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GCSE Grade G Number Millions Hundreds of Thousands Tens of Thousands Thousands Hundreds Tens Unit (ones) Place Value 9 6 7 4 1 0 8 Pencil and Paper Methods for Addition and Subtraction Example 2296 + 1173 Answer Rewrite the numbers in columns making sure the units are all in a column and the tens are all in a column etc 2296 + 1173 3469 The number 9674108 is shown above written on a place value chart 1 Large numbers are read in groups of three The number 9674108 is read as Always start from the units column. nine million, six hundred and seventy four thousand, one hundred and eight. When the total in a column is more than nine, you have to carry a digit into the next column on the left, as shown above. It is important to write down the carried digit or you may forget to include it in the addition. The number system we use is a decimal system. The place value of each number is ten times as large as the place value of the number immediately to the right. Example What is the place value of the 7 in these a. 675 Examples 1. 874 – 215 2. 300 – 163 Answer b. 870031 86714 - 215 659 1. Answer a. In 675 the 7 is in the tens place. We say that the place value of the 7 is ten b. In 870031 the 7 is in the tens of thousands place. We say that the place value of the 7 is tens of thousands. 2 9 1 300 - 163 137 2. Addition - ADD, PLUS, TOGETHER, SUM Always subtract the units column first. Example Add together 4 + 7 Answer 11 Subtraction - MINUS, SUBTRACT, TAKE AWAY, LESS Example 7 – 3 Answer 4 When you have to take a bigger digit from a smaller digit in a column, you must ‘borrow’ a 10 by taking one from the column to the left and putting it with the smaller digit, as shown in the examples above Pencil and Paper Methods for Multiplication and Division Multiplication Multiplication is the same as adding a number again and again For instance 4 x 3 means 4 lots of 3 or 3 + 3 + 3 + 3 Division Dividing a number is the same as sharing Example - Divide 35 by 7 is the same as sharing 35 by 7 We say one number is divisible by another if there is no remainder Example Multiply 543 by 6 Answer x 543 6 3258 2 1 A number is divisible by 2 if it is an even number Starting with 6 x 3 this gives an answer of 18 so you need to carry a digit into the next column on the left. A number is divisible by 5 if it ends in a 0 or 5 Then 6 x 4 gives an answer of 24, then add the 1 which had been carried from the previous column giving 25, again you need to carry a digit into the next column on the left. Order of Operations The 6 x 5 gives an answer of 30, then add the 2 which had been carried from the previous column giving 32. (See BIDMAS Grade C) Division then multiplication then addition then subtraction Example 9÷3+4x2 Answer First divide 9÷3=3 giving 3 + 4 x 2 Then multiply 4x2=8 giving 3 + 8 Then add 3 + 8 = 11 Example Equivalent Fractions Divide 508 by 4 Equivalent fractions are two or more fractions that represent the same part of a whole. Answer Recognise using diagrams equivalent fractions. This is set out as 1 2 7 4 51 02 8 1 2 = 2 4 An equivalent fraction can also be found from multiplying or dividing the numerator and denominator by the same number. First divide 4 into 5 to get 1 and remainder 1. You can see from the diagrams that Then, divide 4 into 10 to get 2 and remainder 2. Finally, divide 4 into 28 to get 7. This gives the answer of 127. Rounding to the Nearest 10 or 100 Example 30 36 is between 30 and 40 40 36 is closer to 40 than 30 36 rounded to the nearest ten is 40 If the number is between, we round up 35 is halfway between 30 and 40, 35 to the nearest ten is 40 Fractions Amy bought 6 sweets. She gave 4 of these to her friends a. What fraction of the sweets did Amy give to her friends b. what fraction did she keep a. Amy gave Negative Numbers - Scales and Number Lines Scales often have positive and negative numbers on them. For example, on a temperature scale 10° below 0° is shown as -10°. Positive numbers such as +4 are often written without the + sign. For instance, +4 is often written as 4. Negative numbers, such as –4, are always written with the – sign. Zero is neither positive nor negative. Ordering Negative Numbers A fraction is part of a whole. The top number is called the numerator. The bottom number is called the denominator. 2 is read as two fifths. 2 is the numerator and 5 is the denominator. 5 2 means you divide a whole into 5 portions and take 2 of them. 5 2 means 2 parts out of every 5 5 2 For instance of these dots are red 5 Example Answer Example 3 × 4 12 → = 4 × 4 16 4 to her friends. 6 2 b. Amy kept 2 for herself. That is, she kept . 6 -8 -7 -6 -5 -4 -3 -2 The further to the right a number is, the larger it is. The further to the left a number is, the smaller it is. < means less than > means greater than Example Insert < or > to make these statements true. a. 6 -6 b. –5 -6 Answer a. 6 > -6 (Since 6 is greater than –6) b. –5 > -6 (Since -5 is greater than –6) -1 0 1 2 3 4 Equivalent Fractions, Decimals and Percentages Place Value Convert from percentage to: The ordinary counting system uses place value, which means that the value of a digit depends upon its place in the number. Decimal Fraction We read 432 as four hundred and thirty two. Divide the percentage by 100 Make the percentage into a fraction with a denominator of 100 and simplify by cancelling down if possible The number 52 has 5 tens and 2 ones. Example 52% = 52 ÷ 100 = 0.52 Convert from decimal to : Percentage Multiply the decimal by 100% Example 0.65 = 0.65 x 100 = 65% Convert from fraction to : Example 52 13 52% = = 100 25 Putting whole Numbers in Order Example Rearrange this list of numbers into order of size starting with the largest number. 86, 104, 79, 88, 114, 200, Answer Fraction 200, 114, 104, 88, 86, 79, If the decimal has one decimal place put it over the denominator 10, if it has 2 decimal places put it over the denominator 100 etc. Then simplify by cancelling down if possible Fractions Example 65 13 0.65 = = 100 20 Examples Percentage Decimal If the denominator is a factor of 100 multiply numerator and denominator to make the denominator 100, then the numerator is the percent Divide the numerator by the denominator Example 3 15 = = 15% 20 100 Or convert to a decimal and change the decimal to a percentage 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are called digits. Example 9 = 9 ÷ 40 = 0.225 40 Example 7 = 7 ÷ 8 = 0.875 = 87.5% 8 Adding and Subtracting Simple Fractions Fractions with the same denominator (bottom number) can easily be added or subtracted. For a Grade F you would also need to simplify the answer. 1. 2. 3 4 + 10 10 Add the numerators (top number). The bottom number stays the same. 4 7 3 + = 10 10 10 7 2 Work out 8 8 Subtract the numerators (top number). The bottom number stays the same. Work out Multiples (Times Tables) Example Write out the first five multiples of 3 Answer 3, 6, 9, 12, 15 Factors A factor of a whole number is any whole number that divides into it exactly. Example List the factors of 20 Answer Checking Calculations 1, 2, 4, 5, 10, 20 38 + 57 = 85, Round to check 40 + 60 = 100 Square Numbers Always check your answer even when using a calculator When you multiply any number by itself, the answer is called the square of the number or the number squared. This is because the answer is a square number. 1 x 1= 1 2x2=4 3x3=9 GCSE Grade F Fractions of Quantities Division 1 1 The word “of” can be replaced by x. For instance of 200 means x 200 10 10 3 Example Find of £270 5 1 Answer of £270 = 270 ÷ 5 = 54 5 3 of £270 = 54 x 3 = 162 (Divide by 5, then multiply by 3) 5 Either by using a calculator of by dividing by the denominator and then multiplying by the numerator There are difference methods for dividing one of these methods is the chunking method. Using Negative Numbers Example The temperature at 6a.m. was -5°C. By 9a.m. the temperature has risen by 7°. What was the temperature at 9a.m. Answer The temperature must rise 5° to 0°. By the time the temperature has risen a further 2°, it would then be 2°C. Adding and Subtracting Negative Numbers We can use a number line to add or subtract To add a positive number, move to the right. To add a negative number, move to the left. Addition and subtraction are inverse operations. To subtract, we must move in the opposite direction to that in which we move to add. To subtract a negative number, move to the right. Writing down some of the multiples of 35 might be useful. 1 x 35 = 35 2 x 35 = 70 10 x 35 = 350 20 x 35 = 700 5 x 35 = 175 1655 - 700 955 - 700 255 -175 80 - 70 10 20 x 35 20 x 35 5 x 35 2 x 35 47 Once the remainder is found, you cannot subtract any more multiples of 35. Add up the multiples (in red) to see how many times 35 has been subtracted. Decimal Place Value Examples -7 Work out 1655 ÷ 35 So 1655 ÷ 35 = 47 remainder 10 To subtract a positive number, move to the left. -8 Example Remember -6 -5 -4 -3 -2 4–1 Begin at 4, move 1 to the left 1–3 Begin at 1, move 3 to the left 1 – (-3) Begin at 1, move 3 to the right -1 – (-3) Begin at -1, move 3 to the right Pencil and Paper Methods -1 0 1 2 4–1=3 4–1=3 4–1=3 4–1=3 3 4 1 can be written as 0.1 and 3 can be written as 0.3 10 10 1 can be written as 0.01 and 41 can be written as 0.41 100 100 Place Value is given by this chart. 100000 10000 1000 100 10 Multiplication There are different methods for multiplication one of these methods is the grid method. Example Work out 243 x 68 without using a calculator. Answer using the grid method. The two numbers are split into hundreds, tens and units. Multiply all the pairs of numbers. X 200 20 3 60 12000 2400 180 8 1600 320 24 Add all the numbers in the boxes which are not shaded. 12000 + 2400 + 180 + 1600 + 320 + 24 = 16524 So 243 x 68 = 16524 For instance, in the number 365.724 the 3 means three hundreds the 6 means six tens the 5 means five ones (or units) the 7 means seven tenths the 2 means two hundredths the 4 means four thousandths Reading and Writing Decimals The number 84.73 is read as eighty four point seven three. The number 84.73 is written with two decimal places 1 1 10 1 100 1 1000 Rounding Pencil and Paper Methods for Adding and Subtracting Decimals To give sensible answers to calculations we often need to approximate. Rounded to 1 d.p. Rounded to 2 d.p. 3.6438 is 3.6 3.6438 is 3.64 Rounded to 1 d.p. Rounded to 2 d.p. To add 3.42 + 15.8 we should set out our work as shown 236.847 is 236.8 236.847 is 236.85 + To round to a given number of decimal places. 1. Keep the number of figures asked for after the decimal point. For instance if asked to round to 3 decimal places, keep 3 figures after the decimal point. 2. Omit all the following figures. If the first figure omitted is 5 or greater, increase the last figure kept by 1. 3.42 15 . 8 19.22 The 3 and the 5 must be lined up and added since these were both units To do the subtraction 7.3 – 1.5 we could set out our work as shown 6 7 .1 3 - 1 .5 5 .8 Rounding to a given number of decimal places is also called approximating to that number of decimal places. The words “decimal places” are often abbreviated to d.p. Example Multiplying and Dividing Decimals by Single-digit Numbers Examples Round 3.6472 to a. 2 d.p. 1. b. 1 d.p. 4.5 x 3 Answer a We round to 2 d.p. which means we want 2 figures after the decimal point. With just 2 figures after the decimal point, 3.6472 is 3.64. The first figure omitted is 7 so we must increase the last figure kept by 1. We then get 2.6472 = 3.65 to 2 d.p. 1 So 4.5 x 3 = 13.5 2. We need 1 figure after the decimal point. With just 1 figure after the decimal point, 3.6472 is 3.6. The first figure omitted is 4 which is not large enough to alter the last figure kept. We then get 3.6472 = 3.6 to 1 d.p._ Decimals On a number line, the smaller a number is, the further to the left it is. To put a list of decimal numbers in order it is a good idea to first write them all with the same number of figures after the decimal point. Example 2.63 2.6 3.421 4.32 3.4 4.320 3.400 3.399 4.098 Now choose the numbers with the smallest number before the decimal point These are 2.630 2.600 Put these in order. Since 600 < 630 then 2.600 < 2.630. The first two numbers are 2.6 2.63 Now choose the numbers with a 3 before the decimal point. These are 3.421 3.400 3.399 Since 399 < 400 < 421, then 3.399 < 3.400 < 3.421 The first five numbers are 2.6 2.63 3.399 3.4 3.421 Left are 4.098 and 4.320 Since 98 < 320, then 4.098 < 4.320 b The numbers in order smallest to largest 2.6 2.63 3.399 3.4 3.421 4.098 8.25 ÷ 5 1. 6 5 5 8.1215 So 8.25 ÷ 5 = 1.65 Reading Calculator Displays There is a maximum numbers of digits that can be displayed on any calculator screen. On a calculator with an 8 digit screen display the answer to the division 7 ÷ 3 is displayed as 2.3333333. Other calculators may display more than or fewer than 8 digits. We often need to give the answer to a calculation to the nearest whole number 3.399 4.098 To put these numbers in order from the smallest to largest Rewrite the numbers as 2.630 2.600 3.421 4.5 x 3 13.5 Examples 1. A calculator displayed the answer to 7 ÷ 3 as 2.3333333 To the nearest whole number the answer to 7 ÷ 3 is 2 2. A calculator displayed the answer to 53 ÷ 7 as 7.5714286 To the nearest whole number the answer to 53 ÷ 7 is 8 Sometimes we need to give the answer to a calculation to the closest but smaller whole number. Example Oranges are 52p each. How many can be bought with £5.00 Answer On the calculator the answer to 500 ÷ 52 is 9.615384615 Only 9 oranges can be bought In the previous example the answer was given as the closest but smaller whole number. Using the Calculator for Calculations 4.32 The calculator is often used for calculations. It is very useful when the numbers are large or have many digits Always have a rough idea of the size of the answer you expect to get. It is very easy to press a wrong key or to forget to press the = key at the end of a calculation Improper Fractions and Mixed Numbers Square Roots A fraction with the numerator (top number) smaller than the denominator (bottom number) is called a proper fraction. 4 An example of a proper fraction is . 5 An improper fraction has a bigger numerator (top number) than the denominator (bottom number). 9 An example of an improper fraction is . It is sometimes called a top heavy fraction. 5 A mixed number is made up of a whole number and a proper fraction. The square root of a given number is a number that, when multiplied by itself, produces the given number. 3 The answer to “what number squared gives 4” is 2, 2 is called the “square root of 4” The sign for a square root is 4 . For instance 4 means “the square root of 4” Squaring and finding the square root are inverse operations. One “undoes” the other For instance 22 = 4 and 4 =2 Square roots are found using the calculator An example of a mixed number is 1 . The answer to “what number cubed gives 64” is 4, 4 is called the “cube root of 64” Examples 14 Convert into a mixed number. 5 Answer 14 means 14 ÷ 5. 5 The sign for a cube root is 3 64 . 4 Dividing 14 by 4 gives 2, with a remainder of 4. (5 fits into 14 two-times with 14 4 =2 5 5 Example 1 Convert 3 into an improper fraction. 4 Answer So Each whole is four – quarters. So 3 whole ones are 12 quarters. 1 So 3 is 12 quarters + 1 quarter = 13 quarters. 4 1 13 3 = 4 4 For instance 3 64 means “the cube root of 64” Cubing and finding the cube root are inverse operations. One “undoes” the other For instance 43 = 64 and 3 64 = 4 Cube roots are found using the calculator 4 left over) 5 GCSE Grade E Percentages of Quantities Equivalent Ratios The word “of” can be replaced by x. For instance 23% of 200 means 23% x 200 Equivalent ratios are equal. For instance 8 : 2 and 4 : 1 are equivalent ratios. 23 Since 23% means 23 out of 100, 23% may be written as 100 We can divide both sides of a ratio by the same number. We can also multiply both sides of a ratio by the same number. Example A ratio is in its simplest form if the numbers in the ratio are the smallest possible whole numbers Find 23% of 200cm Example Answer Write these ratios in their simplest form. 23% of 200cm = 23% x 200 a. 1mm : 5m 23 = 100 x 200cm b. 45p : £2 c. £3.50 : £2. Answer = 46cm Writing Ratios a. 1mm : 5m = 1mm : 5000mm = 1 : 5000 b. = 45p : 200p = 45 : 200 = 9 : 40 45p : £2 The ratio of two quantities x and y is written as x : y. x : y is read as “the ratio of x to y”. The order is important x : y is different from y : x. c. £3.50 : £2 = 3.5 : 2 =7:4 A ratio compares quantities of the same kind. Multiplying and Dividing Decimals Example Example On a bus there are 17 adults and 5 children Find the ratio of a. adults to children b. children to adults. Answer a. In our ratio we must have the number of adults first, then the number of children. The ratio of adults to children is 17 : 5. b. In our ratio we must have the number of children first, then the number of adults. The ratio of children to adults is 5 : 17 Using Ratios Ratios are used in the drawing of plans. The plan will be the same shape as the original only smaller. The ratio used may be 1 : 2 which means that the original will be twice as large as the plan. In other words the plan will be ½ the size of the original. Calculate 16.7 x 0.8 Answer Ignore the decimal point and calculate 167 x 8 X 100 60 7 8 800 480 56 800 + 480 + 56 = 1336 The ratio used may be 1 : 5 which means that the original will be five times as large as the plan. The plan Count how many digits after the decimal point in both numbers and then count from the right. In 16.7 there is 1 and in 0.8 there is 1 so count 2 from the right. So 16.7 x 0.8 = 13.36 would be 1 the size of the original. Negative Numbers 5 Many other ratios could be used. Multiplying and Dividing The rules for multiplying and dividing with negative numbers are: When the signs are the same the answer is positive When the signs are different the answer is negative Examples 1. -2 x -2 = +4 2. -4 x 6 = -24 3. 7 x -2 = -21 4. 24 ÷ -3 = -8 5. -16 ÷ 8 = -2 6. -14 ÷ -7 = +2 Approximation, Estimation Multiplying and Dividing Example To multiply fractions, you multiply the numerators together and you multiply the denominators together. Write 0.0504715 to 4 s.f. Example Answer Counting from the first non zero figure and keeping 4 figures after this 0.0504715 is 0.05047 The first figure omitted is 1 which is not large enough to alter the last figure kept We then get 0.0504715 = 0.05047 to 4 s.f. When we approximate large number to s.f. we may have to insert zeros so the size of the number is unchanged For instance 34592 to 1 s.f. is written as 30000 not as 3. Because the second digit is less than five then we leave the first digit as it is. If the value of the second digit is equal to or greater than five then add one to the first digit. Example 45281 would round to 50000 to 1 significant figure. Work out 1 2 of . 4 5 Answer 1 2 1 ×2 2 1 x = = = 4 5 4 ×5 20 10 To divide a fraction, you turn the fraction upside )finding its reciprocal), and then multiply Example 3 4 3 ×4 12 3÷ =3x = = =4 4 3 3 3 For Grade C you would have to multiply and divide with mixed numbers and improper fractions. Prime Numbers Approximations with Decimals A prime number only has two factors: 1 and the number itself. Rounded to 1 significant figure 3.6482 is 4 The prime numbers up to 50 are Rounded to 2 significant figures 3.6482 is 3.6 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Rounded to 3 significant figures 3.6482 is 3.65 Powers Estimating Answers using Approximation Powers are a convenient way of writing repetitive multiplications. They are also called indices. We can check that the answer to a calculation such as 423 x 76 is about the right size by approximating. 7 x 7 = 72 We can approximate 423 as 400. We can approximate 76 as 80 7 x 7 x 7 = 73 which is 7 to the power 3 or 7 cubed. So we estimate 423 x 76 to be about 32000 7 x 7 x 7 x 7 = 74 which is 7 to the power 4. which is 7 to the power 2 or 7 squared. Always estimate answers when using the calculator. Using Place Value of Multiply and Divide Calculator Errors Using Place value 23 x 10 = 230, 23 x 100 = 2300 When using the calculator, it is a good idea to have a rough idea of the answer. We can often use multiplication by 10 or 100 or 1000 etc. to help us when we are multiplying by 80 or 800 or 8000 etc. It is easy to key a calculation into the calculator incorrectly. If the answer given by the calculator is very different from the rough estimate, the calculation should be keyed in again Fractions We can often use division by 10 or 100 or 1000 etc. to help us when we are dividing by 40 or 400 or 4000 etc. GCSE Grade D Fractions Multiplying Fractions A fraction is written in its lowest terms or as the simplest fraction, if the numbers in the fraction are the smallest possible whole numbers. Multiply the numerators to obtain the numerator of the answer and multiply the denominators to obtain the denominator of the answer. Example 24 Write in its lowest terms 32 Answer When multiplying a mixed number, change the mixed number to an improper fraction before you start multiplying. The largest number that divides into both 24 and 32 is 8 24 ÷ 8 3 = = 32 ÷ 8 4 An amount or quantity can be given as a fraction of another. Example Write £5 as a fraction of £20 Answer 5 As a fraction this is written as . 20 1 This cancels to . 4 1 So £5 is of £20. 4 Adding and Subtracting Fractions To add (or subtract) fractions, the fractions must be the same type Fifths may be added to fifths, quarters may be added to quarters but to add fifths to quarters we must rewrite as the same type. Examples Work out a. Answer 4 9 4 9 × 3 10 b. 2 2 5 ×1 7 8 3 2 is a factor of 4 and 10. 3 is a factor of 3 and 9. 10 2 2 1 Simplifying the fractions, cancelling by 2 and 3 gives × = 3 15 5 12 2 7 b. 2 × 1 Converting the mixed numbers into improper fractions gives 5 5 8 3 9 3 1 Simplifying the fractions by cancelling by 4 and 5 gives × = =4 1 2 2 2 Ratios as Fractions a. × × 15 . 8 A ratio in its simplest form can be expressed as portions of a quantity. Example A garden is divided into lawn and shrubs in the ratio 3 : 2. What fraction of the garden is covered by a. b. Lawn Shrubs We need to find equivalent fractions which have the same denominator. Answer Examples 2 1 3 + = 5 5 5 5 3 2 1 – = = 8 8 8 4 4 3 4 3 To find + first find fractions equivalent to and which have the same denominator 5 4 5 4 4 8 12 16 = = = 5 16 15 20 3 6 9 12 15 = = = = 4 8 12 26 20 4 3 16 15 31 11 Then + = + = =1 5 4 20 20 20 20 The denominator (bottom number) of the fraction comes from adding the numbers in the ratio 2 + 3 = 5 3 a. The lawn covers of the garden 5 2 b. And the shrubs covers of the garden 5 Speed, Distance and Time The relationship between speed, distance and time can be expressed in three ways distance speed = time distance = speed x time distance time = speed In problems relating to speed, you usually mean average speed. Example Paula drove a distance of 270 miles in 5 hours. What was her average speed. Answer Paula’s average speed = distance she drove = 270 = 54 miles per hour (mph) time she took 5 Using Place Value to Multiply and Divide Percentage Increase and Decrease Examples To increase a quantity by a percentage, work out the increase and add it on to the original amount. Use place value to find the answers. a. 1.34 x 10 c. 2.7 x 1000 Answers a. 13.4 Example b. 203.6 x 100 d. 1.34 ÷ 10 (move each digit one place to the left) e. 58 ÷ 100 Increase £6 by 5%. Answer Work out 5% of £6. b. 20360 (move each digit two places to the left) 1% = £6 ÷ 100 = 6 pence c. 2700 5% = 6 pence x 5 = 30 pence or £0.30 (move each digit 3 places to the left) d. 0.134 (move each digit one place to the right) Add £0.30 to the original amount e. 0.58 To decrease a quantity by a percentage, work out the decrease and subtract it from the original amount. (move each digit two places to the right) £6 + £0.30 = £6.30 Example Decrease £8 by 4% Answer 1% = £8 ÷ 100 = 8 pence 4% = 8 pence x 4 = 32 pence or £0.32 Subtract £0.32 from the original amount £8 - £0.32 = £7.68 GCSE Grade C Order of Operations Dividing in a given Ratio There is an order of operations which you must follow when working out calculations. To answer questions like this, you must follow the BIDMAS rule. This tells you the order in which you must do the operations Example Brackets Answer Indices (powers) 1. 2+5=7 Add together the ratios Division 2. 280 ÷ 7 = 40 Divide the total amount by (1) Multiplication 3. 2 x 40 = 80 and 5 x 40 = 200 Multiply the ratios by (2) Addition One family gets 80 apples, the other gets 200 apples. Subtraction Prime Numbers, Highest Common Factor (HCF), Lower Common Multiple (LCM) Example The highest common factor (HCF) of two numbers is the greatest number that is a factor of both of the given numbers 60 – 5 x 32 + (4 x 2) First work out the brackets (4 x 2) = 8 giving 60 – 5 x 32 + 8 The apples in a container are to be divided between two families in the ration 2 : 5 How many apples does each family get if there are 280 apples in the container? Then the index (power) 3 =9 giving 60 – 5 x 9 + 8 The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the given numbers. Then multiply 5 x 9 = 45 giving 60 – 45 + 8 Example Then add 60 + 8 giving 68 – 45 Find Finally subtract 68 – 45 = 23 2 a. the (HCF) of 24 and 54 Fractions b. the (LCM) of 24 and 54 To add (or subtract) mixed numbers we can begin by writing each mixed number as an improper fraction. Answer 24 Examples 2 3 11 7 44 21 65 5 3 +1 = + = + = =5 3 4 3 4 12 12 12 12 1 1 19 3 19 9 10 4 2 3 −1 = − = − = =1 = 1 2 This prime factor tree can be used to rewrite 24 as 2 x 2 x 2 x 3 12 2 6 6 2 6 2 6 6 6 6 3 Another method of adding (or subtracting) mixed numbers is to add (or subtract) the whole numbers then add or subtract the fractions 2 If more than one operation is involved in the same calculation then follow BODMAS (GCSE Grade B) 3 54 This prime factor tree can be used to rewrite 54 as 2 x 3 x 3 x 3 Dividing Fractions 2 To divide by a fraction, multiply by the reciprocal. 27 Examples 3 4 1. 3. 3 4 3 6 4 1 3 1 = × 4 6 1 1 = × 4 2 1 = 8 5 3 12 = × 12 4 5 3 3 = × 1 5 9 = 5 4 =1 5 ÷6 = ÷ ÷ 2. 6÷ 3 4 = = = 6 3 ÷ 1 4 6 4 × 1 3 2 4 × 1 1 3 9 3 3 24 = 2 x 2 x 2 x 3 54 = 2 x 3 x 3 x 3 2 x 3 is common to both numbers Hence the HCF is 6 =8 4. 2 1 3 ÷ 1 4 6 5 3 5 × 3 1 × 1 2 5 = ÷ = = = 25 6 6 25 2 5 The smallest number that both 24 and 54 divide into exactly is 2 x 2 x 2 x 3 x 3 x 3 Hence the LCM is 216 24 54 L C M 2 HCF 2 3 2 3 3 Indices Reciprocals The reciprocal of any number is one divided by the number. Rules of Indices Example 33 x 35 = (3 x 3 x 3) x (3 x 3 x 3 x 3 x 3) = 38 which is 3(3 + 5) m n 1 2 The reciprocal of 0.25 is 1 ÷ 0.25 = 4 The reciprocal of 2 is 1 ÷ 2 = m+n a xa =a When multiplying add the powers You can find the reciprocal by swapping the numerator and denominator 7 ÷ 7 = (7 x 7 x 7 x 7 x 7 x 7) ÷ 7 = 7 x 7 x 7 x 7 x 7 = 7 which is 7 Example am ÷ an = am - n The reciprocal of 6 5 When dividing subtract the powers Examples Use the rules of indices to write the following as single powers of 7 a. 74 x 79 = 74 + 9 = 713 b. 78 ÷ 72 = 78 – 2 = 76 (6 – 1) 2 3 is 3 2 4 7 The reciprocal of is 7 4 Percentage Change To find a percentage increase (or decrease) we compare the actual increase with the initial value % Increase = Actual Increase x 100% Original amount % Decrease = Actual Decrease x 100% Original amount Example A jersey which was priced at £39 is reduced to £28 in a sale. What percentage reduction is this? Answer Actual decrease in price = £39 - £28 = £11 % Decrease = Actual Decrease x 100% Original amount 11 = x 100% 39 = 28% (to the nearest percentage)