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:
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
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
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
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
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.
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:
If the electrician spent x hours on a particular job then
the total charge C in dollars would be
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.
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 (32ºF), and the boiling point at 212ºF, 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 (0ºC), 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 C° for the Celcius temperature and F° for the Fahrenheit temperature.
Lets check and see if our formula is correct.
Convert 32°F to Celcius using the formula. The value of F° is 32.
F° = 32 Substitute the value of F° 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 F° is 212.
F° = 212 Substitute the value of F° into the formula.
Great - this conversion formula is doing well. Finally convert 91°F to Celcius using the formula. The value of F° is 91.
F° = 91 Substitute the value of F° into the formula.
This is good, our thermometers agree with our calculation, since a thermometer only reads to the nearest degree.
Suppose a car is travelling at a constant speed of 100 km/hr:
The relationship between the total distance travelled, d, and the time in hours, t, can be written as a mathematical formula.
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.
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
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: