A formula is a statement in words or symbols showing the relationship between two or more variables, eg degrees Celsius and degrees Fahrenheit. Formulae are useful general statements which can be applied to different numbers on different occasions.
Example
You could write or use a formula to represent the total cost of a number of packs of magazines at £2 per pack and additional individual copies at 40p each. You would need to have a variable P for the number of packs of magazines, a variable N for the number of individual copies required, and a variable C for the cost in pounds.
So for any order of packs and individual copies, the total cost is found by:
(P x £2) + (N x 40p)
The cost of individual copies is in pence, but the cost of the packs and the total cost are both in pounds. When constructing a formula you need to make sure that all the units are the same. To keep them all in pounds, write 40p as £0.4. Then the formula is:
C = (P x 2) + (N x 0.4)
or
C = 2 P + 0.4N
To find the total cost for an order of seven packs and three individual copies, replace P and N with the appropriate numbers. In this case P is seven and N is three.
C = (7 x 2) + (3 x 0.4)
= 14 + 1.2
= 15.2
So the total cost is £15.20.
Many GCSE syllabuses are examined by testing several components which are said to be ‘weighted’. For example, in GCSE science (1999-2000 syllabuses) the practical tasks are marked out of 30 and the score, when doubled, counts towards the final mark. So here, the practical tasks have double weighting. Then a score out of three for spelling, punctuation and grammar (SPaG) is included. A formula is applied to the pupils’ marks to produce a final mark out of 63.
There are three variables: the practical mark, the SPaG mark, and the total mark.
Call the practical mark P, the SPaG mark S, and the total mark T.
The formula is:
T = (P x 2) + S or T = 2 P + S
One pupil got 24 for the practical mark and 2 for the SPaG. So,
P = 24, S = 2
and T = (2 x 24) + 2
= 50
So this pupil got a mark of 50 out of a possible 63.
Worked examples
Example one
An examination consisted of two papers. Paper 1 is marked out of 80 with a SPaG mark out of four added. Paper 2 is marked out of 120, with a further SPaG mark out of four added. The final mark is obtained by adding together the four marks and giving this total as a percentage, rounded to the nearest whole number.
In Paper one, a pupil gets 57 out of 80 and for SPaG three out of four. In Paper two he gets 78 out of 120, and for SPaG two out of four. What will his final mark be, expressed as a percentage?
There are two stages required to answer this question.
Stage one
Paper one is marked out of 80 and the SPaG mark out of four, so the maximum possible mark is 84.
Paper two is marked out of 120 and SPaG mark out of four, so the maximum possible mark is 124.
The total maximum possible maximum mark from both papers is 84 + 124 = 208.
The formula required to obtain candidates' final marks out of 208 is:
Final mark is (mark + SPaG for paper one) + (mark + SPaG for paper two)
Inserting the pupil’s marks into the formula gives:
Final mark = (57 + 3) + (78+2)
= 60 + 80
= 140
Therefore the pupil achieved 140 out of 208 or 140/208.
Stage two
Convert 140/208 to a percentage.
140 ÷ 208 = 0.67307692 = 67.31%
= 67% to the nearest whole number
Therefore the pupil's final mark is 67%.
Example two
A teacher sets her pupils three tests in year 7. She weights the marks in the second test by three and those of the third test by five. Each test is marked out of 100, so the total possible marks are 100 + 300 + 500 = 900.
Call the marks gained in the first test T1, those in the second test T2 , and in the third test T3.
The total mark T is found by the formula:
T = T1 + (T2 x 3) + (T3 x 5)
or
T = T1 + 3T2 + 5T3
A pupil gets 15 for the first test, 21 for the second test, and 58 for the third test.
T1 = 15, T2 = 21 and T3 = 58
T = 15 + (21 x 3) + (58 x 5)
= 15 + 63 + 290
= 368
So his total mark is 368 out of 900.
To turn this into a percentage:
368 ÷ 900 = 0.4088
= 40.88% or 41% to the nearest whole number.
Avoiding common errors
Most common errors can be avoided by:
- ensuring calculations are carried out in the correct order (carry out multiplication and division before addition and subtraction)
- knowing that a symbol (usually a letter) is simply a shorthand way of writing a longer statement, eg (A) represents the marks out of 65 on test one, (B) represents the number of books ordered, and
- ensuring that you know what the units are before you calculate, so you don’t confuse pounds with pence, metres with centimetres, etc.