**MTH302 Business Mathematics & Statistics Short Notes - MTH302 Short Questions Answers**

**BUSINESS MATH AND STATISTICS**

**MTH302 Short notes by Malik Zaka**

** **

TYPES OF EMPLOYEES

- regular
- part time
- incentive base

GROSS EARNINGS/SALARY

Gross earning includes the following?

- basic salary
- allowances
- house rent
- conveyance allowance

iii. utilities allowances

TAXATION RULES ON ALLOWNCES

If allowances are 50% of basic salary, the amount is treated as tax free. Any allowance that exceed this allowance are considered taxable, both for the employee as well as the company.

PROVIDENT FUND

A company can establish a provident fund for the benefit of the employees. By law, 1/11^{th} of basic salary per month is deducted by the company from the gross earning of the employees. An equal amount i.e. 1/11^{th} of basic salary per month is contributed by the company to the provident fund to the account of the employee. Total becomes 2/11^{th} of the basic salary.

Example:

Basic = 10000

Allow = 5000

Provident fund= ?

Employee contribution to provident fund = 1/11 x 10000 = 909.1

Company contribution to provident fund = 1/11 x 10000=909.1

Total provident fund = 909.1+909.1 = 1818.2

GRATUITY FUND

A company can establish a gratuity trust fund for the benefit of the employees. There is a saving of 1/11^{th} of basic salary on behalf of the employee in gratuity fund.

LEAVES

CL = 18 days per year

EL = 18

SL = 12

Total cost of leaves as percent of gross salary = **18.2%**

SOCIAL CHARGES

Medical / group insurance = 5% of gross salary

Education, club member ship, Misc. = 5.8% of gross salary

Leaves = 18.2% of gross salary

Total social charges = ** 29% **of gross salary

GROS REMUNERATION

It is pay or salary typically monetary payment for services rendered, as in an employment like

- basic salary
- house rent allowances

iii. conveyance

- utilities
- provident fund
- gratuity fund

vii. leaves

viii. group insurance

- miscellaneous charges

PERCENTAGE

Percentage is formed by Xing a number called the base by a percent called the rate.

% = base x rate

AVERAGE = sum / n

WEIGHTED AVERAGE

It is one type of ** arithmetic mean** of a asset of data in which some elements of the sets carry more importance (weight) than others.

Example:

Unit hours

A 6 300

B 3 200

C 1 100

First convert weight in fractions

6+3+1= 10

6/10 = .6

3/10 = .3

1/10 = .1

weighted average = sum of fractions x hours

= (.6x300) + (.3x200)+ (.1x100) = 250

PERCENTAGE CHANGE

Change = final value – initial value

Percentage change = change / initial value x 100%

STOCK

It is share in the ownership of a company

STOCK YIELD/

It can refer to the rate of income generated from a stock in the form of regular dividends.

EARNING PER SHARE (EPS)

EPS = total profits / number of shares

PRICE EARNING RATIO: = market value of shares / EPS

NET CURRENT ASSET VALUE PER SHARE

= current asset – total liabilities / number of outstanding share

CURRENT ASSETS

The value of all assets that are reasonably expected to be converted into cash with in one year

LIABILITIES

A company’s legal debts or obligations that arise during the course of business operations

MARKET VALUE

The price at which investors buy or sell a share of stock at a given time

FACE VALUE

Original cost of a share of stock which is shown on the certificate

DIVIDENT

A company distributes a part of the profit it terms as dividend

DISCOUNT

It is rebate or reduction in price

NET COST PRICE = list price – discount

SIMPLE INTEREST I = PTR/100

P = Principal

R= rate

T = time in years

I = interest

COMPOUND INTEREST=S-P , where

Money Accrued after n years =S= P(1+R/100)^N

P = Principal

R = rate

N = no of years

S = compound interest

ANNUITY

Annuity is sequence of payment/installment

Annuity = C x [(1+i)^{n} – 1 / i]

C= payment per period / amount of annuity

i = interest rate

n = number of payments

ACCUMULATED VALUE

The accumulated volves of an annuity is the total payments mode including the interest.

R = amount of annuity

N = number of payments

I = interest rates

S = accumulated vlue

A = discounted / present worth of an annuity

S = r [(1+i)^{n} – 1 / i]

Accumulation factor for n payments

[(1+i)^{n} – 1 / i]

accumulated value = payment per period x accumulation factor for n payment

DISCOUNTED FACTOR RATE

When future value is converted into present worth, the rate at which the calculations are made.

Example.

Rate of interest = 4.25% = 0.0425

No of periods = 18

Amount of annuity = 1000 Rs.

Accumulation factor = ?

Accumulated value = ?

Discounted value = ?

AF = (1+0.0425) -1 / 0.0425 = 26.24

S = 10000 X 26.24 = 260,240 Rs.

DV = first of all we find discount factor

DF = (1-1/(1+i)^{n} / i)

= (1-1/(1+i)^{n} / i)

^{= }(1-1/(1+0.0425)^{18} / 0.0425) = 12.4059

DV= 10000x12.4059 = 124059 Rs.

MATRIX

A matrix is a rectangular array of numbers. The plural of matrix is matrices like

A = ( -1 9

-3 4)

DIMENSION

Dimension order of a matrix = rows x columns

RATIO

A ratio is a comparison between things. If in a room there are 30 men and 15 women then the ratio of men to women is 2 to 1. this is written as 2:1 and read is “two is to one”. “:” is the notation for a ratio.

PROPORTION

A proportion is an equation with the ratio on each side. It is a statement that two ratios are equal. 3:4 = 6:8 or ¾ = 6/8 is an example of proportion

**MIDDLEMAN**

A middle man is a person who buys a product directly from the manufacturer, and then either sells the product at retail prices to the public, or sells the product at wholesale prices to a distributor.

**Trade Discount**

Amount of discount = d × L

Where, d = Percentage of Discount

L = List Price

Net Price = L – Ld = L(1 – d)

Net Price = List Price – Amount of Discount

^{ }

** MARKUP**:-

Markup is an amount added to a cost price while calculating a selling price.

** **

** **

** Markup as Percentage of Cost (MUC**:-

Here markup is some percentage of cost price. For simplicity, it is also named as %Markup on cost. The relation between %markup on cost, cost price and selling price is:

Selling Price = Cost price + (Cost price × %Markup on cost)

= Cost price (1 + %Markup on cost)

Markup as Percentage of Sale price (MUS):

Here markup is some percentage of selling price. For simplicity, it is also named as %Markup on sale. The relation between %markup on sale, cost price and selling price is:

Selling Price = Cost price + (Selling price × %Markup on sale)

Cost price = Selling price – (Selling price × %Markup on sale)

= Selling price (1 – %Markup on sale)

Rs Markup:

Markup in terms of rupees is called Rs markup. The relations between Rs markup, cost price and selling price are:

- Selling Price = Cost price + Rs Markup
- Rs Markup = %Markup on cost × Cost price
- Rs Markup = %Markup on sale × Selling price

^{ }

For example:

The cost price of certain item is 80Rs and its selling price is 100Rs. Then

Rs Markup = Selling price – Cost price

= 100 – 80

= 20 Rs

** MARKDOWN**:-

Markdown is a reduction from the list/cost price.

** DISCOUNT**:-

Discount is a reduction in price which the seller offers to the buyer.

^{ }

** SERIES TRADE DISCOUNT**:-

This refers to the giving of further discounts as incentives for more sales. Usually such discount is offered for selling product in bulk.

L = List price = 100

D = discounts

Net price = L(1-D1)(1-D2)(1-D3)

Single equivalent discount rate = L – Netprice = ?%

Rs. Discount = (0.2787)(20000)

= 5,574 Rs

Find the single discount rate that is equivalent to the series 15%, 10% and 5%. TradeDiscount Net price = (1-d = 100(1 -15%) (1 - 10%) (1 - 5%) =100(0.85) (0.9) (0.95) = 100(0.7268) = 72.68 % Discount = 100 - 72.68 = 27.62% |

^{ }

** CASH DISCOUNT**:-

Cash Discount is allowed on Invoices, Returned Goods, Freight, Sales Tax and A common business phrase for a cash discount is "3/10, net/30," meaning that a 3% discount is offered if the amount due is paid within 10 days; otherwise 100% of the amount due is payable in 30 days

**CASH DISCOUNT-EXAMPLE**

Invoice was dated May 1^{st}. The terms 2/10 mean that 2% discount is offered if invoice is paid up to 10^{th}May.

What is the net payment for invoice value of Rs. 50,000 if paid up to 10^{th} May?

Cash Discount

N = L(1 – d)

= 50,000(1-0.02)

= 50,000(0.98)

= 49,000 Rs.

**DISCOUNT PERIODS**

Discount Periods are periods for the buyer to take advantage of Discount Terms.

**CREDIT PERIODS**

Credit Periods are periods for the buyers to pay invoices within specified times.

**PARTIAL PAYMENTS**

When you buy on credit and have cash discount terms, part of the invoice may be paid within the specified time. These part payments are called ** Partial Payments**.

You owe Rs. 40,000.

Your terms were 3/10 (3% discount by 10^{th} day).

Within 10 days you sent in a payment of Rs. 10,000.

Rs. 10,000 was a part payment.

How much is your new balance?

First we will find the amount that if 3% discount is given on it, the net amount is 10000Rs.

Let that amount is t. Then

10000 = t (1 – 0.03)

This implies, t = 10000

(1 – 0.03)

Thus, t = 10309Rs

This means that although you pay 10,000Rs, due to 3% cash discount 10309Rs among 40,000Rs is paid.

Hence the new balance = 40000 – 10309 = 29691Rs.

**MARKETING TERMS**

There are a number of marketing terms.

First of these is the ** Manufacturer Cost**. This is the cost of manufacturing.

Next is the price charged to middlemen in “The Distribution Chain”.

The Distributor>Wholesaler>Retailer is a chain.

The next term is the ** Selling Price**. This is the price charged to Consumers

by Retailers. It may or may not be the same as list price.

**Operating Expenses**

Expenses the company incurs in operating the business, e.g. rent, wages and utilities is called operating Expenses

** Selling Price**:-

Selling Price is composed of Cost and Rs Markup.

Selling Price (S) = Cost (C) + Rs Markup (M)

** MARGIN**:-

While determining Sale Price, a company includes the operating expenses and profit to their own cost. This amount is called the margin of the company. It is usually calculated as percentage but can also be expressed as rupees. It is also named as **markup on sale.**

Margin or markup on sale = Selling price - Cost price ×100%* *

Selling Price

Selling price = Cost price + Rs Margin

Margin and markup confuse many. By margin, company evaluates that for every rupee generated in sales, how much is left over to cover basic operating costs and profit. Markup represents the amount added to a cost to arrive at a selling price

Markup on cost = Selling price – Cost price ×100%* *

Cost price

**Note: Remember unless it is mentioned that markup is on sale, simple markup means markup on cost.**

**MARKUP AND PERCENT ON COST**

Tanveer’s flower business sells floral arrangements for Rs. 35.

To make his desired profit, Tanveer needs a 40% Markup on cost.

What do the flower arrangements cost Tanveer?

What is the Rs. Markup?

Rs. Markup and Percent Markup on Cost

Sale price S = Cost C + {C ×Markup on cost (MUC)}

S = C + 0.40(C)

35 = 1 .40(C)

C = 35/1,4 = 25 Rs.

Rs Markup = 25 x 0.4

= 10 Rs.

^{ }

**Selling Price = Cost price + (Selling price × %Markup on sale)**

**CONVERTING MARKUPS**

Convert 50% Markup ** (MU) on Cost **to %

Formula for converting %Markup on Sale (mus) to %Markup on Cost Price (muc) is:

% Markup on Selling Price (mus) = %Markup on Cost / (1 + %Markup on Cost)

mus = muc/(1+muc)

**Solution**

% Markup on Sale (mus) = 0.5 / (1+0.5) = 0.5/1.5

mus = 0.3333 = 33.33%

**Converting Markups**

Converting 33.33% **MU on ****Sale**

to %**MU on C**

Convert % Markup on Cost (muc) to % Markup on selling price (mus):

% Markup on cost = % Markup on S / (1 - % Markup on S)

muc = mus / (1-mus)

**Solution**

Markup on cost = 0.3333/(1 – 0.333)

= 0.3333/0.6666 = 0.5

= 50%

**MARKDOWN**

Reduction from original selling Price is called Markdown.

**Formula**

%Markdown = (Rs. Markdown / Selling Price (original))** ×**100%

** MARKDOWN-EXAMPLE** 1

Store A marked down a Rs. 500 shirt to Rs. 360.

What is the Rs. Markdown?

What is the %markdown?

**Rs. Markdown**

Let S = Sale price

Rs. Markdown = Old S – New S

= Rs. 500 – Rs. 360

= Rs. 140 Markdown

**% Markdown**

% Markdown = Markdown **×**100%

Old S

% Markdown = 140**×**100%

500

= 0.28**×**100%

= 28 %

**PROJECT FINANCIAL ANALYSIS**

Financial analysis is the analysis of the accounts and the economic prospects of a firm, which can be used to monitor and evaluate the firm's financial position, to plan future financing, and to designate the size of the firm and its rate of growth.

**COST ESTIMATES**

cost estimates cover calculations based on quantities and unit rates.

**REVENUE ESTIMATES**

Along with costs even revenues are calculated. These calculations are similar to component costs.

**FORECASTS OF COSTS**

Forecasting requires a technique for projections. One of such technique, Time Series Analysis, will be covered later in this course.

** **

** **

**FORECASTS OF REVENUES**

These will be done similar to the forecast of costs. Here also the method must be determined first. Once the methodology is clear, the worksheets can be prepared easily.

**NET CASH FLOWS**

The difference between Revenue and Cost is called the Net Cash flow. This is an important calculation as the entire Project Operation and Performance is based on its cash flows.

**BENEFIT COST ANALYSIS**

This is the end result of the Project Analysis. The ratio between Present Worth of Benefits and Costs is called the Benefit Cost (BC) ratio.

I**NTERNAL RATE OF RETURN**

Internal Rate of Return or IRR is that Discount Rate at which the Present Worth of Costs is equal to the Present Worth of Benefits. IRR is the most important parameter in Financial and Economic Analysis.

**BREAK-EVEN ANALYSIS**

In every project where investment is made it is important to know how long it takes to recover the investment. It is also important to find the breakeven point where the Cash Inflow becomes equal to Cash Outflow. After that point the company has a positive cash flow (i.e. there is surplus cash after meeting expenses).

** BEP in units = Fixed Costs_____**

** Contribution Margin per unit**

BEP in Rs** **calculates the revenue that must be obtained to reach break even point.

** BEP in Rs = Fixed Costs × Net Sales**

** Contribution Margin**

* ***BEP in Rs = Fixed costs × Selling Price per unit**

** Contribution Margin per unit**

**BEP as % of capacity = **** ****BEP in units_____**** × ****100 %**

Production capacity

**Excel Functions for Financial Analysis**

**AMORDEGRC(cost,date_purchased,first_period,salvage,period,rate,basis)**

If an asset is purchased in the middle of the accounting period, the prorated depreciation is taken into account.

**AMORLINC(cost,date_purchased,first_period,salvage,period,rate,basis)**

** **

Returns the depreciation for each accounting period. If an asset is purchased in the middle of the accounting period, the prorated depreciation is taken into account.

** **

**CUMIPMT**

Returns the cumulative interest paid between two periods.

**CUMPRINC**

Returns the cumulative principal paid on a loan between two periods

**DB****(cost,salvage,life,period,month)**

Returns the depreciation of an asset for a specified period using the fixed-declining balance method.

** DDB**(

Returns the depreciation of an asset for a specified period using the double declining balance method or some other method you specify

** MIRR**(

Returns the modified internal rate of return for a series of periodic cash flows. MIRR considers both the cost of the investment and the interest received on reinvestment of cash.

**IRR(values,guess)**

** **Returns the internal rate of return for a series of cash flows

**PV(rate,nper,pmt,fv,type)**

** **Returns the present value of an investment

**NPV(rate,value1,value2, ...)**

Returns the net present value of an investment based on a series of periodic cash flows and a discount rate

**XNPV(rate,values,dates)**

Returns the net present value for a schedule of cash flows that is not necessarily periodic.

**SLN(cost,salvage,life)**

** **Returns the straight-line depreciation of an asset for one period

** SYD**(

Returns the sum-of-years' digits depreciation of an asset for a specified period

** SYD = (cost-salvage) x (life – per + 1) x 2**

** (life)(life + 1)**

**VDB(cost,salvage,life,start_period,end_period,factor,no_switch)**

Returns the depreciation of an asset for any period you specify, including partial periods, using the double-declining balance method or some other method you specify. VDB stands for variable declining balance.

** XIRR**(

Returns the internal rate of return for a schedule of cash flows that is not necessarily periodic.

**LINEAR EQUATIONS**

Linear equations have following applications in Merchandising Mathematics:

- Solve two linear equations with two variables
- Solve problems that require setting up linear equations with two

variables

- Perform linear Cost-Volume-Profit and break-even analysis employing:
- The contribution margin approach
- The algebraic approach of solving the cost and revenue functions

** **

**Production Capacity (PC)**

It is the number of units a firm can make in a given period.

**Contribution Margin**

Contribution Margin is the Rs. amount that is found by deducting** Variable Costs **from

Contribution margin per unit =CM =Sale price per unit – Variable cost per unit

** Contribution Rate (CR)**

**Contribution rate = Contribution Margin × 100% = CM × 100%**

** Net sales S**

** Contribution rate = Contribution Margin per unit × 100%= CM × 100%**

** Sale price per unit S**

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