1
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Practice problem Chapter 1:
Exercise 1: The demand for a good priced at $50 is 420 units, and when the
price is $80 demand is 240 units. Assuming that the demand function takes
the form Q = aP + b, find the values of a and b.
Exercise 2: The demand and supply functions of a good are given by P = −3QD + 48 1 𝑃= 𝑄𝑠+23 2
Find the equilibrium quantity if the government imposes a fixed tax of $4 on each good.
Exercise 3: The demand and supply functions for two interdependent commodities are given by QD1 = 100 − 2P1 + P2 QD2 = 5 + 2P1 − 3P2 QS1 = −10 + P1 QS2 = −5 + 6P2
where QDi, QSi and Pi denote the quantity demanded, quantity supplied and
price of good i respectively. Determine the equilibrium price and quantity for this two-commodity model.
Exercise 4: The demand and supply functions of a good are given by P = −5QD + 80 P = 2QS + 10
where P, QD and QS denote price, quantity demanded and quantity supplied respectively.
(1) Find the equilibrium price and quantity
(2) If the government deducts, as tax, 15% of the market price of each good,
determine the new equilibrium price and quantity.
Exercise 5: The supply and demand functions of a good are given by P = QS + 8 P = −3QD + 80 2
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
where P, QS and QD denote price, quantity supplied and quantity demanded respectively.
(a) Find the equilibrium price and quantity if the government imposes a fixed tax of $36 on each good.
(b) Find the corresponding value of the government’s tax revenue.
Exercise 6 (*): The demand and supply functions of a good are given by P = −3QD + 60 P = 2QS + 40
respectively. If the government decides to impose a tax of $t per good, show
that the equilibrium quantity is given by 𝑄 = 4 − 15 𝑡 and write down a
similar expression for the equilibrium price.
(a) If it is known that the equilibrium quantity is 3, work out the value of t.
How much of this tax is paid by the firm?
(b) If, instead of imposing a tax, the government provides a subsidy of $5 per
good, find the new equilibrium price and quantity. Chapter 2: Exercise 1:
(a) If the demand function of a good is given by P = 80 − 3Q
find the price when Q = 10 and deduce the total revenue.
(b) If fixed costs are 100 and variable costs are 5 per unit find the total cost when Q = 10.
(c) Use your answers to parts (a) and (b) to work out the corresponding profit. Exercise 2:
(a) Given the following demand functions, express TR as a function of Q and
hence sketch the graphs of TR against Q: P = 4𝑃 7 P = 𝑄 P = 10 − 4Q
(b) Given the following total revenue functions, find the corresponding demand functions: TR = 50Q − 4Q2 TR = 10 3
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(c) Given that fixed costs are 500 and that variable costs are 10 per unit,
express TC and AC as functions of Q.
(d) Given that fixed costs are 1 and that variable costs are Q + 1 per unit,
express TC and AC as functions of Q.
Exercise 3: The total cost, TC, of producing 100 units of a good is 600 and
the total cost of producing 150 units is 850. Assuming that the total cost
function is linear, find an expression for TC in terms of Q, the number of units produced.
Exercise 4: The total cost of producing 500 items a day in a factory is
$40000, which includes a fixed cost of $2000.
(a) Work out the variable cost per item.
(b) Work out the total cost of producing 600 items a day.
Exercise 5: A taxi firm charges a fixed cost of $10 together with a variable cost of $3 per mile.
(a) Work out the average cost per mile for a journey of 4 miles.
(b) Work out the minimum distance traveled if the average cost per mile is to be less than $3.25.
Exercise 6: Find an expression for the profit function given the demand function 2Q + P = 25 32
and the average cost function 𝐴𝐶= +5 𝑄
Find the values of Q for which the firm (a) breaks even (b) makes a loss of 432 units (c) maximises profit.
Exercise 7: If fixed costs are 30, variable costs per unit are Q + 3, and the demand function is P + 2Q = 50
show that the associated profit function is π = −3Q2 + 47Q − 30.
Find the break-even values of Q and deduce the maximum profit.
* INDICES AND LOGARITHMS 4
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Which of the following production functions are homogeneous? For those
functions which are homogeneous write down their degrees of homogeneity
and comment on their returns to scale. (a) Q = 500K1/3 L1/4 (b) Q = 3LK + L2 (c) Q = L + 5L2K3 Chapter 3: I. Percentage
Exercise 1: A firm has 132 female and 88 male employees.
(a) What percentage of staff are female?
(b) During the next year 8 additional female staff are employed. If the
percentage of female staff is now 56%, how many additional male staff were recruited during the year?
Exercise 2: Find the new quantities when
(a) $16.25 is increased by 12%
(b) the population of a town, currently at 113 566, rises by 5%
(c) a good priced by a firm at $87.90 is subject to a sales tax of 15%
(d) a good priced at $2300 is reduced by 30% in a sale
(e) a car, valued at $23 000, depreciates by 32%.
Exercise 3: A student discount card reduces a bill in a restaurant from $124
to $80.60. Work out the percentage discount.
Exercise 4: A TV costs $900 including 20% sales tax. Find the new price if tax is reduced to 15%.
Exercise 5: An antiques dealer tries to sell a vase at 45% above the $18 000
which the dealer paid at auction.
(a) What is the new sale price?
(b) By what percentage can the dealer now reduce the price before making a loss? Exercise 6:
(a) Current monthly output from a factory is 25 000. In a recession, this is
expected to fall by 65%. Estimate the new level of output.
(b) As a result of a modernisation programme, a firm is able to reduce the
size of its workforce by 24%. If it now employs 570 workers, how many
people did it employ before restructuring? 5
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(c) Shares originally worth $10.50 fall in a stock market crash to $2.10. Find the percentage decrease.
Exercise 7: Total revenue from daily ticket sales to a theme park is $1 352
400. A total of 12 000 tickets were sold and 65% of these were child’s tickets
with a 30% discount of the adult price. Work out the cost of an adult ticket.
Exercise 8: The cost of a computer is $6000 including 20% sales tax. In a
generous gesture, the government decides to reduce the rate to just 17.5%.
Find the cost of the computer after the tax has changed.
Exercise 9: A coat originally costing $150 is reduced by 25% in a sale and,
since nobody bought the coat, a further reduction of 20% of the sale price is applied.
(a) Find the final cost of the coat after both reductions.
(b) Find the overall percentage reduction and explain why this is not the same as a single reduction of 45%.
Exercise 10: A furniture store has a sale of 40% on selected items. A sales
assistant, Carol, reduces the price of a sofa originally costing $1200. (a) What is the new price?
The manager does not want this sofa to be in the sale and the following day
tells another sales assistant, Michael, to restore the sofa back to the original
price. He does not know what the original price was and decides to show of
his mathematical knowledge by taking the answer to part (a) and multiplying it by 1.4.
(b) Explain carefully why this does not give the correct answer of $1200.
(c) Suggest an alternative calculation that would give the right answer.
Exercise 11: During 2014 the price of a good increased by 8%. In the sales
on 1 January 2015 all items are reduced by 25%.
(a) If the sale price of the good is $688.50, find the original price at the beginning of 2014.
(b) Find the overall percentage change.
(c) What percentage increase would be needed to restore the cost to the
original price prevailing on 1 January 2014? Give your answer to 1 decimal place.
Exercise 12: Find the single percentage increase or decrease equivalent to
(a) a 10% increase followed by a 25% increase
(b) a 34% decrease followed by a 65% increase
(c) a 25% increase followed by a 25% decrease. 6
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Explain in words why the overall change in part (c) is not 0%.
Exercise 13: Use the index numbers listed in the table to find the percentage change in output from (a) 14Q1 to 14Q4 (b) 13Q1 to 14Q4 (c) 13Q1 to 14Q1 Output 13Q1 13Q2 13Q3 13Q4 14Q1 14Q2 14Q3 14Q4 Index 89.3 98.1 105 99.3 100 106.3 110.2 105.7
Exercise 14: Table gives the annual rate of inflation during a 5-year period.
If a nominal house price at the end of 2000 was $10.8 million, find the real
house price adjusted to prices prevailing at the end of the year 2003. Round
your answer to three significant figures. Year 2000 2001 2002 2003 2004 Annual 1.8% 2.1% 2.9% 2.4% 2.7% rate of inflation
Exercise 15: Table 3.13 shows the monthly index of sales of a good during
the first four months of the year. Month Jan Feb Mar Apr Index 100 120 145 150
(a) Which month is chosen as the base year?
(b) If sales in February are 3840, what are the sales in April?
(c) What is the index number in May if sales are 4256?
Exercise 16: Table shows the index numbers associated with transport costs
during a 20-year period. The public transport costs reflect changes to bus and
train fares, whereas private transport costs include purchase, service, petrol,
tax and insurance costs of cars. Year 1985 1990 1005 2000 2005 Public 100 130 198 224 245 transport Private 100 125 180 199 221 transport
(1) Which year is chosen as the base year? 7
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(2) Find the percentage increases in the cost of public transport from (a) 1985 to 1990 (b) 1990 to 1995 (c) 1995 to 2000 (d) 2000 to 2005
(3) Repeat part (2) for private transport.
Exercise 17: Table shows the prices of a good for each year between 2009 and 2014. Year 2009 2010 2011 2012 2013 2014 Price ($) 40 48 44 56 60 71
(a) Work out the index numbers, correct to 1 decimal place, taking 2010 as the base year.
(b) If the index number for 2015 is 135, calculate the corresponding price.
You may assume that the base year is still 2010.
(c) If the index number in 2011 is approximately 73, find the year that is used as the base year.
Exercise 18: Table shows government expenditure (in billions of dollars) on
education for four consecutive years, together with the rate of inflation for each year. Year 2004 2005 2006 2007 Spending 236 240 267 276 Inflation 4.7 4.2 3.4
(a) Taking 2004 as the base year, work out the index numbers of the nominal
data given in the third row of the table.
(b) Find the values of expenditure at constant 2004 prices and hence
recalculate the index numbers of real government expenditure.
(c) Give an interpretation of the index numbers calculated in part (b).
Exercise 19: Index numbers associated with the growth of unemployment
during an 8-year period are shown in the table Year 1 2 3 4 5 6 7 8 8
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG Index 100 95 105 110 119 127 1 Index 100 112 118 2
(a) What are the base years for the two indices?
(b) If the government had not switched to index 2, what would be the values of index 1 in years 7 and 8?
(c) What values would index 2 have been in years 1, 2, 3, 4 and 5?
(d) If unemployment was 1.2 million in year 4, how many people were unemployed in years 1 and 8?
Exercise 20: The prices of a good at the end of each year between 2003 and
2008 are listed in the table, which also shows the annual rate of inflation Year 2003 2004 2005 2006 2007 2008 Price 230 242 251 257 270 284 Inflation 4% 3% 2.5% 2% 2%
(a) Find the values of the prices adjusted to the end of year 2004, correct to 2
decimal places. Hence, calculate the index numbers of the real data with 2004
as the base year. Give your answers correct to 1 decimal place.
(b) If the index number of the real price for 2009 is 109 and the rate of
inflation for that year is 2.5%, work out the nominal value of the price in
2009. Give your answer rounded to the nearest whole number.
(c) If the index number of the real data in 2002 is 95.6 and the nominal price
is $215, find the rate of inflation for 2002. Give your answer correct to 1 decimal place. II. Compound Interest
Exercise 1: A bank offers a return of 7% interest compounded annually. Find
the future value of a principal of $4500 after 6 years. What is the overall
percentage rise over this period?
Exercise 2: Find the future value of $20 000 in 2 years’ time if compounded quarterly at 8% interest. 9
MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Exercise 3: The value of an asset, currently priced at $100 000, is expected to increase by 20% a year.
(a) Find its value in 10 years’ time.
(b) After how many years will it be worth $1 million?
Exercise 4: How long will it take for a sum of money to double if it is
invested at 5% interest compounded annually?
Exercise 5: A piece of machinery depreciates in value by 5% a year.
Determine its value in 3 years’ time if its current value is $50 000.
Exercise 6: A principal, $7000, is invested at 9% interest for 8 years.
Determine its future value if the interest is compounded (a) annually (b) semi-annually (c) monthly (d) continuously
Exercise 7: Which of the following savings accounts offers the greater return?
Account A: an annual rate of 8.05% paid semi-annually.
Account B: an annual rate of 7.95% paid monthly.
Exercise 8: Find the future value of $100 compounded continuously at an annual rate of 6% for 12 year
Exercise 9: How long will it take for a sum of money to triple in value if
invested at an annual rate of 3% compounded continuously?
Exercise 10: If a piece of machinery depreciates continuously at an annual
rate of 4%, how many years will it take for the value of the machinery to halve?
Exercise 11: Determine the EAR if the nominal rate is 7% compounded continuously.
Exercise 12: Current annual consumption of energy is 78 billion units and
this is expected to rise at a fixed rate of 5.8% each year. The capacity of the
industry to supply energy is currently 104 billion units.
(a) Assuming that the supply remains steady, after how many years will demand exceed supply?
10 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(b) What constant rate of growth of energy production would be needed to
satisfy demand for the next 50 years?
Exercise 13: Find the value, in 2 year time, of $4000 invested at 5%
compounded annually. In the following 2 years, the interest rate is expected
to rise to 8%. Find the final value of the investment at the end of the 4-year
period and find the overall percentage increase. Give your answers correct to 2 decimal places.
Exercise 14: Find the APR of a loan if the monthly interest rate is 1.65%.
Give your answer correct to 2 decimal places.
Exercise 15: A principal of $7650 is invested at a rate of 3.7% compounded
annually. After how many years will the investment first exceed $12 250?
Exercise 16: A principal of $70 000 is invested at 6% interest for 4 years.
Find the difference in the future value if the interest is compounded quarterly
compared to continuous compounding. Round your answer to 2 decimal places.
Exercise 17: Midwest Bank offers a return of 5% compounded annually for
each and every year. The rival BFB offers a return of 3% for the first year
and 7% in the second and subsequent years (both compounded annually).
Which bank would you choose to invest in if you decided to invest a principal for (a) 2 years? (b) 3 years?
Exercise 18: A car depreciates by 40% in the first year, 30% in the second
year and 20% thereafter. I buy a car for $14 700 when it is 2 years old.
(a) How much did it cost when new?
(b) After how many years will it be worth less than 25% of the amount that I paid for it?
Exercise 19: Simon decides to buy a new sofa which is available at each of
three stores at the same fixed price. He decides to borrow the money using
each store’s credit facility.
Store A has an effective rate of interest of 12.6%.
Store B charges interest at a rate of 10.5% compounded continuously.
11 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Store C charges interest at a rate of 11.5% compounded quarterly.
From which store should Simon buy his sofa to minimise the total cost?
Exercise 20: World oil reserves are currently estimated to be 600 billion
units. If this quantity is reduced by 8% a year, after how many years will oil
reserves drop below 100 billion units?
Exercise 21: The nominal rate of interest of a store card is 18% compounded monthly.
(a) State the monthly interest rate.
(b) Find the equivalent annual rate of interest if the compounding is
continuous. Round your answer to 2 decimal places. III. Geometric series
Exercise 1: An individual saves $5000 in a bank account at the beginning of
each year for 10 years. No further savings or withdrawals are made from the
account. Determine the total amount saved if the annual interest rate is 8% compounded: (a) annually. (b) semi-annually.
Exercise 2: Determine the monthly repayments needed to repay a $125 000
loan which is paid back over 20 years when the interest rate is 7%
compounded annually. Round your answer to 2 decimal places.
Exercise 3: A prize fund is set up with a single investment of $5000 to
provide an annual prize of $500. The fund is invested to earn interest at a rate
of 7% compounded annually. If the first prize is awarded 1 year after the
initial investment, find the number of years for which the prize can be
awarded before the fund falls below $500.
Exercise 4: A person invests $5000 at the beginning of a year in a savings
account that offers a return of 4.5% compounded annually. At the beginning
of each subsequent year, an additional $1000 is invested in the account. How
much will there be in the account at the end of ten years?
Exercise 5: A person borrows $100 000 at the beginning of a year and agrees
to repay the loan in ten equal installments at the end of each year. Interest is
charged at a rate of 6% compounded annually.
12 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(a) Find the annual repayment.
(b) Work out the total amount of interest paid and compare this with the total
interest paid when repaying the loan in five equal annual installments instead of ten.
Exercise 6: A regular saving of $500 is made into a sinking fund at the start
of each year for 10 years. Determine the value of the fund at the end of the
tenth year on the assumption that the rate of interest is (a) 11% compounded annually.
(b) 10% compounded continuously.
Exercise 7: Monthly sales figures for January are 5600. This is expected to
fall for the following 9 months at a rate of 2% each month. Thereafter sales
are predicted to rise at a constant rate of 4% each month. Estimate total sales
for the next 2 years (including the first January).
Exercise 8: Determine the monthly repayments needed to repay a $50 000
loan that is paid back over 25 years when the interest rate is 9% compounded
annually. Calculate the increased monthly repayments needed in the case when
(a) the interest rate rises to 10%.
(b) the period of repayment is reduced to 20 years. IV. Investment appraisal
Exercise 1: Determine the present value of $7000 in 2 years’ time if the
discount rate is 8% compounded (a) quarterly. (b) continuously.
Exercise 2: A small business promises a profit of $8000 on an initial
investment of $20 000 after 5 years.
(a) Calculate the internal rate of return.
(b) Would you advise someone to invest in this business if the market rate is 6% compounded annually?
Exercise 3: An investment company is considering one of two possible
business ventures. Project 1 gives a return of $250 000 in 4 years’ time
whereas Project 2 gives a return of $350 000 in 8 years’ time. Which project
13 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
should the company invest in when the interest rate is 7% compounded annually?
Exercise 4: A builder is offered one of two methods of payment:
Option 1: A single sum of $73 000 to be paid now.
Option 2: Five equal payments of $15 000 to be paid quarterly with the first instalment to be paid now.
Advise the builder which of er to accept if the interest rate is 6% compounded quarterly.
Exercise 5: A financial company invests £250 000 now and receives £300
000 in three years’ time. Calculate the internal rate of return.
Exercise 7: You are given the opportunity of investing in one of three
projects. Projects A, B and C require initial outlays of $20 000, $30 000 and
$100 000 and are guaranteed to return $25 000, $37 000 and $117 000,
respectively, in 3 years’ time. Which of these projects would you invest in if
the market rate is 5% compounded annually?
Exercise 8: Determine the present value of an annuity that pays out $100 at the end of each year (a) for 5 years (b) in perpetuity
if the interest rate is 10% compounded annually.
Exercise 9: An investor is given the opportunity to invest in one of two projects:
Project A costs $10 000 now and pays back $15 000 at the end of 4 years.
Project B costs $15 000 now and pays back $25 000 at the end of 5 years.
The current interest rate is 9%.
By calculating the net present values, decide which, if either, of these projects is to be recommended.
Exercise 10: A proposed investment costs $130 000 today. The expected
revenue flow is $40 000 at the end of year 1, and $140 000 at the end of year
2. Find the internal rate of return.
14 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Exercise 11: Find the present value of $450 in 6 years’ time if the discount
rate is 9.5% compounded semi-annually. Round your answer to 2 decimal places.
Exercise 12: A project requires an initial investment of $7000, and is
guaranteed to yield a return of $1500 at the end of the first year, $2500 at the
end of the second year and $ x at the end of the third year. Find the value of
x, correct to the nearest $, given that the net present value is $838.18 when
the interest rate is 6% compounded annually.
Exercise 13: Determine the present value of an annuity, if it pays out $2500
at the end of each year in perpetuity, assuming that the interest rate is 8% compounded annually.
Exercise 14 (*): A firm decides to invest in a new piece of machinery which
is expected to produce an additional revenue of $8000 at the end of every
year for 10 years. At the end of this period the fi rm plans to sell the
machinery for scrap, for which it expects to receive $5000. What is the
maximum amount that the firm should pay for the machine if it is not to
suffer a net loss as a result of this investment? You may assume that the
discount rate is 6% compounded annually.
Exercise 15: A project requires an initial investment of $50 000. It produces
a return of $40 000 at the end of year 1 and $30 000 at the end of year 2. Find
the exact value of the internal rate of return.
Exercise 16: An annuity pays out $20 000 per year in perpetuity. If the
interest rate is 5% compounded annually, find
(a) the present value of the whole annuity.
(b) the present value of the annuity for payments received, starting from the end of the 30th year.
(c) the present value of the annuity of the first 30 years.
Exercise 17: A project requires an initial outlay of $80 000 and produces a
return of $20 000 at the end of year 1, $30 000 at the end of year 2, and $ R at
the end of year 3. Determine the value of R if the internal rate of return is 10%. Chapter 4:
15 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG I. Marginal functions
Exercise 1: If the demand function is P = 100 − 4Q
find expressions for TR and MR in terms of Q. Hence estimate the change in
TR brought about by a 0.3 unit increase in output from a current level of 12 units.
Exercise 2: If the demand function is P = 80 − 3Q show that MR = 2P – 80
Exercise 3: A monopolist’s demand function is given by P + Q = 100
Write down expressions for TR and MR in terms of Q and sketch their
graphs. Find the value of Q which gives a marginal revenue of zero and
comment on the significance of this value.
Exercise 4: If the average cost function of a good is 𝐴𝐶= 15+ 𝑄 2𝑄+9
find an expression for TC. What are the fixed costs in this case? Write down
an expression for the marginal cost function.
Exercise 5: A firm’s production function is Q = 50L − 0.01L2
where L denotes the size of the workforce. Find the value of MPL in the case when
(a) L = 1 (b) L = 10 (c) L = 100 (d) L = 1000
Exercise 6: If the demand function is 𝑃=3000−2√𝑄
find expressions for TR and MR. Calculate the marginal revenue when Q = 9
and give an interpretation of this result.
Exercise 7: A firm’s demand function is given by 𝑃=100−4√𝑄−3𝑄
(a) Write down an expression for total revenue, TR, in terms of Q.
(b) Find an expression for the marginal revenue, MR, and find the value of MR when Q = 9.
16 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
(c) Use the result of part (b) to estimate the change in TR when Q increases
by 0.25 units from its current level of 9 units and compare this with the exact change in TR.
Exercise 8: The fixed costs of producing a good are 100 and the variable 𝑄 costs are 2 + per unit. 10
(a) Find expressions for TC and MC.
(b) Evaluate MC at Q = 30 and hence estimate the change in TC brought
about by a 2 unit increase in output from a current level of 30 units.
(c) At what level of output does MC = 22?
Exercise 9: A firm’s production function is given by 𝑄=5√𝐿−0.1𝐿
(a) Find an expression for the marginal product of labour, MPL.
(b) Solve the equation MPL = 0 and briefl y explain the signifi cance of this value of L. 6
Exercise 10: A firm’s average cost function takes the form 𝐴𝐶=4𝑄+𝑎+ 𝑄
and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.
Exercise 11: The total cost of producing a good is given by 𝑇𝐶=250+20𝑄
The marginal revenue is 18 at Q = 219. If production is increased from its
current level of 219, would you expect profit to increase, decrease or stay the
same? Give reasons for your answer.
Exercise 12: Given the demand and total cost functions 𝑃=150−2𝑄 𝑎𝑛𝑑 𝑇𝐶=40+0.5𝑄2
find the marginal profit when Q = 25 and give an interpretation of this result. II. Elasticity
Exercise 1: Given the demand function P = 500 − 4Q2
calculate the price elasticity of demand averaged along an arc joining Q = 8 and Q = 10.
Exercise 2: Find the price elasticity of demand at the point Q = 9 for the demand function
17 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG P = 500 − 4Q2
and compare your answer with that of Question 1.
Exercise 3: Find the price elasticity of demand at P = 6 for each of the following demand functions: (a) P = 30 − 2Q (b) P = 30 − 12Q (c) 𝑃=√(100−2𝑄)
Exercise 4: (a) If an airline increases prices for business class flights by 8%,
demand falls by about 2.5%. Estimate the elasticity of demand. Is demand
elastic, inelastic or unit elastic?
(b) Explain whether you would expect a similar result to hold for economy class flights.
Exercise 5: The demand function of a good is given by: 1000 Q= 𝑃2
(a) Calculate the price elasticity of demand at P = 5 and hence estimate the
percentage change in demand when P increases by 2%.
(b) Comment on the accuracy of your estimate in part (a) by calculating the
exact percentage change in demand when P increases from 5 to 5.1.
Exercise 6: (a) Find the elasticity of demand in terms of Q for the demand function P = 20 − 0.05Q
(b) For what value of Q is demand unit elastic?
(c) Find an expression for MR and verify that MR = 0 when demand is unit elastic.
Exercise 7: Consider the supply equation Q = 4 + 0.1P2
(a) Write down an expression for dQ/dP.
(b) Show that the supply equation can be rearranged as 𝑃=√(10𝑄−40)
Differentiate this to find an expression for dP/dQ.
(c) Use your answers to parts (a) and (b) to verify that
18 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝑑𝑄 1 = 𝑑𝑃 𝑑𝑃/𝑑𝑄
(d) Calculate the elasticity of supply at the point Q = 14.
Exercise 8: If the supply equation is 𝑄=7+0.1𝑃+0.004𝑃2
find the price elasticity of supply if the current price is 80.
(a) Is supply elastic, inelastic or unit elastic at this price?
(b) Estimate the percentage change in supply if the price rises by 5%.
Exercise 9: Find the elasticity for the demand function Q = 80 − 2P − 0.5P2
averaged along an arc joining Q = 32 to Q = 50. Give your answer to two decimal places.
Exercise 10: Consider the supply equation P = 7 + 2Q2
By evaluating the price elasticity of supply at the point P = 105, estimate the
percentage increase in supply when the price rises by 7%.
Exercise 11: If the demand equation is Q + 4P = 60
find a general expression for the price elasticity of demand in terms of P. For
what value of P is demand unit elastic?
Exercise 12: A supply function is given by Q = 40 + 0.1P2
(1) Find the price elasticity of supply averaged along an arc between P = 11
and P = 13. Give your answer correct to 3 decimal places.
(2) Find an expression for price elasticity of supply at a general point, P. Hence:
(a) Estimate the percentage change in supply when the price increases by 5%
from its current level of 17. Give your answer correct to 1 decimal place.
(b) Find the price at which supply is unit elastic. III. Optimisation
Exercise 1: If the demand equation of a good is P = 40 − 2Q
19 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
find the level of output that maximises total revenue.
Exercise 2: A firm’s short-run production function is given by Q = 30L2 − 0.5L3
Find the value of L which maximises APL and verify that MPL = APL at this point.
Exercise 3: The demand and total cost functions of a good are 4P + Q − 16 = 0
And 𝑇𝐶=4+2𝑄−3𝑄210+ 320 𝑄 respectively.
(a) Find expressions for TR, π, MR and MC in terms of Q.
(b) Solve the equation 𝑑𝜋𝑑𝑄=0
and hence determine the value of Q which maximises profit.
(c) Verify that, at the point of maximum profit, MR = MC.
Exercise 4: The supply and demand equations of a good are given by 3P – QS = 3 and 2P + QD = 14 respectively.
The government decides to impose a tax, t , per unit. Find the value of t
which maximises the government’s total tax revenue on the assumption that
equilibrium conditions prevail in the market.
Exercise 5: A manufacturer has fi xed costs of $200 each week, and the
variable costs per unit can be expressed by the function, VC = 2Q − 36.
(a) Find an expression for the total cost function and deduce that the average cost function is given by 200 𝐴𝐶= +2𝑄−36 𝑄
(b) Find the stationary point of this function and show that this is a minimum.
(c) Verify that, at this stationary point, average cost is the same as marginal cost.
Exercise 7: A firm’s short-run production function is given by 𝑄=3√𝐿
20 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
where L is the number of units of labour.
If the price per unit sold is $50 and the price per unit of labour is $10, find the
value of L needed to maximise profits. You may assume that the fi rm sells
all that it produces and you can ignore all other costs.
Exercise 8: The average cost per person of hiring a tour guide on a week’s
river cruise for a maximum party size of 30 people is given by
AC = 3Q2 − 192Q + 3500 (0 < Q ≤ 30)
Find the minimum average cost for the trip.
Exercise 9: An electronic components firm launches a new product on 1st
January. During the following year a rough estimate of the number of orders,
S, received t days after the launch is given by 𝑆=𝑡2−0.002𝑡3
What is the maximum number of orders received on any one day of the year?
Exercise 10: A firm’s demand function is P = 60 − 0.5Q
If fixed costs are 10 and variable costs are Q + 3 per unit, find the maximum profit.
Exercise 11: If fixed costs are 15 and the variable costs are 2Q per unit, write
down expressions for TC, AC and MC. Find the value of Q which minimises
AC and verify that AC = MC at this point.
Exercise 12: Daily sales, S , of a new product for the fi rst two weeks after
the launch is modelled by 𝑆=𝑡3−24𝑡2+180𝑡+60 (0 =< t =< 13)
where t is the number of days. Find and classify the stationary points of this function
Exercise 13: If the demand function of a good is 𝑃=√(1000−4𝑄). i F nd the
value of Q which maximises total revenue.
Exercise 14: A firm’s total cost and demand functions are given by
TC = Q2 + 50Q + 10 and P = 200 − 4Q respectively.
(a) Find the level of output needed to maximise the firm’s profit.
(b) The government imposes a tax of $t per good. If the firm adds this tax to
its costs and continues to maximise profit, show that the price of the good