GCSE Maths notes

A few notes on some of the maths topics

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Rearranging Formulae

When we change the subject of a formula, it is very important that we always do the same thing to both sides of the equation.

The question asked us to make z the subject of the formula x = y - z divided by 3. 

It is often best to remove the fraction first, so we multiply everything by 3: 

3x = 3y -  z

Add z to both sides: 

3x + z = 3y

Subtract 3x from both sides: 

z = 3y - 3x

And finally, factorise: 

z = 3 (y - x)

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When estimating the answer to a problem, we usually round each of the numbers to one significant figure (1 s.f.)

The question asked us to estimate the answer to the problem: 

(364 x 21) divided by 39

So we start by writing each of the numbers correct to 1 s.f.:   

(400 x 20) divided by 40  

= 8000 divided by 40 

= 200

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Stem and Leaf Diagrams

Stem and leaf diagrams are used to represent a set of data. When drawing or interpreting a stem and leaf diagram, it is important that you use a key so that you know exactly what is being represented.

In the question, the key tells us that the numbers represent prices (5|7 means 57p), so the data should read as:

3| 2                                          32p   

4| 1  4  4                                  41p  44p 44p

5| 2  7  7  9  9                          52p 57p 57p 59p 59p

6| 1  4                                      61p 64p

7| 0                                          70p

So we can see that the most expensive cake was 70p.

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Expressing as a Percentage

To express one quantity as a percentage of another, we first write the numbers as a fraction and then change the fraction to a percentage (by multiplying by 100).

Remember that the numbers must have the same units - if you dont convert them first youll end up with the wrong answer!

So the question express 32 cm as a percentage of 4 m should be changed to express 32 cm as a percentage of 400 cm 

32 divided by 400 x 100 = 8%

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Speed, Distance & Time

Emma ran for 30 minutes at 16 km/h. 30 minutes is the time, and 16 km/h is the speed, so we need to calculate the distance.

We use the formula

distance = speed x time

Distance = 16 x 0.5 = 8 km

Remember that the units must be the same. The speed was given in kilometers per hour, so we need to convert the time to hours (30 minutes = 0.5 hours), and give our answer in km.

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Remember that when we multiply powers of the same number, we add, and when we divide powers of the same number, we subtract.

a to the power of 2 multiplied by a to the power of 7 multiplied by a to the power of 3 is a multiplication, so we add the powers.

The powers 2, 7 and 3 can be added together: 2 + 7 + 3 = 12.

So, a to the power of 2 multiplied by a to the power of 7 multiplied by a to the power of 3  = a to the power of 12.

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Factorising Quadratics

The question asked us to factorise the expression x squared - 6x + 8

We are looking for an answer of the form (x + p)(x + q) where p + q = -6 and pq = +8.

-2 + -4 = -6 and -2 x -4 = +8

so the answer is                                

x squared - 6x +8 = (x - 2)(x - 4).

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Dividing to a Given Ratio

When dividing in a given ratio we need to know the total number of parts.

Eg. divide 40 sweets between Jo and Tom in the ratio 3:5.

Jo required 3 parts of the ratio and Tom required 5, so the total is:

3 + 5 = 8

40 divided by 8 = 5, so Jo receives 3 multiplied by 5 = 15 sweets.

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When rounding to a given number of decimal places we need to look at the digit after the decimal place that has been asked for. If this number is 5 or more, we round up and if it is less than 5 we round down.

So, when we are asked to round 6.78512 to two decimal places, we note that the digit in the third decimal place is 5, so we round up.

Therefore, 6.78512 = 6.79 (2d.p.)

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