural number without considering the decimal point. Now in the quotient the decimal point is put to give as many decimal places as there are in the dividend.

d) Division by a decimal fraction

Both the dividend and the divisor is multiplied by a power of 10 so that the divisor becomes a whole number. Now the dividend is divided by the divisor.

Recurring Decimal

If in a decimal fraction, a figure or a set of figures is repeated continuously then it is called non-terminating or recurring decimal.

In a recurring decimal, if one figure is only repeated then it is expressed by putting a dot or bar over it. If a set of numbers is repeated then it is expressed by putting a dot on the first and last digit of the set or by putting a bar over it.

Pure Recurring Decimal

In a decimal fraction, if all the figures after the decimal point is repeated then it is called a pure recurring decimal.

Mixed Recurring Decimal

A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.

 

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HCF and LCM

Factors and Multiples

If number x divides another number y then x is a factor of number y.
In this case, y is a multiple of x.

Highest Common Factor (HCF)

The HCF of two or more than two numbers is the greatest number that divided each of them exactly.

Methods of finding the HCF of two numbers

a) Factorization Method - Each one of the given numbers are expressed as the product of prime factors. The product of least powers of common prime factors gives the HCF of the numbers.

b) Division Method - Suppose we have to find the HCF of two given numbers. We divide the larger number by the smaller one. Then the divisor is divided by the remainder. This process of dividing the preceding divisor by the remainder is repeated until zero is obtained as the remainder. The last divisor is the required HCF

Method of finding the HCF of more than two numbers

Suppose we have to find the H.C.F. of three numbers, then, HCF of three numbers is the [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers.

Least Common Multiple (LCM)

The least number that is exactly divisible by each one of the given numbers is called the LCM.

Methods for finding the LCM of a given set of numbers

a) Factorization Method - Each one of the given numbers is resolved into product of prime factors. The product of the highest powers of all factors is the required LCM.

b) Division Method - The given numbers are arranged in a row in any order. These numbers are divided by a number which divides at least two numbers exactly. The numbers which are not divisible are carried forward. This process is repeated until all the numbers are divided. The product of the divisors is the required LCM.

Product of two numbers = Product of their HCF and LCM

HCF and LCM of Fractions

a) HCF = HCF of numerator/LCM of denominator

b) LCM = LCM of numerator/HCF of denominator

HCF and LCM of Decimal Fractions

In the given set of numbers, zeros are annexed to some numbers so that the number of decimal places becomes the same. Considering these numbers without decimal point, the HCF or LCM is found out. Now in the result as many decimal places are marked off as there were in the given numbers.

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