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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