Introduction to Business Mathematics (1349)
Q. 1 (a) In an election, there were two candidates one of them received 65% of the votes cast
and secured a majority of 1500 votes more than his competitor. How many people casted the votes? (8+7+5 = 20)Total votes secured by the winning candidate = 1500
Percent of votes secured by the winning candidate = 65%
How m any people casted the votes = (1500/65) *100
= 2308
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(b) Distinguish rate
from ratio. There men invested Rs.180000, Rs.120000 and Rs.60000 respectively.
How should they share out of profit of Rs.36000?
A ratio is a comparison of two numbers. A ratio can be written
using a colon, 3:5 , or as a fraction 35 . A rate , by contrast, is a
comparison of two quantities which can have different units. For example 5
miles per 3 hours is a rate, as is 34 dollars per square foot.
First : Second : Third
180000:120000:60000
18:12:6
3:2:1
Total shares = 3+2+1
= 6
Profit = 36000
Share of first man = (36000/6)*3
= 18000
Share of second man= 12000
Share of third man = 6000
c) The saving income
ratios of two persons are 1/9 and 2/3. Who is saving more?
Percent of saving of first person = 1/9
= 11.11%
Percent of saving of second person = 2/3
= 66.67%
Second person is saving more.
Q. 2 (a) A man need to borrow Rs.800,000 for two
years. Which of the following loan is more advantageous to him: (8+7+5 = 20)
i) 4.1% simple
interest; or
= (800000/100)*4.1%*2
= 65600
ii) 4% per annum
compounded semi-annually.
= 65945
First loan is more advantageous
(b) An annuity of Rs.
5000 payable at the end of each year amounts to Rs. 27082 in five years. Find
the rate of interest of the annuity.
|
Present value |
0 |
|
Future value |
27082 |
|
Annual payment |
5000 |
|
Years |
5 |
|
Interest rate |
4.00% |
(c) Jeremy said that if
the means and extremes of a proportion are interchanged, the resulting ratios
form a proportion. Do you agree with Jeremy? Explain why or why not?
Alright, let's talk about extremes and means of a geometric
proportion. So let's just set up a geometric proportion. A over B is equal to C
over D. So the means of this proportion are being and see those are the means
and the extremes of this proportion are A. And D. Also the extremes. So what
happens if we switch those around? Well there are properties of proportions
that tell us that we can do that. So if we interchange them first, let's just
flip both of these ratios upside down. Let's just interchange A. And B. B over
A. And then if we also interchange D. And C, we have D over C. That is still
the same proportion. This is still true. So remember anytime you have a
proportion, I'm gonna rewrite the original one over here. A over B. Is equal to
C over D. The cross property across products property tells us that this means
that A times D. Is equal to B. Times C. So that is true. And well if I come
over here with this one where I've interchanged my means and my extremes and I
do the cross products property, I still get B times C. And that's equal to A
Times D. This means the exact same thing as that. So certainly I can switch
take both my ratios flip them upside down. I still have a true proportion. Now
going back to my original A over B. Is equal to C over D. What if I just switch
my two means. In other words, what if I say a oversea is equal to is equal to B
over D. Is that still true? Well sure if you look at this, we do our cross
multiplication A times D. Is equal to C. Times B. C times B. Of course is the
exact same thing as B times C. Because multiplication is communicative. So this
is still a true proportion. We could also interchange A. And D. And we would
still get the same result. Let's say I had D over B. And that is going to be
equal to C. Over A. When I do my cross multiplication, I'm going to get D.
Times. This is an A. It doesn't look like an A. Does it? When you raise this,
let's not be make a better A. Here. D. Times A. Is equal to B times C. Again D.
Times A is still equal to A times D. Because multiplication is communicative.
So yes, all of these workouts, we still get a true proportion.
Q. 3 (a) Define depreciation. Give mathematical
characteristics of original cost method of depreciation. Write the formula to
find value of the asset after n year. (7+6+7
= 20)
The term depreciation refers to an accounting method used to
allocate the cost of a tangible or physical asset over its useful life.
Depreciation represents how much of an asset's value has been used. It allows
companies to earn revenue from the assets they own by paying for them over a
certain period of time.
Because companies don't have to account for them entirely in the
year the assets are purchased, the immediate cost of ownership is significantly
reduced. Not accounting for depreciation can greatly affect a company's
profits. Companies can also depreciate long-term assets for both tax and
accounting purposes.
Depreciation can be compared with amortization, which accounts for
the change in value over time of intangible assets.
Assets such as machinery and equipment are expensive. Instead of
realizing the entire cost of an asset in year one, companies can use
depreciation to spread out the cost and match depreciation expenses to related
revenues in the same reporting period. This allows a company to write off an
asset's value over a period of time, notably its useful life.
Companies take depreciation regularly so they can move their
assets' costs from their balance sheets to their income statements. When a
company buys an asset, it records the transaction as a debit to increase an
asset account on the balance sheet and a credit to reduce cash (or increase
accounts payable), which is also on the balance sheet. Neither journal entry
affects the income statement, where revenues and expenses are reported.
At the end of an accounting period, an accountant books depreciation
for all capitalized assets that are not fully depreciated. The journal entry
consists of a:
Debit to depreciation expense, which flows through to the income
statement
Credit to accumulated depreciation, which is reported on the
balance sheet
As noted above, businesses can take advantage of depreciation for
both tax and accounting purposes. This means they can take a tax deduction for
the cost of the asset, reducing taxable income. But the Internal Revenue
Service (IRS) states that when depreciating assets, companies must spread the
cost out over time. The IRS also has rules for when companies can take a
deduction.
Special Considerations
Depreciation is considered a non-cash charge because it doesn't
represent an actual cash outflow. The entire cash outlay might be paid
initially when an asset is purchased, but the expense is recorded incrementally
for financial reporting purposes. That's because assets provide a benefit to
the company over a lengthy period of time. But the depreciation charges still reduce
a company's earnings, which is helpful for tax purposes.
The matching principle under generally accepted accounting
principles (GAAP) is an accrual accounting concept that dictates that expenses
must be matched to the same period in which the related revenue is generated.
Depreciation helps to tie the cost of an asset with the benefit of its use over
time. In other words, the incremental expense associated with using up the
asset is also recorded for the asset that is put to use each year and generates
revenue.
The total amount depreciated each year, which is represented as a
percentage, is called the depreciation rate. For example, if a company had
$100,000 in total depreciation over the asset's expected life, and the annual
depreciation was $15,000. This means the rate would be 15% per year.
Buildings and structures can be depreciated, but land is not
eligible for depreciation.
Threshold Amounts
Different companies may set their own threshold amounts for when to
begin depreciating a fixed asset or property, plant, and equipment (PP&E).
For example, a small company may set a $500 threshold, over which it
depreciates an asset. On the other hand, a larger company may set a $10,000
threshold, under which all purchases are expensed immediately.
Accumulated Depreciation
Accumulated depreciation is a contra asset account, meaning its
natural balance is a credit that reduces its overall asset value. Accumulated
depreciation on any given asset is its cumulative depreciation up to a single
point in its life.
As stated earlier, carrying value is the net of the asset account
and the accumulated depreciation. The salvage value is the carrying value that
remains on the balance sheet after which all depreciation is accounted for
until the asset is disposed of or sold.
It is based on what a company expects to receive in exchange for
the asset at the end of its useful life. An asset’s estimated salvage value is
an important component in the calculation of depreciation.
The IRS publishes depreciation schedules detailing the number of
years an asset can be depreciated for tax purposes, based on various asset
classes.
Types of Depreciation
There are several methods that accountants commonly use to
depreciate capital assets and other revenue-generating assets. These are straight-line,
declining balance, double-declining balance, sum-of-the-years' digits, and unit
of production. We've highlighted some of the basic principles of each below.
Straight-Line
Using the straight-line method is the most basic way to record
depreciation. It reports an equal depreciation expense each year throughout the
entire useful life of the asset until the entire asset is depreciated to its
salvage value.
Straight Line Depreciation
Let's assume that a company buys a machine at a cost of $5,000. The
company decides on a salvage value of $1,000 and a useful life of five years.
Based on these assumptions, the depreciable amount is $4,000 ($5,000 cost -
$1,000 salvage value).
The annual depreciation using the straight-line method is
calculated by dividing the depreciable amount by the total number of years. In
this case, it amounts to $800 per year ($4,000 / 5). This results in a
depreciation rate of 20% ($800 / $4,000).
(b) If you want to earn
an annual rate of 10% on your investments, how much (to the nearest cent)
should you pay for a note that will be worth $5,000 in 9 months?
STEP 1: Convert 6 months into years.
9/12=.75 years
STEP 2: Find an interest by using the formula , where I is
interest, P is total principal, i is rate of interest per year, and t is total
time in years.
In this examplee P = $5000, i = 10%=0.1 and t = 0.75 years, so
I=P*i*t
=5000*0.1*0.75
= $375
Total amount(A)=P+I=5000+375
=$5375
(c) Explain what is
meant by an ordinary annuity. Explain why no interest is credited to an
ordinary annuity at the end of the first period.
An ordinary annuity is a series of equal payments made at the end
of consecutive periods over a fixed length of time. While the payments in an
ordinary annuity can be made as frequently as every week, in practice they are
generally made monthly, quarterly, semi-annually, or annually. The opposite of
an ordinary annuity is an annuity due, in which payments are made at the
beginning of each period. These two series of payments are not the same as the
financial product known as an annuity, though they are related.
Examples of ordinary annuities are interest payments from bonds,
which are generally made semiannually, and quarterly dividends from a stock
that has maintained stable payout levels for years. The present value of an
ordinary annuity is largely dependent on the prevailing interest rate.
Because of the time value of money, rising interest rates reduce
the present value of an ordinary annuity, while declining interest rates
increase its present value. This is because the value of the annuity is based
on the return your money could earn elsewhere. If you can get a higher interest
rate somewhere else, the value of the annuity in question goes down.
Present Value of an Ordinary Annuity Example
The present value formula for an ordinary annuity takes into
account three variables. They are as follows:
PMT = the period cash payment
r = the interest rate per period
n = the total number of periods
Given these variables, the present value of an ordinary annuity is:
Present Value = PMT x ((1 - (1 + r) ^ -n ) / r)
For example, if an ordinary annuity pays $50,000 per year for five
years and the interest rate is 7%, the present value would be:
Present Value = $50,000 x ((1 - (1 + 0.07) ^ -5) / 0.07) = $205,010
Q. 4 (a) You want to purchase an automobile for
$21,600. The dealer offers you 0% financing for 48 months or a $3,000 rebate.
You can obtain 4.8% financing for 40 months at the local bank. Which option
should you choose? Explain. (8+12 = 20)
Hence, monthly payments to local bank are lower so, it is better to
take loan from it and take rebate from the dealer.
(b) If $1,000 is
invested at 8% compounded (i) annually, (ii) semiannually,
(iii) quarterly, (iv) monthly, what is the amount after 5 years? Write answers
to the nearest cent.
1)
PV =
FV*(1+r)n
FV =
28000
a)
annually
n = 6, r
= 3% = 0.03
PV =
28000*(1+0.03)6 = 23449.56
b)
quarterly
n = 6*4
= 24, r = 3%/4 = 0.03/4 = 0.0075
PV =
28000*(1+0.0075)6 = 23403.28
2)
3%
compounded annually
Rule of
72 = 72/3 = 24 years
Actual
2 =
1*(1+0.03)n
1.03n =
2
apply
log on both sides
n*LN1.03
= LN 2
n = LN
2 / LN1.03 = 23.45
3)
EAR =
(1+r/n)n - 1
r = 19%
= 0.19
n = 12
EAR =
(1+0.19/12)12 - 1 = 0.207451 = 20.75%
4)
FV =
PV*(1+r)n
PV =
6000
r =
13.2% = 0.0132/12 = 0.011 = 1.1%
n = 3*12
= 36
FV =
6000*(1+0.011)36 = 8895.96
Q. 5 (a) Define commission rate. How it is
different from wage? What is commission on Rs.3000 @ 
. (8+7+5 = 20)
Commission rate is the payment associated with either a fixed
payment or percentage of a sale. Professions that work on commission, such as
insurance brokers, real estate agents and car salespeople, receive payments
when they produce a sale.
With a regular salary, you pay an employee a set amount of wages.
Salaries are given regardless of whether the employee sells anything or not. On
the other hand, commission is determined by an employee's sales. Some
businesses choose to offer a base salary and commissions.
Commission = 3000*33(1/2)%
= 1005
(b) Define the term
successive discount. Give formula of computing discount rate for given quantity
discount offer. What is discount rate in offer “Buy two get three”?
Successive discount means discount on the discount. (just like what
Compound Interest rate = interest on interest) Now let us say the original
price of a music CD is Rs. 100. A shopkeeper offers 10% discount on this music
CD and then again offers 20% discount on the new price.
The formula for total discount in case of successive-discounts : If
the first discount is x% and 2nd discount is y% then, Total discount = ( x + y
- xy / 100 ) % Successive discount is the discount offered on the discount. It
is similar to compound interest (interest on interest).
(c) Which is the better
investment and why: 9% compounded quarterly or 9.25% compounded annually?
Dear Student,
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copy paste h jo dusre student k pass b available h. Agr ap ne university
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