Mathematics for Economics and Business

lan Jacques

Chapter 3 Mathematics of Finance - all with Video Answers

Educators

Section 1

Percentages

01:35

Problem 1

Calculate
(a) $10 \%$ of $\$ 2.90$
(b) $75 \%$ of $\$ 1250$
(c) $24 \%$ of $\$ 580$

Sanchit Jain

Numerade Educator

02:49

(a) A firm's annual sales rise from 50000 to 55000 from one year to the next. Express the rise as a percentage of the original.
(b) The government imposes a $15 \%$ tax on the price of a good. How much does the consumer pay for a good priced by a firm at $\$ 1360$ ?
(c) Investments fall during the course of a year by $7 \%$. Find the value of an investment at the end of the year if it was worth $\$ 9500$ at the beginning of the year.

Niamat Khuda

Numerade Educator

02:09

Problem 3

(a) The value of a good rises by $13 \%$ in a year. If it was worth $\$ 6.5$ million at the beginning of the year, find its value at the end of the year.
(b) The GNP of a country has increased by $63 \%$ over the past 5 years and is now $\$ 124$ billion. What was the GNP 5 years ago?
(c) Sales rise from 115000 to 123050 in a year. Find the annual percentage rise.

Niamat Khuda

Numerade Educator

02:35

Problem 4

(a) Current monthly output from a factory is 25000 . In a recession, this is expected to fall by $65 \%$. Estimate the new level of output.
(b) As a result of a modernization 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.

Niamat Khuda

Numerade Educator

03:12

Problem 5

Find the single percentage increase or decrease equivalent to
(a) an increase of $30 \%$ followed by an increase of $40 \%$
(b) a decrease of $30 \%$ followed by a decrease of $40 \%$
(c) an increase of $10 \%$ followed by a decrease of $50 \%$.

Niamat Khuda

Numerade Educator

02:39

Problem 6

Calculate the index numbers for the data shown in Table 3.1, this time taking 1999 as the base year.

Niamat Khuda

Numerade Educator

02:44

Problem 7

Use the index numbers listed in Table $3.5$ to find the percentage change in output from
(a) $05 \mathrm{Q} 1$ to $05 \mathrm{Q} 4$
(b) $04 \mathrm{Q} 1$ to $05 \mathrm{Q} 4$
(c) $04 \mathrm{Q} 1$ to $05 \mathrm{Q} 1$

Niamat Khuda

Numerade Educator

00:45

Problem 8

Table $3.10$ shows the average annual salary (in thousands of dollars) of employees in a small firm, together with the annual rate of inflation for that year. Adjust these salaries to the prices prevailing at the end of 1991 and so give the real values of the employees' salaries at constant '1991 prices'. Comment on the rise in earnings during this period.

Chandler Austin

Numerade Educator

03:32

Problem 9

Express the following percentages as fractions in their simplest form:
(a) $35 \%$
(b) $88 \%$
(c) $250 \%$
(d) $17^{1} / 2 \%$
(e) $0.2 \%$

Niamat Khuda

Numerade Educator

00:58

Problem 10

Calculate each of the following:
(a) $5 \%$ of 24
(b) $8 \%$ of 88
(c) $48 \%$ of 4563
(d) $112 \%$ of 56

Niamat Khuda

Numerade Educator

01:19

Problem 11

Write down the scale factors corresponding to
(a) an increase of $19 \%$
(b) an increase of $250 \%$
(c) a decrease of $2 \%$
(d) a decrease of $43 \%$

Niamat Khuda

Numerade Educator

03:09

Problem 12

Find the new quantities when
(a) $£ 16.25$ is increased by $12 \%$
(b) the population of a town, currently at 113566, rises by $5 \%$
(c) a good priced by a firm at $£ 87.90$ is subject to value added tax at $15 \%$
(d) a good priced at $£ 2300$ is reduced by $30 \%$ in a sale
(e) a car, valued at $£ 23000$, depreciates by $32 \%$

Niamat Khuda

Numerade Educator

03:02

Problem 13

A shop sells books at ' $20 \%$ below the recommended retail price (r.r.p.)'. If it sells a book for $£ 12.40$ find
(a) the r.r.p.
(b) the cost of the book after a further reduction of $15 \%$ in a sale
(c) the overall percentage discount obtained by buying the book from the shop in the sale compared with the manufacturer's r.r.p.

Niamat Khuda

Numerade Educator

01:23

Problem 14

A TV costs $£ 940$ including $17.5 \%$ VAT. Find the new price if VAT is reduced to $8 \%$.

Niamat Khuda

Numerade Educator

01:23

Problem 15

An antiques dealer tries to sell a vase at $45 \%$ above the $£ 18000$ 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?

Niamat Khuda

Numerade Educator

05:22

Problem 16

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 $0 \%$.

Niamat Khuda

Numerade Educator

09:18

Problem 17

Table $3.11$ 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.
(1) Which year is chosen as the base year?
(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.
(4) Comment briefly on the relative rise in public and private transport costs during this 20 -year period.

Aishwarya Krishnakumar

Numerade Educator

01:31

Problem 18

Table $3.12$ shows the number of items (in thousands) produced from a factory production line during the course of a year. Taking the second quarter as the base quarter, calculate the associated index numbers. Suggest a possible reason for the fluctuations in output.

Niamat Khuda

Numerade Educator

02:52

Problem 19

Table $3.13$ shows government expenditure (in billions of dollars) on education for four consecutive years, together with the rate of inflation for each year.
(a) Taking 1994 as the base year, work out the index numbers of the nominal data given in the second row of the table.
(b) Find the values of expenditure at constant 1994 prices and hence recalculate the index numbers of real government expenditure.
(c) Give an interpretation of the index numbers calculated in part (b).

Karan Sood

Numerade Educator

03:59

Problem 20

Index numbers associated with the growth of unemployment during an 8-year period are shown in Table 3.14.
(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 $8 ?$

Stephanie Carter

Numerade Educator

01:49

Problem 21

(Excel) Table $3.15$ shows the annual salaries (in thousands of dollars) of four categories of employee during a 3-year period. In year 0 the firm employed $24,250,109$ and 7 people of types $A, B, C$ and D, respectively. Calculate the Laspeyre index of the total wage bill in years 1 and 2 taking year 0 as the base year. Comment briefly on these values.

Carson Merrill

Numerade Educator

00:46

Problem 22

(Excel) In the Laspeyre index, the quantities used for the weights are those of the base year. If these are replaced by quantities for the current year, then the index is called the Paasche index. In Problem 21, suppose that the numbers of employees of types $A, B, C$ and $D$ in year 1 are $30,240,115$ and 8 respectively. For year 2 the corresponding figures are $28,220,125$ and 20 . By adding two extra columns for the products $P_{1} Q_{1}$ and $P_{2} Q_{2}$ to the spreadsheet of Problem 21, calculate the Paasche index for years 1 and 2 . Compare with the Laspeyre index calculated in Problem $21 .$

State one advantage and one disadvantage of using the Laspeyre and Paasche methods for the calculation of combined index numbers.

Alejandro Ruiz

Numerade Educator