Practice problem mid m4b

Exercise 13: Use the index numbers listed in the table to find the percentage change in output from Exercise 14: Table gives the annual rate of inflation during a 5-year period.. a Work

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

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

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:

(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

Find the values of Q for which the firm

(a) breaks even

(b) makes a loss of 432 units

Find the break-even values of Q and deduce the maximum profit

* INDICES AND LOGARITHMS

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

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?

(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

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

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

Exercise 15: Table 3.13 shows the monthly index of sales of a good during

the first four months of the year

(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

(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

(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

(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

(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

(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

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

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?

(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

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