BUSINESS MATHEMATICS

Currency Conversion

The buying rate for the forex bureau is the selling rate for the customer and the selling rate is the buying rate for the customer.

Examples:

1. The Midland Forex Bureau is offering the following rates for pounds sterling.

Buy at Ush 2,600/=

Sell at Ush 2,50o/=

a) How many pounds would you get for:-

i) 30,000/= ii) 450,000/= iii) 20 millions?

b) How much shillings would you get for:-

i) £200 ii) £10,000 iii) £500,000?

2. John paid for a mattress worth shs. 52,500/= and its cloth cover costing Shs. 36,500/=

Given that U.S $ 1. = Ush 1,800/= and £ 1 = Ush. 1.4, Calculate in pounds sterling:

i) the price of the mattress

ii) the total cost of the mattress and its cover

3. Convert Us $ 400 (US dollars) to pounds sterling, (£), if Us$1 = Ugs. 1,800/= and £1 = Ush 2,500/=.

INTEREST

Simple Interest: Simple interest = P x R x T

100

Where P is the principal amount;

R is the rate and

T is the time

Example1.

a) Karu invested Shs. 7,200/= at 12% simple interest for 3 years. How much interest did he earn?

b) Find the amount of money Karu had after this time.

2. Find the simple interest earned on Shs. 12,000/= in 4 years at a rate of 15%.

3. Find the time required for Shs. 7,260/= to earn Shs. 2,904/= simple interest at a rate of 8% per annum?

4. Find the rate at which Shs, 800,000/= should be deposited in order to earn Shs. 150,000 simple interest in 3 years.

5. A man borrowed Shs. 200,000/= from the Bank at the simple interest rate of 2.5% per annum. He paid back the money in 24 equal monthly instalments over a period of two years. How much money did he pay every month?

Compound Interest

If we leave the interest in our bank account when the bank pays us, that interest becomes part of the principal the following year and starts earning interest as well. Whn the interest is added on to the principal in this way we say the money has been invested at compound interest.

Example 1: Calculate

a) the amount

b) the interest on shs. 45,000/= invested for 2 years at 8% compound interest.

Solution: Interest after 1 year = PRT

100

= Shs/ 45,000 x 8 x 1

100

= Shs. 3,600/=

New amount = 45,000 + 3,600/=

= 48,600/=

Note: (1 + 8/100) 48,600 = 108 x 48,600

100

= Shs. 52,488/=

Conclusion:

The Interest after one year is P x R x 1 = PR

100 100

The amount after one year is P + PR = P(1 + R/100)

100

The amount is multiplied by 1 + R/100 each year. So after 2 years the amount is P(1 + R/100)².

So after 3 years the amount is P(1 + R/100)³.

Therefore the amount A after n years is P(1 + R/100)ⁿ.

2. Calculate the amount to be repaid on the following loans given that the interest rate in Compound interest.

a) Shs 250,000/= at 2% p.a for 6 years,

b) $ 4000 at 3% per month for a year.

c) £ 750 at 12½% p.a for 2 years.

d) 1.3 million shillings at 23% p.a for 10 years.

3. Mr. Kapere deposited Shs. 2,421 million into his Savings account at the bank at Compound interest rate of 8,5% per annum. Determine the number of years his money will take to accumulate to shs. 2.85 millions.

DEPRECIATION

An asset loses value calculated as a proportion of its value at the beginning of the period.

Example 1:

Calculate the value of a car that cost £16,000 and has depreciated at 5% p.a for 4 years.

2. Find the value of a microwave oven that cost $ 1,200 and has depreciated at 4% p.a for 3 years.

3. Calculate the value of a piece of machinery that cost £120 and has depreciated at 10% p.a for 7 years.

TAXATION

The amount of tax paid depends on the amount of money earned. Before any tax is paid, however, there are allowances deducted from the amount of income and then income tax is taken from the remainder. These allowances such as transport, health, children’s allowance, married allowance – (generally personal allowances).

In some countries there is an allowance for pension and life assurance contributions. The total income is called the gross income. After the allowances have been deducted from this income, what is left is called the net income or taxable income.

Example 1. The table below shows how tax is levied on the taxable income of working citizens in a certain country.

Income (shs) per annum	Tax rate (%)
i) first 80,000/=	7.5
ii) 80,001 – 160,000 (next 80,000)	12.5
iii) 160,001 - 240,000 (next 80,000)	20.0
iv) 240,001 - 320,000 (next 80,000)	30.0
v) 320,001 - 400,000	36.5
vi) 400,001 - 480,000	45.0
vii) Anything over 480,001	52.6

A man’s gross annual income is Shs. 964,000. His allowances including insurance were:

- Housing Shs. 14,500 per month

- Marriage; one tenth of his gross annual income

- Medical shs. 50,700/=

- Transport Shs. 10,000 per month

- An insurance premium of shs. 68,900/- per annum

- Family allowances for up to four children as follows:

Shs. 3,400/= for each child above the age of 18

Shs.4,200/= for each child above 10 but below 18 years

Shs. 5,400/= for each child below 9 years old.

Given that he has a family of five children with three of them below the age of 8, one aged 16 and the eldest child 20 years old, determine:

a) his taxable income

b) the income tax he pays annually as a percentage of his gross annual income.

Solution

a)	Gross income	=	964,000/=
Less	- Housing (12 x 14,500)	=	174,000/=
	- Marriage allowance	=	96,400/=
	- Medical	=	50,700/=
	- Transport (12 x 10,000)	=	120,000/=
	- Insurance premium	=	68,900/=
	- Child allowance (3 under 8)	=	16,200/= (5,400 x 3)
	- Child allowance (16 years old)	=	4,200/=
	- Child allowance (above 18)	=	3,400/=
Difference =	(964,000 - 533,800)	=	430,200/=

Therefore the taxable income is 430,200/=

c) Income Tax

1	First 80,000	=	7.5% x 80,000	=	6,000/=
2	Second 80,000	=	+12.5% x 80,000	=	10,000/=
3	Third 80,000	=	+20% x 90,000	=	16,000/=
4	Fourth 80,000	=	+30% x 80,0000	=	24,000/=
5	Fifth 80,000	=	36.5% x 80,000	=	29,200/=
6	Remainder	=	45% x (430,200 - 400,000) (45% x 30,200)	=	13,590/=
			Income Tax	=	98,790/=

Therefore Percentage = 98,790 x 100 = 10.2%

Of tax 964,000 of the Gross income.

Example 11

In a certain country, income tax is levied as follows:

Taxable Income(Shs.)	Rate %
0 - 10,000	10
10001 - 20,000	25
20001 - 30,000	30
30001 - 40,000	45
Anything over 40001	50

A person’s monthly gross income has certain allowances deducted from it before it is subject to taxation (this includes family relief and insurance). The allowances are as follows:-

Married man	Shs. 1,800/=
Un Married man	Shs. 1,200/=
Each Child below 11 years	Shs. 500/=
Above 11 years but below 18	Shs. 700/=
Insurance premium	Shs. 1,200/=

Peter earns Shs. 64,000/=. He is married with three children aged between eleven and eighteen, and two children below eleven years. Given that he is insured and has claimed transport allowance of Shs. 1,700/=, Calculate the income tax he pays under the income tax rates above.

HIRE PURCHASE

People often buy goods such as radios, bicycles and cars on credit. Often they make a down payment or deposit and pay the rest of the money in monthly installments. This method of buying goods is called Hire Purchase.

Example 1:

The following is an advertisement

GET YOURSELF A PHOTOCOPIER

CHEAPLY WHILE STOCK LASTS.

TERMS: CASH AT USH. 960,000

OR HIRE PURCHASE: DEPOSIT 15% OF

MARKED PRICE AND PAY EITHER

USH. 75,000 WEEKLY FOR 12 WEEKS

OR USH 245,000 MONTHLY FOR

4 MONTHS.

Calculate;

i) the saving a customer would make by buying the photocopier on cash terms rather than through weekly hire purchase.

ii) The percentage profit made on the monthly hire purchase if the wholesale cost of a photocopier is 17.5% below the cash price.

Solution

FURTHER EXAMPLES.

1. In a show room, the price of a car is given as Shs. 5,800,000/=. During a sale a discount of 15% is allowed.

a) How much does a customer pay for the car?

b) After the car has been bought, its value depreciates by 25% in the first year and by 20% during the second year. Find the price of the car after;

i) one year ii) two years

2. A farmer bought a machine for Shs. 2,200,000/=. If the machine depreciates at the rate of 15% per annum, find the value of the machine after two years.

3. Salesgirl Lare receives Shs. 152,000/= per month in salary plus a commission of 7% on all the sales she makes. If her daily sales average is Shs. 180,000/= for 25 days in a month that she works, find Lara’s total earnings for the month.

4. Goods worth Shs, 2,400,000/= are insured at 4% per annum for 6 years. The premium is collected in monthly installments:

a) What is the monthly insurance payment?

b) What is the total annual premium paid in this period?

5. Two business partners A and B contributed shs. 2,240,000/= and Shs. 2,560,000/= respectively, to start a business. They agreed to share the profits as follows.

30% to be shared equally

30% to be shared in the ratio of their contributions.

40% to be retained for the running of business

If their total profits for the year 1998 was Shs. 1,728,000/=,

a) Calculate the

i) Amount that was received by each partner

ii) Amount retained for running business

b) Express what B received as a percentage of his contribution tot he business.

6. In a certain country, income tax is levied as follows:

A person’s monthly gross income has certain allowances deducted from it before it is subject to taxation. This includes family relief and insurance. The allowances are as follows:

Married man Shs. 1,800/=

Un married man Shs. 1,200/= each child below 11 years shs. 500/= and each child above 11 years but below 18 years shs. 700/=. Insurance premium Shs. 1,200/=

Peter earns Shs. 64,000/= . He is married with three children aged between eleven and eighteen and he claimed transport allowances of shs. 1,700/=. Calculate the income tax he pays under the income tax rates below.

Taxable income	Rate(%)
0 - 10,000	10
10,001 - 20,000	25
20,001 - 30,000	30
30,001 - 40,000	45
40,001 - and above	50

7. A certain country’s income tax structure is such that a person’s gross monthly income has certain allowances deducted from it before it is subjected to taxation. The allowance spelt out are as follows:

Marriage allowance one-twenth of the gross monthly income.

Family relief and insurance Shs. 120,000/- per annum.

Water and electricity Shs. 12,500/= per month

Medical (Self and family) shs. 240,000/= per annum.

Transport shs. 800/= per day

Family allowance for four children only at the following rates.

Shs. 5,800/= for each child above the age of 16. Shs. 7,200/= for a child above 10 years but below

16 years and Shs. 9,000/= for a child below 10 years. Joy has a family of four children with two of

then below the age of 9, the elder child is 20 and the other 14 years. Given that she earns Shs.

680,000/=. Calculate,

a) The taxable income and the income tax she pays under the income tax rates below.

Taxable income (Shs.)	Tax Rate(%)
0 - 15,000	8.50
15,000 - 84,000	16.50
84,001 - 170,000	24.00
170,001 - 285,000	30.00
285,001 - 435,000	37.50
Above 435,000	48.50

b) Determine the percentage of her gross monthly income paid in tax.

8. A certain country’s income tax structure is such that a person’s gross monthly income has certain allowances deducted from it before it is subjected to taxation. The allowances spelt out are as follows:

Marriage allowance one-twentieth of gross monthly income.

Family relief and Insurance shs. 120,000/= per annum.

Water and electricity Shs. 12,500/= per month

Housing Shs. 35,000/=

Medical (Self and Family)Shs. 240,000/= per annum.

Transport shs.800/= per day.

Family allowances for four children only at the following rates.

Shs. 5,800/= per child above the age of 16.

Shs. 7,200/= for a child above 10 years but below 16 years

Shs. 9,000/= for a child below 10 years. Joy has a family of four children with 1 of them below the age of 9, the elder child is 20 and the other 14 years.

Given that she earns Shs. 680,000/=, calculate

a) The taxable income and the income tax she pays under the income tax rates below.

Taxable Income	Tax Rates (%)
0 - 15,000	8.50
15,000 - 84,000	16.50
84,001 - 170,000	24.00
170,001 - 285,000	30.00
285,001 - 435,000	37.50
Above 435,000	48.50

9. Mary who is not married earns 961,500/= per year. She has a personnel allowance of Shs. 68,000/= and earned income allowance of 15% of the gross income. Income tax is charged at 25% on the first Shs. 285,000/= and 30% on her remaining income. Calculate the amount of income tax due.

10. The income tax rates of a certain country are as follows.

Income (Shs.)	Rates
0 - 394,000	Tax free
394,001 - 694,000	30%
694,001 and more	36%

Find the income of a man who paid Shs. 385,200/= of box.

11. A certain amount of money was invested at compound interest at a rate of 10% for 5 years. Given that at the end of the period, the owner received Shs. 500,000/= find the amount originally deposited.

GHS	LINEAR PROGRAMMING	S.4
		AUGUST 2003

In solving linear programming problems, there are two stages:

1. To interpret the information given as a series of simultaneous inequalities and display them graphically.

2. To investigate some characteristic of the points in the unshaded solution set.

Questions

For question 1 to 3 draw an accurate graph to represent the inequalities listed, using shading to show the unwanted regions.

1. x + y ≤ 11: y ≥ 3; y ≤ x

Find the point having whole number coordinates and satisfying these inequalities which gives;

A) the maximum value of x + 4y

B) the minimum value of 3x + y

2. 3x + 2y ‹ 24; x + y ‹ 12: y › ½x; y ‹ 1. Find the point having whole number coordinates and satisfying these inequalities which gives;

a) the maximum value of 2x + 3y

b) the minimum value of x + y

3. 3x + 2y ≤ 60; x + 2y ≤ 30; x ≥ 10; y ≥ 0

a) the maximum value of 2x + y

b) the maximum value of xy

4. A boy is given $1.20 to buy some peaches and apples. Peaches cost 20c each, apples 10c each. He is told to buy at least 6 individual fruits, but he must not buy more apples than peaches.

Let x be the number of peaches he buys.

Let y be the number of apples he buys.

a) Write down three inequalities which must be satisfied.

b) Draw a linear programming graph and use it to list the combinations of fruit that are open to him

5. A girl is told to buy some melons and oranges. Melons are 50c each and oranges 25c each, and she has $2 to spend. She must not buy more than 2 melons and she must buy at least 4 oranges. She is also told to buy at least 6 fruits all together.

Let x be the number of melons

Let y be the number of oranges.

a) Write down four inequalities which must be satisfied.

b) Draw a graph and use it to list the combinations of fruit that are open to her.

6. A chef is going to make some fruit cakes and sponge cakes. He has plenty of all ingredients except for flour and sugar. He has only 2000g of flour and 1200g of sugar. A fruit cake uses 500g of flour and 100g of sugar. A sponge cake uses 200g of flour and 200g of sugar. He wishes to make more than 4 cakes all together.

7. A man has a spare time job spraying cars and vans. Vans take 2 hours each and cars take 1 hour each. He has 14 hours available per week. He has an agreement with one firm to do 2 of their vans every week. Apart from that he has no fixed work.

His permission to use his back garden contains the clause that he must be at least twice as many cars as vans.

Let x be the number of vans sprayed each week

Let y be the number of cars sprayed each week.

a) Write down three inequalities which must be satisfied.

b) Draw a graph and use it to list the possible combinations of vehicles which he can spray each week.

8. The manager of a football team has $100 to spend on buying new players. He can buy defenders at $6 each or forwards at $8 each. There must be at least 6 of each sort. To cover for injuries he must buy at least 13 players all together.

Let x represent the number of defenders he buys and

Let y represent the number of forwards.

a) In what ways can he buy players?

b) If the wages are $10 per week for each defender and $ 20 per week for each forward, what is the combination of players, which has the lowest wage bill?

9. A tennis-playing golfer has $ 15 to spend on golf balls (x) costing $1 each and tennis balls (y) costing 60c each. He must buy at least 16 altogether and he must buy more golf ball than tennis balls.

a) What is the greatest number of balls he can buy?

b) After using them, he can sell golf balls for 10c each and tennis balls for 20c each. What is his maximum possible income from sales?

10. A travel agent has to fly 1000 people and 35,000kg of baggage from London to Paris. Two types of aircraft are available: A which takes 100 people and 2000kg of baggage, or B which takes 60 people and 3000kg of baggage. He can use no more than 16 aircraft altogether. Write down three inequalities which must be satisfied if he uses x of A and y of B.

a) What is the smallest number of aircraft he could used?

b) If the hire charge for each aircraft A is $10,000 and each aircraft B is $12,000, find the cheapest option available to him.

c) If the hire charges are altered so that each A costs $10,000 and each B costs $20,000, find the cheapest option now available to him.

11. A farmer has to transport 20 people and 32 sheep to a market. He can use either Fiats (x) which take 2 people and 1 sheep, or Rolls Royces (y) which take 2 people and 4 sheep. He must not use more than 15 cars altogether.

a) What is the lowest total numbers of cars he could use?

b) If it costs $10 to hire each Fiat and $30 for each Rolls Royce. What is the cheapest solution?

12. A shop owner wishes to buy up to 20 televisions for stock. He can buy either type C for $150 each or type B for $300 each. He has a total of $4500 he can spend. He must have at least 6 of each type in stock. If he buys x of type A and y of type B, write down 4 inequalities which must be satisfied and represent the information on a graph.

a) If he makes a profit of $40 on each of type A and $ 100 on each of type B, how many of each should he buy for maximum profit?

b) If the profit is $80 on each of type A and $100 on each of type B. How many of each should he buy now?

13. A farmer needs to buy up to 5 cows for a new herd. He can buy either brown cows (x) at $50 each or black cows (y) at $80 each and he can spend a total of no more than $1600. He must have at least 9 of each type.

On selling the cows he makes a profit of $50 on each brown cow and $60 on each black cow. How many of each sort should he buy for maximum profit?

14. The manager of a car park allows 10m²_of parking space for each car and 30m² for each lorry. The total space available is 300m². He decides that the maximum number of vehicles at any time must not exceed 20 and he also insists that there must be at least as many cars as lorries. If the number of cars is x and the number of lorries is y, write down three inequalities which must be satisfied.

a) If the parking charge is $1 for a car and $ 5 for a lorry, find how many vehicles of each kind he should admit to maximize his income.

b) If the charges are changed to $2 for a car and $3 for a lorry, find how many of each kind he would be advised to admit.

15. A factory manager is planning to buy two types of machines. Type X needs 3m² of floor space, type Y needs 2m² and she has 40m² available. The cost of type X is £20, of type Y £100 and she can spend up to £900. Find the greatest number of machines she can buy.

16. A farmer is going to sow oats and barley. He estimates that oats need 4 men per hectare, barley 6 men and he has 26 men available. Oats cost £12 per hectare, barley £8 per hectare. And he is prepared to spend up to £48. Find the greatest possible area of land he can sow.

17. A dealer is going to buy radio and television sets. She intends to buy a total of 100 sets. A radio set costs £40, a television set £120 and she is prepared to spend £10400. The profit on a radio set is £16, on a television £32. Find the maximum profit she can make.

18. Two foods A and B contain respectively 4 and 6 units of protein per kg and 5 and 3 units of starch per kg. The cost of A is 40p a kg, of B is 50p a kg. If the minimum daily intake is to be at least 16 units of protein and 11 units of starch, what is the cheapest way of meeting those conditions?

19. A shepherd is planning a sheepfold using an existing wall as one side of the sheepfold. He will use hurdles 2m in length and there will be no overlap of hurdles. The breadth of the sheepfold must be greater than 20m, and the length greater than 40m. He has 45 hurdles. List the possible lengths and breadths of the sheepfold, and find which has the greatest area.

20. A tobacconist proposes to buy up to 500 pipes. He has the choice of two kinds, one at £1 each and the other at £3 each, and he can spend up to £1100. The profit on a £3 pipe is twice that on a £1 pipe. How many of each kind should he buy to make as large a profit as possible?

21. The number of units of vitamins A and B per kg of two breakfast cereals X and Y are shown in the table.

Vitamin A

Vitamin B

The minimum daily intake required is 120 units of A and 40 units of B. What is the least total weight of breakfast cereal a man must eat to have enough of these vitamins?

22. A woman has 7 guests in her house and is prepared to spend £2 on papers for them. The Financial Gazette costs 20p and The sporting Record costs 25p. She wishes each guest to have at least one of these papers, but obviously, no guest wants more than one copy of either paper. Six guests she knows insist on The Financial Gazette. List all possible solutions to her problem of how many of each paper to buy.

23. A factory manager wishes to install two types of machines, small and large. Small machines need 2 operators and occupy 4m² of floor space: large machines need 3 operators and 8cm² of floorspace. There are up to 56 operators available and 136m² of floorspace. The profit per week is £3 on a small machine and £5 on a large. Find the greatest weekly profit.

24. The Post Office is planning to sell books of stamps containing only 6p and 8p stamps. The cost of the book must not exceed £2, and must contain at least 30 stamps. What is the greatest umber of 8p stamps there could be in the book? If the stamps are printed in multiples of four, and there must be some 8p stamps in the book, how many of each can be included in the book?