Overall Exercise

Work Book DMT 1023, Business Mathematics

Exercise 1:

1. Find the distance between these following pairs of points.

a) A(2, 0) and B(1, 8)

b) A(3, -4) and B(5, 6)

c) A(0, 3) and B(10, 12)

2. Find an equation of the line passing through the following points.

a) (1, -1) and (2, 3)

b) (2, -5) and (7, 9)

c) (-4, 9) and (11, 2)

3. Find an equation of the line passing through the given point and having the given slope

m.

a) (1, 5); m = 1

b) (3, 2); m = -2

c) (-1, 9); m = 0

4. Find the slope value for the following points.

a) (2, 4) and (3, -1)

b) (1, 3) and (5, 6)

c) (5, 5) and (10, 5)

5. Determine whether the lines through the given pairs of points are perpendicular, parallel

or neither.

a) A(-2, 5), B(4, 2) and C(-1, -2), D(3, 6)

b) A(2, 3), B(2, -2) and C(-2, 4), D(-2, 5)

c) A(-2, 4), B(2, 4) and C(1, -2), D(3, -2)

6. Determine whether the pair of lines given are perpendicular, parallel or neither.a) y = 2x + 3 ; y = 2x – 9

b) 2x – 3y = 8 ; x + 4y = 5

7. Ten books are sold at RM80. Twenty books are sold at RM60. Find the demand

equation.

8. At the price of RM25 per shirt, no shirts are being sold. If the price offered at RM40, 20

shirts are sold. Find the supply equation.

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9. Find an equation of the line that passes through the point (2, -7) and that is perpendicular

to the line x – 2y – 10 = 1.

10. Find an equation of the line that passes through the point (-5, -9) and that is parallel to

the line 4x + 3y – 2 = 20.

11. Find an equation of the line that passes through the point (11, -8) and that is

perpendicular to the line that passes through the points (3, -2) and (5, 4).

12. Every year, TNB will buy 50 units power supply regardless of the price. Sketch the

demand graph.

13. STM has to pay Telekom RM500 regardless of the number of calls made and the

duration. Sketch the supply graph.

14. The demand function is given by 4x = 30 – y. (x = quantity and y = price).

a) Find the quantity demanded if the price is at

i) RM 6

ii) RM 10

iii) RM 26

b) Find the price per unit if the quantity demanded is

i) 3

ii) 5

iii) 7

a) What is the highest price that can be offered for this product?

b) Find the quantity demanded if the product is not charged any payment.

15. The fixed cost of a product is RM 5000 and the variable cost is RM7.50. For each unit,

the qoods are sold at RM10. Find

a) the cost function,

b) the revenue function,

c) the profit function,

d) the break-even point.

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16. Find the equilibrium price and quantity for the following market:

a)

b)

c)

17. Given the line PQ: 3x – 12y + 24 = 0. Find

a) the slope of PQ,

b) x-intercept and y-intercept,

c) the value of y when x = 12,

d) the equation of a line through (2, -4) and is parallel to PQ,

e) the equation of a line through (2, -3) and is perpendicular to PQ.

18. The sales manager of Sunshine Book Store wants to plot sales versus time for the last

five years. By using the information below, plot the graph and find the equation of

the trend line. What sales figure can be predicted for the sixth and seventh year?

x, time(years) 1 2 3 4 5

y, sales(RM’000) 20 30 40 50 60

19. The supply equation for a commodity is given by 4y – 6x = 120, where y is

measured in RM and x is measured in units of 100.

a) Sketch the corresponding curve,

b) What is the lowest price at which the supplier will make the

commodity available in the market?

c) How many units will be marketed when the unit price is RM60?

20. Nippon Inc. manufactures its products at a cost of $4 per unit and sells them for $10

per unit. If the fixed cost for the firm is $12,000 per month, determine the break - even

point for the firm.

21. Using the data in (20), answer the following questions:

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a) What is the profit/loss sustained by the firm if

i) 1500 units are produced and sold per month?

ii) 3000 units are produced and sold per month?

b) What is the number of units the firm should produce in order to

realize a minimum monthly profit of $9000?

22. The management of a firm must decide between two manufacturing processes for a

certain product. The monthly cost of the first process is given by C1(x) = 20x + 10 000,

where x is the number of units produced, whereas the monthly cost of the second

process is given by C2(x) = 10x + 30 000. Both costs are measured in dollars.

a) If the projected sales are 800 units at a unit price of $40, which

process should the management choose?

b) Which process should be chose if the projected sales are:

i) 1500 units,

ii) 3000 units.

Exercise 2:

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1. A complete demand function is given by the equation

where P is the price of the good, Y is income, Pr is the price of a related good and T is a

taste.

a) Graph the demand function when Y = 8000, Pr = 8 and T = 4.

b) What type of good is the related good? (substitute or complementary good)

c) What happens to the graph, if T increases to 8, indicating greater preference for

the good?

d) Construct the graph.

2. A person has $140 to spend on two goods (X, Y) whose respective prices are RM4 and

RM6.

a) Draw a budget line showing all different combinations of the two

goods that can be bought with the given budget.

b) What happens to the original budget line

i) if the budget falls by 25%,

ii) if the price of X doubles,

iii) if the price of Y falls to RM4.

3. Given:

Y = C + I

C = 50 + 0.8Y

I = Io = 50

Find the equilibrium level of income.

4. Given:

Y = C + I + G

C = 25 + 0.75Y

I = Io = 50

G = Go = 25

Find the equilibrium level of income.

5. Given:

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Y = C + I + G

C = Co + bY

I = Io

G = Go

where

Co = 135

b = 0.8

Io = 75

Go = 30

a) Find the equation for the equilibrium level of income in the reduced form.

b) Solve for the level of income

i) directly,

ii) with the reduced form.

6. Given:

Y = C + I

C = Co + bYd

I = Io

Yd = Y – T

where

Io = 40

Co = 100

b = 0.6

T = 50

Find

a) the reduced form,

b) the numerical value of equilibrium level of income, Y.

7. Given:

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Y = C + I

C = Co + bYd

T = To + tY

Yd = Y – T

where

I = Io = 30

Co = 85

b = 0.75

t = 0.2

T0 = 20

Find

a) the reduced form,

b) the numerical value of Y.

Exercise 3:

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1. Determine the characteristic of the roots of the quadratic equations.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

2. Solve the quadratic equations using factorization method.

a)

b)

c)

d)

e)

f)

g)

3. Solve the quadratic equations using formula.

a)

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b)

c)

d)

e)

4. Solve the quadratic equations using appropriate method.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

5. Sketch the parabolas showing the intercept and the maximum/minimum points.

a)

b)

c)

d)

e)

f)

g)

6. Find the market equilibrium point for these equations:

a)

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b)

7. The monthly total revenue from a certain product sales of x unit is given as:

R(x) in RM

How many units should be sold every month to earn maximum revenue: What is the

maximum revenue?

8. The average cost function to produce x unit of a certain product is given as:

R(x) in RM

How many units should be produced to minimize the average cost? Find the minimum

cost.

Exercise 4:

1. Find (a) the 15th term,

(b) the sum of the first 15 terms,

for the following sequence

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i) 12, 17, 22,…

ii) 4, 10, 16,…

iii) 1, 3, 9,…

iv) 2, 8, 32,…

2. Find the number of terms in the following sequence

i) 5, 11, 17, …, 83

ii) 4, 8, 12, …, 40

iii) 2, 4, 8, …, 128

iv) 40, 20, 10, …., 2.5

3. Find the sum of

i) the first 50 positive integer,

ii) the first 20 positive even numbers,

iii) all odd integers from 23 to 151 inclusive.

4. The first term of an arithmetic sequence is -5 and the last term is 40. If the

common difference is 3, find

i) the number of terms in the sequence,

ii) the sum of all the terms in the sequence.

5. Ahmad has to pay RM1 at the end of the first month, RM5 at the end of the second

month, RM25 at the end of the third month and so on for eight months. Calculate the total

amount he has to pay.

6. To settle the purchase of a house, Ali has to pay a down payment of RM10 000 plus

RM2000 at the end of the first month, RM1900 at the end of the second month,

RM1800 at the end of the third and so on for twenty months. What is the total

payment?

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7. The first term of an arithmetic sequence is 4. The sum of the first eight terms is 74 and

the sum of all terms in this sequence is 365. Find the

a) common difference,

b) number of terms,

c) last term.

8. The 5th term of an arithmetic sequence is 10 and the sum of the first 10 terms is 115.

Find the

a) first term and the common difference,

b) sum of the first 20 terms.

9. 2x + 2, x + 4, and x are the first three terms in the geometric sequence with every

terms of positive number. Find the

a) value of x,

b) first term and the common ration.

10. You buy a radio and agree to pay as follows: RM 150 for the first month, RM140 for the

second month, RM130 for the third month and so on for fifteen months.

a) What is the 15th payment?

b) What is the total payment for the purchase?

Exercise 5:

1. Differentiate the following functions using first principle method:

a)

b)

c)

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d)

2. Differentiate the following function with respect to x:

a)

b)

c)

d)

e)

f)

g)

3. Find the differentiation for the following equation:

a)

b)

c)

d)

e)

f)

4. The total cost of a product in ringgit, C(x) is given as where x

is the level of output. Find

a) the marginal cost function,

b) the marginal cost when x = 2 and interpret the answer.

5. The daily total revenue of a product in ringgit, R(x) is given as

where x is the daily level of output. Find

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a) the marginal revenue function,

b) the marginal revenue when x = 2 and interpret the answer.

6. The weekly total profit in ringgit of a product, P(x) is given as where x

is the level of output per week. Find

a) the marginal profit function,

b) the marginal profit when x = 10, and interpret the answer.

7. A firm finds that the number of radios that can be sold per month at a price p ringgit

is given by the monthly demand function x = 600 – 2p where x is the number of radios

demanded. Find

a) p as a function of x,

b) the monthly revenue function,

c) the level of production at which marginal revenue is zero.

Exercise 6:

1. Find the simple interest earned and the simple amount for the following investment:

a) RM 10 000 for 5 years at 12% per annum,

b) RM 20 000 for 4 years 6 months at 10% per annum,

c) RM15 000 for 5 ¼ years at 9% per annum.

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2. A student has received RM30 000 loan from a wealthy aunt in order to finance his 4-

year college program. The terms are that the student repays his aunt in full at the

end of 8 years with a simple interest computed at a rate of 4 percent per year.

Determine the interest which must be paid on the 8-year loan.

3. RM60 000 is invested for 3 years in a bank earning a simple interest rate of 9% per

annum. Find the simple amount at the end of the investment period.

4. Paul borrowed RM480 from his friend John. After 3 years, he paid back RM642. Find

the simple interest rate that was charged on him.

5. Suppose RM1000 is invested at an annual interest rate of 5%. Compute the amount

accumulated after 4 years if the interest is

a) simple interest,

b) compounded annually,

c) compounded semi-annually.

6. A sum of RM40 000 is invested at a rate of 12% per year compounded annually. If the

amount is for a period of 5 years, what will the compound amount equal? How much

interest will be earned during the 5 years?

7. The sum of RM50 000 has been placed in an investment which earns interest of 12

percent per year, compounded quarterly. What is the total interest for the year?

8. Find the future value and compound interest on an investment account of $700

compounded quarterly at 10% for 3 years.

9. Crystal Smith deposited $4000 in a saving account paying 2% interest compounded

daily. Find the future value of the money and the compound interest she earned at

the end of 5 years.

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10. Matt bought a CD for $3000 that pays 6% interest and is compounded semiannually.

Find the future value and the interest he earned if he cashed in the CD at the end of 7

years.

11. Liza has a savings account that has a principal of $3250. If she is getting an interest rate

of 3% annually, find the future value and the interest earned if she keeps the money in

the account for 2 years.

12. Martha Moon invested $14 200 in an account that pays 6% interest compounded

monthly. Find the future value and the interest she earned on her investment if she

kept it for 8 years.

13. Wong’s Cellular Service borrowed $19 000 at 8.5% for 3 years to purchase a van.

Find the interest and maturity value of the loan as well as the monthly payment.

14. Lily invested RM100 every month for five years in an investment scheme. She was

offered 5% compounded monthly for the first three years and 9% compounded monthly

for the rest of the period. Find the accumulated amount at the end of five years.

15. The table below shows the monthly deposits that were made into an investment account

that pays 12% compounded monthly.

Year Monthly deposits

1993

1994

1995

RM100

RM200

RM300

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16. RM300 was invested every month in an account that pays 10% compounded annually for

eighteen months. Calculate the amount in the account after eighteen months.

17. James intends to give scholarship worth RM5000 every year for six years. How much

must he deposit now into an account that pays 7% per annum to provide this

scholarship?

18. Shirley wants to provide a scholarship of RM3000 each year for the next three years. The

scholarship will be awarded at the end of each year to the best student in the University

of Malaya. If money is worth 10% compounded annually, find the amount that must be

invested now.

19. Under a contract, Jenny has to pay RM100 at the beginning of each month for fifteen

months. What is the present value of the contract if money is worth 12% compounded

monthly?

20. Find the present value of an annuity of RM 500 every year for 5 years if the first payment

is made in 2 years. Assume money is worth 6% compounded annually.

Exercise 7:

1. An asset costing RM28 000 has a life expectancy of 6 years and zero salvage value.

Using the straight line method, compute:

a) the annual depreciation,

b) the book value at the end of 3 years.

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2. Construct a depreciation schedule using a straight line method for a van that costs

RM70 000 and has a salvage value of RM7000 at the end of 6 years.

3. Ah Long bought a set of furniture for RM15 000. The estimated savage value at the end

of 5 years is RM500. Find the book value of the furniture at the end of 2 years using the

straight line method.

4. An equipment costing RM2000 is purchased. Using a declining balance rate of 20%,

a) prepare a depreciation schedule for the first 3 years of use,

b) what is the book value at the end of 6 years?

5. The cost of a machine is RM28 000 and has a life expectancy of 5 years and salvage

value of RM8000. Using the declining balance method, compute the

a) annual depreciation rate,

b) book value at the end of 4 years.

6. Sofian bought a new car at RM44 000. He uses the declining balance method for

computing depreciation. If he uses an annual depreciation rate of 15%, calculate the

accumulated depreciation of the car at the end of 4 years.

7. The cost of an asset is RM15 000 and its salvage is RM2000 at the end of 5 years

of its useful life. Construct a depreciation schedule of the machine using the sum of years

digits method.

8. A new machine is purchased for RM8000. The estimated useful life of the

machine is 9 years and its salvage value is RM700. Calculate the depreciation for the

a) first year,

b) third year,

using the sum of years digits method.

9. A machine costing RM5600 has a life expectancy of 6 years with a salvage value

of RM1300. Construct a depreciation schedule using the

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a) straight line method,

b) sum of years digits method.

10. A fishing boat was bought at RM55 000. It is estimated to have a useful life of 8 years

with a salvage value of RM7000. Using the declining balance method, find

a) the annual rate of depreciation,

b) the book value of the boat at the end of fifth year.

11. Find the yearly depreciation, using the straight line method, of 10 security

uniforms costing $200 each and having a lifetime of 4 years. There is no scrap value.

12. Find the book value at the end of 60 000 miles of a bus costing $80 000 if it has an

expected lifetime of 100 000 miles and a scrap value of $8000. Use the straight line

method.

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Overall Exercise

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