Chapter 13 Further Applications (2)

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Further Applications (2) Contents 13.1 Euler’s Work in Algebra 13.2 Travel-graphs 13.3 Applications of Percentage and Rate

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13 Further Applications (2) 13.1 Euler’s Work in Algebra A. Introduction Definition 13.1: A prime number is a natural number greater than 1 and it can only be divided by 1 and itself. Notes: 2 is the smallest prime number and it is the only even prime number. Home Content

For example, (a) 13 is divisible by 1 and 13 only, so 13 is a prime number; (b)

12 ÷ 3 = 4 12 is divisible by 3. Therefore, 12 is not a prime number.

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13 Further Applications (2) 13.1 Euler’s Work in Algebra Definition 13.2: A composite number is a natural number which is the product of two positive integers other than 1 and itself. Notes: 1 is neither a prime number nor a composite number. If two integers are both divisible by a certain number, then that number is called the common factor of the two integers. Home Content

Consider two numbers, 8 and 12. Since both numbers are divisible by 4, we say that 4 is a common factor of 8 and 12.

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13 Further Applications (2) 13.1 Euler’s Work in Algebra Definition 13.3: A natural number is called a perfect number if it is the sum of all its factors excluding itself. Notes: The smallest perfect number is 6, the factors of 6 are 1, 2, 3 and 6. Since 1 + 2 + 3 = 6, 6 is a perfect number.

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In about 300 BC, the famous Greek mathematician, Euclid ( 歐幾里得 ) devised a simple method for generating perfect number.

If the number 2n – 1 is prime, then 2n – 1(2n – 1) is a perfect number.

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13 Further Applications (2) 13.1 Euler’s Work in Algebra In 1747, the great Swiss mathematician, Leonard Euler ( 歐拉 ), found the eighth perfect number after he had showed that 231 – 1 = 2 147 483 647 is a prime number. He also proved that all even perfect numbers are of the form 2n – 1(2n – 1). Euler also discovered some special properties of perfect numbers:

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•

Every even perfect number ends with a ‘6’ or an ‘8’.

•

All even perfect numbers are triangular numbers.

•

Every even perfect number (except 6) is the sum of consecutive odd cubes. Today, mathematicians still cannot find any odd perfect number.

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13 Further Applications (2) 13.1 Euler’s Work in Algebra B. Euler’s Formula For thousands of years, mathematicians have believed and accepted the following theorem: For any positive number k ≥ 2, there does not exist a polynomial f (n) with degree k such that f (n) are prime numbers for all natural numbers n. Euler claimed that he found a simple formula to generate prime numbers. Home Content

Euler’s formula: n2 + n + 1 is a prime number for all natural n. What will happen when substituting n = 40 to the Euler formula?

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13 Further Applications (2) 13.2 Travel-graphs Speed is a rate to measure the distance travelled by an object per unit of time.

Speed =

Distance travelled Time taken

A travel-graph can be used to show the relation between the distance travelled by an object and the travelling time. Home Content

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13 Further Applications (2) 13.2 Travel-graphs For example, the following travel – graph show the distance travelled by a train in an hour.

Home Travel-graph is also called distance-time graph.

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Fig. 13.1

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13 Further Applications (2) 13.3 Applications of Percentage and Rate A. Taxation In Hong Kong, there are three major types of tax for individuals: 1.

Salaries tax is charged on the income received from an employment.

2.

Property tax is charged on the rents received from a property.

3.

Profits tax is charged on the profits earned from a business, trade or profession,

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13 Further Applications (2) 13.3 Applications of Percentage and Rate (a) Salaries tax The salaries tax is calculated as follows: 1. Calculate the annual income. 2. Calculate the deductions and allowances to get the net chargeable income.

The allowance depends on the family composition.

Net chargeable income = Annual income – Deductions – Allowances 3. Calculate the payable salaries tax for the net chargeable income. Home Content

Notes: (a)

The annual income includes salary, commission, bonus, share option gain and so on.

(b)

The deductions include expenses of self-education, home loan interest and so on.

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13 Further Applications (2) 13.3 Applications of Percentage and Rate After deducting the allowance and deductions, the payable salaries tax is calculated on the amount of net chargeable income. The net chargeable income is split into four parts with different tax rates. The sum of all these parts is the actual amount of payable salaries tax. The salaries tax rates for the year 2004 to 2005 are given below.

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Net chargeable income

Tax rate

1

On the first $30 000

2%

2

On the next $30 000

8%

3

On the next $30 000

14%

4

On the remaining

20% Table 13.5

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13 Further Applications (2) 13.3 Applications of Percentage and Rate (a)

Property tax

Property tax is charged on rents received from a piece of land, building or flat in Hong Kong. This is computed at the standard rate of 16% for the year 2004 to 2005 on the net assessable value of the property and is charged annually. The net assessable value is calculated in the following way: Net assessable value = Annual rent – Allowance

The allowance is 20% of the annual rent. Then Home Content

In calculating the net assessable value, rates may be deducted if the owner is responsible for the payment of the rates of the property

Property tax = Net assessable value × 16%

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13 Further Applications (2) 13.3 Applications of Percentage and Rate (a)

Profits tax

Profits tax is charged on the profits earned from a business, trade or profession. The net assessable value is calculated as below: Net assessable value = Annual gross profits – Operating expenses The profits tax is calculated as below: Profits tax = net assessable value × Profits tax rate Home Content

Notes: There are two different tax rates: •

For corporations, the profits tax rate is 17.5%.

•

For unlimited company, the profits tax rate is 16%.

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13 Further Applications (2) 13.3 Applications of Percentage and Rate B. Taxi - rates The question on how to charge a taxi fare $C is a complicated problem. It consists of four parts:

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1.

The minimum charge for the fist two kilometres (M).

•

The charge for extra distance travelled after 2 km (D).

•

The charge for waiting time (W).

•

Additional charge (A). The additional charges include baggage, pets, tunnel fare, etc.

Thus, C = M + D + W + A, where M and A are constants, D depends on the extra distance travelled after 2 km and W depends on the waiting time in minutes.

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13 Further Applications (2) 13.3 Applications of Percentage and Rate The charge for distance travelled by the taxi can be represented by the following graph.

Home Content Fig. 13.18

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13 Further Applications (2) 13.3 Applications of Percentage and Rate C. Water Bills The water bill is charged at four different levels, according to the levels of consumption:

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Units of consumption

Charge per m3

On the first 12 m3

Free

On the next 31 m3

$4.16

On the next 19 m3

$6.45

Remainder

$9.05 Table 13.6

For example, if one uses 10 m3 of water, then it is free of charges; if one uses 20 m3 of water, then the charge = (20 – 12) × $4.16 = $33.28.

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