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## Constructing Mathematical Formulae

A mathematical formula is an equation expressing one variable as a combination of other variable(s) using algebraic operations such as add, subtract, multiply, divide, raise to a power, take the natural logarithm or take the cosine, or some combination of operations.

### Filling the tank

We all know how to figure out the cost of our petrol purchase.  Just multiply the cost per litre by the number of litres.  For example, if the price of petrol is \$2.42 per litre, we multiply the number of litres by 2.42 to get the cost. Here are some examples of the cost of certain purchases at that price:

 Quantity (litres) Cost (dollars) 10 2.42*10 25 2.42*25 30 2.42*30 x 2.42*x The things which change (are variable) are the quantity and the total cost.  We can write a mathematical formula to show the relationship.  The total cost in dollars, C, of x litres of petrol at \$2.42/litre is
C = 2.42x

As you know, petrol price fluctuates. So this formula could not be used if the petrol price changes, say to \$2.51/litre. The formula would then be
C = 2.51x

We can write a very general mathematical formula to take into account the fluctuating (variable) cost per litre. If the price of petrol is p dollars per litre, then the total cost in dollars, C, of x litres is
C
= px

This formula can then be used to calculate any amount of petrol bought at whatever price charged.

For example, if the petrol price is \$2.75/litre, to purchase 28 litres of petrol, p =2.75 and x = 28
C = px = 2.75 * 28 = 77
So the total cost is \$77.00.

It is important to note that C = 2.42x and C = px are both formulas while C = 2.75 * 28 is not. All equations have one variable C. The first two equations have extra variables and are formulas whereas the third equation, C = 1.75 * 28, does not.

### Electrician Charges

An electrician charges \$25 for a call-out within the city limit and \$65 per hour on the job. The \$25 is a fixed cost regardless of how much time the electrician spends on the job. To work out the charge he needs to multiply the number of hours by his hourly rate then add the fixed charge.  Here are some examples of total costs for calling this electrician:

 Hours on the job Total costs 2 2*65+25 = 155 3 3*65+25 = 220 4 4*65+25 = 285 5 5*65+25 = 350 x x*65+25 or 65x + 25

If the electrician spent x hours on a particular job then the total charge C in dollars would be
C
= 65x + 25. This is a mathematical formula for the cost C.

This formula can be used to calculate the cost of using this electrician, by replacing x (substituting into x) the number of hours spent on a job.

For example if the electrician spent 1.5 hours on the job, then x = 1.5. Substituting this value into the formula, C= 65x + 25 = 65*1.5 + 25 = 76.50, and the total cost will be \$76.50.

### Temperature Conversion

Two commonly used temperature scales are Fahrenheit and Celsius. The Fahrenheit scale was invented by the German scientist Daniel Gabriel Fahrenheit (1686 -1736), who set the freezing point of water at 32 degrees Fahrenheit (32�F), and the boiling point at 212�F, 180 degrees apart. The Celsius scale was invented by the Swedish astronomer Anders Celsius (1701-1744), who set the freezing point of water at 0 degree celsius (0�C), and the boiling point at 100�C. Celcius and Fahrenheit temperatures can be related by a mathematical formula.  In words, take the Fahrenheit temperature and subtract 32.  Multiply this answer by 5. Now divide the answer by 9  This gives us the Celcius temperature. Now we can write a mathematical formula for this.  First we need a couple of variables, one for each temperature.  We'll use for the Celcius temperature and for the Fahrenheit temperature. Lets check and see if our formula is correct.

Convert 32°F to Celcius using the formula. The value of is 32.

= 32   Substitute the value of into the formula. Good, the formula correctly converts the freezing point of water. Now check the boiling point of water.  Convert 212°F to Celcius using the formula. The value of is 212.

= 212   Substitute the value of into the formula. Great - this conversion formula is doing well.  Finally convert 91°F to Celcius using the formula. The value of is 91.

= 91   Substitute the value of into the formula. This is good, our thermometers agree with our calculation, since a thermometer only reads to the nearest degree.

### How far?

Suppose a car is travelling at a constant speed of 100 km/hr:

 Time (hours) Distance travelled (km) 1 100*1 = 100 2 100*2 = 200 3 100*3 = 300

The relationship between the total distance travelled, d, and the time in hours, t, can be written as a mathematical formula.

See if you can complete the formula:   d =
Remember to use computer notation.

If the speed is changed to 80 km/h, then   d =

In general we can write a formula to express the distance travelled, d, in terms of the constant speed, v, and the time, t.
See if you can do it.   d =

### Income Tax

If you have ever calculated your tax to be sure Inland Revenue got it right, you may have used the calculation tables in the tax guide to calculate your tax.  Their tables show the tax for every possible income.  It is much easier to use a formula.  According to the NZ Inland Revenue Department (IRD) 2005 IR3 Tax Guide, if a person’s income for the year is between \$38,000 and \$60,000 inclusive, his/her income tax will be \$7,410 plus 33% for each dollar in that bracket.

 Income (\$) Amount taxed at 33% (\$) Total tax (\$) 40000 40000 - 38000 7410+(40000 - 38000)*0.33 48000 48000 - 38000 7410+(48000 - 38000)*0.33 50000 50000 - 38000 7410+(50000 - 38000)*0.33 60000 60000 - 38000 7410+(60000 - 38000)*0.33 I I - 38000 7410+(I - 38000)*0.33

Let I be the income, where 38000 I 60000. Then the tax, T, to pay is

T = 7410 + (I – 38000) * 0.33, which can be simplified:

T = 7410 + (I – 38000) * 0.33
T = 7410 + 0.33I - 12540
T = 0.33I – 5130

This is a formula for tax to pay, T, in terms of Income earned, I.

Now suppose Mary earned \$58,325. Her tax to pay is \$14,117.25:
T = 0.33I – 5130 = 0.33*58325 – 5130 = 14,117.25

Bill's income was \$40,000.  Enter his tax to pay:  \$