Circumferance= Pi x Diameter
A taxi firm charges 50p per mile plus a fixed charge of £2.00. Write down a formula for the cost (C) of hiring this taxi to travel ‘n’ miles.
- Travelling 1 mile would cost £2 + 50p.
- Travelling 2 miles would cost £2 + 2 x 50p.
- Travelling 3 miles would cost £2 + 3 x 50p.
So travelling for 'n' miles will cost £2 + n x 50p.
The formula is C = £2 + (n x 50p)
The formula for finding the circumference of a circle is C = 2 x Pi x r. So it is easy to find the circumference if we know the radius. What happens, though, if we know the circumference but want to know the radius? In this case we can rearrange to make r the subject of the formula.
C = 2 x pi x r, so we divide both sides by 2 x pi, C over 2 x pi = r.
Collecting like terms: To simplify an expression, we collect like terms. Like terms include letters and numbers that are the same.
Look at the expression 4x + 5x -2 - 2x + 7
- The x terms can be collected together to give 7x.
- The numbers can be collected together to give 5.
So 4x + 5x -2 - 2x + 7 simplified is 7x + 5.
Different terms: To answer some exam questions you will have to simplify an expression that has many different terms or letters.
Have a look at this typical exam question. You will notice that there are three different terms in this question: x, y and z.
Multiplying out Brackets
Remember that a term outside the bracket (or sometimes in another bracket) is multiplying everything inside the bracket.
You also need to remember some basic algebra shorthand:
2a means 2 times a.
ab means a times b.
a2 means a times a.
Have a look at this example:
- Multiply out: 3(4x - 7)
- First multiply: 3 × 4x = 12x
- Then multiply: 3 × -7 = - 21
- Therefore: 3(4x - 7) = 12x - 21
Expressions that use negative numbers + Multipying
Multiply out the expression: -3(2n - 8)
First, multiply -3 × 2n = - 6n; Then multiply -3 × -8 = 24; -6n + 24
Therefore, -3(2n - 8) = - 6n + 24
When we multiply out a pair of brackets, everything in the second bracket has to be multiplied by everything in the first bracket.
One way to multiply out a pair of brackets is by taking each term in the first bracket and multiplying it against the second bracket.
Multiply out these two brackets: (x + 4) (x + 3)
- Multiply everything in the first bracket by the second bracket:
- x (x + 3) + 4 (x + 3)
- = x2 + 3x + 4x + 12
- = x2 + 7x + 12
Brackets + Powers
If you are asked to multiply out an expression such as (a - 5)2, remember that squaring means multiplying by itself.
Multiply out (a - 5)2
Write down the expression: (a - 5)2
Expand the expression by writing it out: (a - 5) (a - 5)
Everything in the first bracket is multiplied by the second bracket: a(a - 5) - 5(a - 5)
Expand the brackets, by multiplying everything inside the brackets by the term outside the bracket:
1: a × a = a2 2: a × -5 = -5a 3: - 5 × a = -5a 4: - 5 × - 5 = 25
Expression: a2 - 5a - 5a + 25, Simplified: a2 - 10a + 25