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MTH302 Business Mathematics & Statistics Short Notes - MTH302 Short Questions Answers
BUSINESS MATH AND STATISTICS
MTH302 Short notes by Malik Zaka
TYPES OF EMPLOYEES
GROSS EARNINGS/SALARY
Gross earning includes the following?
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
iii. conveyance
vii. leaves
viii. group insurance
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:
^{ }
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
TRADE DISCOUNT-EXAMPLE 2 Find the single discount rate that is equivalent to the series 15%, 10% and 5%. TradeDiscount Net price = (1-d_{1})(1-d_{2})(1-d_{3}) = 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.
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 %MU on Sale
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.
INTERNAL 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(cost,salvage,life,period,factor)
Returns the depreciation of an asset for a specified period using the double declining balance method or some other method you specify
MIRR(values,finance_rate,reinvest_rate)
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(cost,salvage,life,per)
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(values,dates,guess)
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:
variables
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 Sales or revenues and ‘contributes’ to meeting Fixed Costs and making a ‘Net Profit’Contribution Margin = Net Sales – Variable Cost = S – VC
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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