Maths - GCSE

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Percentages

  • Finding the percentage of something - with a calculator: ( / = divide)

Example: Find 23% of £62.50.

1. 23 / 100 x 62.50 = £14.38

 

  • Finding the percentage of something - without a calculator:

Example: Find 35% of £80.

1. 10% is £8.     2. 30% is 3 lots of 8 = £24    3. 5% is half of 8 = £4

Answer - 35% is 24 + 4 = £28.

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Percentages

  • How to change a fraction to a percentage - with a calculator:

Example: What is 17 out of 83 as a percentage?

1. 17 / 83 x 100 = 20.5%

  • How to change a fraction to a percentage - without a calculator:

Example: What is 7 out of 20 as a percentage?

1. We have 7 / 20 but we need to make it ? /100.

2. So 20 x 5 =100 - then multiply the top by 5 is well and you get 35 / 100.

3. This is equal to 35%.

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Fractions

  • How to add and subtract fractions with the same denominator:

Example: What is  1 / 5 + 2 / 5?

1. The top numbers are added but the bottom number stays the same - this happens when the denominator of both fractions are the same. Therefore the answer to this sum is 3 / 5. When subtracting, this process is the same, but you subtract the numerators instead of add.

  • How to add and subtract fractions with different denominators:

Example: What is 1 / 5 + 7 / 10?

1. We first need to find an equivilant fraction for 1 / 5, to make the denominator /10. The eqivilant is 2 / 10. So the new question is  2 / 10 + 7 / 10 = 9 / 10.

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Fractions

  • How to add difficult unlike fractions:

Example: What is 1 / 2 + 1 / 3?

1. The difficulty here is we cannot convert halves into thirds or thirds into halves. We have to convert both fractions. Firstly, list the first few fractions that are equivilant to a half:

1/2 - 2/4 - 3/6 - 4/8 - 5/10

Then list the first few fractions that are equivilant to a third:

1/3 - 2/6 - 3/9 - 4/12 - 5/15

2. Look for the first fraction in both lists which have the same denominators - this is 3/6 and 2/6.

Then add these fractions 3 / 6 + 2 / 6 = 5 / 6.

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Fractions

  • How to multiply fractions:

Example: What is  1 / 3 x 1 / 4?

This is a simple question - multiply the two numerators (1x1=1). Then the two denominators (3x4=12). The answer to this question is therefore 1/12.

  • How to multiply mixed number fractions:

Example: What is 3 1/4 x 2/3?

1. First write the mixed fraction as an improper fraction. This is 13/4 x 2/3. Cancel the improper fraction down by dividing the top and bottom by 2. E.g. 13 x 2 / 4 x 3. Replace the 2 with a one and the 4 with a 2. Then multiply 13 x 1 = 13 and 2 x 3 = 6 (13/6). Then change this back into a mixed number = 2 1/6.

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Area of shapes

  • How to work out the area of a parallelogram:

A parallelogram is any quadrilateral with two pairs of parallel lines.

Equation:- Area of a parallelogram = base x height

  • How to work out the area of a triangle:

Equation:- Area of a triangle = base x height / 2

  • How to work out the area of a trapezium:

Equation:- Area of a trapezium = (a + b) / 2 x h

  • How to work out the area + circumference of a circle:

Equation:- Area of a circle = 3.14 r2    C = d x 3.14

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Dividing

  • Dividing decimals without a calculator:

Example: What is 53.6 / 4?

1.

  • Dividing harder sums with decimals:

Example: What is 56.7 / 0.14?

1. Treat this like a fraction - 56.7/ 0.14.

2. Then get rid of the decimals by multiplying top and bottom by 100. This gives us an equivalent, decimal - free division, of 5670/ 14. Then replace the question with this sum of 5670 divided by 14. Do the bus stop method for this (as shown above) and the answer is 405.

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Negative numbers

  • How to add and subtract negative numbers:

Rules:-

1. If two signs are the SAME replace with a + sign.

2. If two signs are DIFFERENT replace with a - sign.

Examples:

1. 3 - -2 = 5

2. 6 + -3 = 3

3. 5 - (3 - 6) = (Do what's in the bracket first - 3 - 6 = -3) Then 5 - -3 = 8.

4. 2 + +2 = 4

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Negative numbers

  • How to multiply and divide negative numbers:

Rules:

1. If two signs are the SAME the answer has a + sign.

2. If two signs are DIFFERENT the answer has a - sign.

Examples:

1. -3 x -2 = 6

2. 10 / -2 = -5

3. 3 x -2 = -6

4. -12 / -2 = 6

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Rounding - significant figures

  • How to round numbers to a significant number:

Example: Write the number 5.47651 to 1 significant number.

1. The first significant number is 5, the second 4, third 7 and so on. Therefore, the answer to this question is 5. The decider for this question is 4 - the next number along. Because it is 4 or less, the answer stays as it is and does not go up.

Example: Give 0.0045902 to 3 significant figures.

2. Ignore the zeros at the beginning - therefore the first significant number is 4, the second is 5 and so the third is 9. The decider is 0 and because it is 4 or less, the number stays as it is and so the final answer is 0.00459 to 3 significant figures.

Example: Round 30 895 to 2 significant figures.

3. The second significant number is 0 and the decider number is 8. This is above 5 and so the number is rounded up.This makes the final answer of 31 000.

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Exam questions

Estimate the answer to the following calculation. Show the values you use. (/= divide)

                                                            29 / 11.44 x 0.48

First, round each number to one significant number. Starting with 29 - rounded is 30. 11.44 rounded to one significant figure is 10 and 0.48 rounded to 1 significant figure is 0.5. Then you need to put these numbers into the calculation. 10 x 0.5 = 5 and 30 / 5 = 6.

Ali has been working at CGP for 2 years, Bob 5 years and Clarry for 6 years. Their manager decides to give them a bonus in direct proportion to their years of service. If Bob gets a bonus of £750, work out the bonuses recieved by Ali and Clarry.

This is a ratio question in disguise. First, write out the names and the ratios - Ali    Bob    Clarry to the ratio 2 : 5 : 6. We know that 5 parts is equal to £750, so to find 1 part, you need to do 750 divided by 5, which is £150. Ali will get 2 x £150 = £300. Clarry will get 6 x £150 = £900.

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Fractions

  • How to find a fraction of a number:

Example: What is 5/8 of 16?

1. This just means 5/8 x 16. (OF means x). Then we do a trick with the number 16 - turn it into 16/1. This means that you now have the sum of 5/8 x 16/1. This is now just simple multiplication. But remember to cancel down if asked to. The final answer is 10/1, which is just 10.

  • How to divide fractions:

Example: What is 1/3 divided by 1/2?

1. The first step is to turn the second fraction upside down, to make the new sum 1/3 x 2/1. The final answer is therefore 2/3.

Example: What is 5/8 divided by 3?

1. Turn the 3 into a fraction - 3/1. Then follow the same steps as before by turning the fraction upside down - 1/3 and then multiply.

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Fractions

  • How to divide fractions with mixed numbers:

Example: What is  4 1/6 divided by 3/4?

1. First, turn the mixed number into an ordinary fraction. Do this by multiplying 4 x 6 = 24 and then adding the one to make 25/6. This is what we use instead, making the new sum 25/6 divided by 3/4. Then turn the second fraction upside down and multiply - the answer is 50/9. But you can change this into a mixed number. You can do this by 50 divided by 9 which is 5 r5. So you put down the 5 as your whole number and 5/9. The final answer is 5 5/9.

  • How to order fractions:

Example: Put these fractions in order of size, from smallest to largest

3/4  5/8  2/3  5/6 - the first thing to do is find a common denominator for all of these fractions.To find this, you need to find the lowest common multiple of 4, 8, 3 and 6.In this case it is 24. Then convert the top numbers - the final fractions are 18/24, 15/24, 16/24 and 20/24.Then put them in order - smallest to largest. Therefore the final order is 5/8, 2/3, 3/4 and 5/6.

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Ratio

  • Ratio basics:

A ratio is a way to compare numbers. Basically, a ratio tells you: The amount of one thing compared to the amount of another thing.

You can simplify a ratio by dividing both sides by the same number OR your calculator can help by pressing the first number (then the fraction button) then the second number and pressing =.

Example of an exam question: Write the ratio 8 : 12 in the form 1 : n.

1. To do this, you need to divide it all by 8 - so you get 1 : 1.5.

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Ratio

  • How to divide in a given ratio:

Example: Ali and Bob are sharing out the fish below. The ratio of Ali's share to Bob's share is 2 : 3. There are 10 fishes altogether. How many fish do they each get?

1. In order to answer this question, it is always easier to divide the fish into equal groups. In this case, divide the fish into five equal groups of 2. Then divide the fish this way. So Ali will have 2 groups of this and Bob will have 3. Therefore, Ali will get 4 fish and Bob will get 6 fish.

Example: Ali and Bob are sharing out the 21 coins below. The ratio of Ali's share to Bob's share is 3 : 4. How many coins do they each get?

1. First imagine that there were 7 coins - Ali would get 3 coins and Bob would get 4. So what you need to do is divide the 21 coins into equal piles/ groups - there are 3 coins in each pile. So Ali gets three of the piles and Bob gets 4. Therefore, Ali gets 9 coins and Bob gets 12.

Example: Ali and Bob have won £30 on the lottery. They want to split their winnings in the ratio 2 : 1. How much does each person get?

1. Divide £30 by 3. (Always divide the total amount into parts by adding the ratio together). = £10. Then multiply by each share - 2 x 10 = £20 (This is what Ali gets). And 1 x 10 = £10. (This is what Bob gets.

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Ratio

The rule for dividing in a given ratio is:-

Divide into 'parts', then multiply by each share

  • To recap overall:
  • Find the number of 'parts'... by adding the numbers in the ratio.
  • Find the size of one part... by dividing the total amount by the number of parts.
  • Multiply each side of the ratio by the size of one part.

Another example: The ratio of pork to beef is 3 : 5. The total number of pieces of meat is 48. How many pork chops are there?

1. Divide 48 into 8 parts (because 3 + 5 = 8) 1 part = 48 divided by 8 = 6.

2. Multiply 1 part by the 'share' for pork chops. Pork chops make up 3 parts = 3 x 6 = 18. There are 18 pork chops altogether.

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Algebra - the basics

  • Algebra - using letters to stand for numbers.
  • Term -  a collection of numbers and letters multiplied or divided by each other. E.g. 5f, 6g.

In a term, you never write the 'times sign'. E.g. xy = x (x) y    2ab = 2 x a x b   p2 / q = p x p / q.

  • Expression - a group of terms that are added or subtracted. E.g. 5f - 2ab or p2 + 8.
  • Equation - terms or expressions joined by an equals sign. E.g. 3a + 2.
  • Formula - an equation that tells you how to work something out. E.g. s = d / t.
  • Like terms - terms that are similar. E.g. 3c + 2c = 5c or 4 - c + 3 = 7 - c.

Example of harder sums : bc + 3c -1 +3bc - 4. The first thing to do is work out the type of term. So 'bc' is a 'bc' term, +3c is a 'c' term, -1 is a number term, +3bc is a 'bc' term and -4 is a number term. Then collect the 'like terms': = 4bc + 3c - 5.

  • Recap - First circle all the terms (including the plus or minus sign in front). Then Add up or subtract the 'like terms'. As you go, remember to cross out the terms you have dealt  with.
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Algebra - expanding the brackets

  • How to expand single brackets:

Example: 3 (a + 2). You need to multiply everything in the bracket by 3. So the answer is 3a + 6.

Example: -5 (2x + 1). You need to multiply everything in the bracket by -5. So the answer is -10x -5.

  • How to expand double brackets:

Example: (2y - 4) (3y + 1). You need to multiply everything in the first bracket by everything in the second bracket. So you do 2y multiplied by 3y, then you multiply 2y by +1, then you multiply -4 by 3y and then -4 multiplied by +1. There is an acronym to remember this - FOIL = First - Outside - Inside - Last. So the answer is 6y2 + 2y - 12y  - 4. You can also simplify this by doing 12y - 2y = 10y. So the simplified answer would be 6y2 - 10y - 4.

Example: (4a - 5) (2a - 3). By using the rule of FOIL - you multiply 4a by 2a, 4a by -3, -5 by 2a and   -5 by -3. So the answer is 8a2 - 12a - 10a + 15. Or simplified is 8a2 - 22a + 15.

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Algebra - expanding the brackets (2)

How to expand squared brackets:

Example: (3x + 2) 2. Most people on this question would square the individual numbers and get the answer of 9x +4 but this is wrong. Squaring something means multiplying a number by itself. In this case a bracket - this question actually means (3x + 2) (3x + 2). Then you just expand the brackets in the normal way. So the final answer is 9x2 + 6x + 6x + 4. Simplified is 9x2 + 12x + 4.

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Algebra - Factorising

  • Factorising is the opposite of expanding brackets - instead of getting rid of the brackets, you add them in.

Example: 5x + 5. The first step is to find a number that both terms divide by / go into. In this case, the answer is 5. Put this number on the outside of the pair of brackets. E.g.  5 (       ). Then you need to work out what goes in the inside of the brackets. To do this; ask the question.... 5 multiplied by what gives 5x. This is x. Then, 5 multiplied by what gives + 5. This is + 1. Giving the final answer 5 (x + 1).

Example: 2x - x2. The number that both of these divide by / go into is x. So put this on the outside of the brackets. The final answer is x (2 - x).

Method to factorising:

1. Take out the biggest number that goes into all terms.

2. Take out any letters that are in all the terms.

3. Inside the brackets, put what will multiply back to the original numbers.

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The Power Rules

The 10 power rules:

Rule 1:- When multiplying, you add the powers.

Rule 2:- When dividing, you subtract the powers.

Rule 3:- When raising one power to another, you multiply the powers.

Rule 4:- Anything to the power 1, is just itself.

Rule 5:- Anything to the power 0, is 1.

Rule 6:- 1 to any power is just 1.

Rule 7:- With fractions, apply the power to the top and bottom.

Rule 8:- Negative powers mean 1 / (over) positive power.

Rule 9:- Fractional powers are roots.

Rule 10:- Two-stage fractional powers are a power and a root.

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