Place Value Thousandths Hundredths Tenths Decimal Point Ones Tens Hundreds Thousands 000s 00s 0s s. 0 00 000 Know the meanings of these column headings is very important. It tells us the value of each digit. This can be especially useful when rounding to significant figures, multiplying and dividing by 0,00,000, converting between fractions decimals and percentages and long multiplication and division. For example 7 9. 8 7 This digit is not just worth 7. It has a value of 700. This digit is not just 9. It is worth 9 ones. he same as nine whole ones or ninety tenths, or 0.9 tens. This digit is not worth 7. It has a value of 7 thousandths or 7 000
MULTIPLICATION You must learn your tables Just do it!! Long Multiplication GRID METHOD Example 8 X 88 X 0 8 0 800 0 0 8 Add the numbers up 800 0 0 + 8 88 Example X 9 9 X 00 0 0 000 00 9 900 0 80 Sum of each row 80 9 Pitfall: Don t forget zeros Line up columns for addition
CHINESE METHOD 8 X 88 8 0 8 8 Pitfall: Make sure you get the diagonals the right way. This is a quick and reliable method to use DECOMPOSITION METHOD Example X 8 Method 0 X 0 X + 8 8 Example 7 X 08 Method 0 X 80 7 X + 8 08 Pitfall: This only works if you are multiplying by one digit (i.e. single digits)
DIVISION Method - Bus stop method Example 80 8 8³ 0 s into 8 go remainder, carry the s into go remainder, carry the s into 0 go 8 So 80 8 Example 0 ² s into go s into go 0 carry the s into go So 0
Example : 78 Method Write out the times table up to 9x 9 9 8 8 07 0 7 7 8 9 into 7 goes 0 carry the 7 into 78 goes remainder 9, carry the 9 into 9 goes. So 78 Example 7 9 Method Write out the 9 times table up to 9x9 9 8 7 7 9 7 0 9 7 9 9 into goes 0 carry the 9 into goes remainder, carry the 9 into 7 goes remainder 9, carry the 9 9 into 9 goes. So 7 9 7
Multiplying by 0, 00 and 000 8 x 0 80 (Add a zero) 8 x 00 800 (Add zeros) 8 x 000 8000 (Add zeros).8 x 0 8. (Move the decimal point one place to the right).8 x 00 8. (Move the decimal point two places to the right).8 x 000 8 (Move the decimal point three places to the right) Dividing by 0, 00 and 000 8 0 8. (Move the digits one place to the right) 8 00.8 (Move the digits two places to the right) 8 000.8 (Move the digits three places to the right) 00 0 0 (Move the digits one place to the right) 00 00 (Move the digits two places to the right) 00 000. (Move the digits three places to the right) 8
BIDMAS B Brackets I Indices D Division M Multiplication A Addition S Subtraction When you do long calculations you must work them out according to the order of operations. BIDMAS helps you to remember the order. Example + ( x 8) Brackets first + Division + Addition 9 Example ² x (-7) Brackets first ² x 8 Indices 9 x 8 Multiplication 7 9
Fractions Equivalent Fractions Equivalent fractions have the same value, even though they may look different. These fractions all have the same value: 9 8 If you multiply or divide the numerator and denominator by the same number the fraction keeps its value: 9 Finding equivalent fractions: 9 8 Simplifying Fractions: 0 9 9 Converting Mixed Number and Improper Fractions Mixed number improper fraction + 7 7 7. Multiply the denominator by the integer For example:. Add the product to the numerator + 7 0
Converting Improper Fractions to Mixed Numbers To convert between improper fractions you divide and find the remainder. remainder 0 r Multiplying Fractions Multiply the numerators and the denominators: 9 0 0 9 0 0 Always simplify Dividing Fractions When dividing fraction turn the second fraction on it s head ( Flip it ) and then multiply. 0 0 0 Flip the second fraction on it s head then multiply becomes
Multiplying Fractions and Integers When multiplying fractions and integers change the integer (whole number) into a fraction with a denominator of. Multiplying and Dividing Mixed Numbers When multiplying and dividing mixed numbers we first convert them into improper (top-heavy) fractions. 8 88 8 9 8 9 8 8 88 8 9 remainder 8 Cross-Cancelling By looking for common factors we can speed up difficult questions of multiplying fractions. Cancel by a factor of 7 8 9 9 Cancel by a factor of Cancel by a factor of Cancel by a factor of
Adding and Subtracting Fractions To add and subtract fractions you have to use your equivalent fraction skills. When fractions have the same denominator (bottom numbers) we can add the numerators (top numbers). 8 When fractions have the same denominator we can add the numerators. + 8 0 + 8 7 0 Simplify Adding and Subtracting Mixed Numbers: Method : The first method for adding and subtracting mixed numbers is to change first into improper fractions. 9 9 7 7 9 7
Adding and Subtracting Mixed Numbers: Method : Another method of adding and subtracting mixed numbers is to add and subtract the whole numbers (integers) and fractions separately. This method is quicker, but it can be confusing when you combine the whole numbers and fractions together at the end. Whole numbers (integers) It can be confusing... Fractions Combine whole numbers and fractions + 0 Final answer It can be confusing when the fractions total is an improper fraction (top-heavy) or a negative number.
Non Calculator Finding the Percentage of a Value % Divide by 00 % Divide by 00 and double it % Find 0% and halve it / % Halve % % Calculate % and % and add 0% Divide by 0 % Find 0% and % and add 0% Find 0% and double it % Halve and halve again (or divide by ) 0% Find 0% and multiply by 0% Halve it 7% Find 0% and % and add Example Find % of 0%.0, 0% x.0 7.0 %., % x..0 + 0.00 Note for money there must be decimal places (or none) so the acceptable answers are 0.00 or 0. Example Find / % of 0%., % 0.8 / % 0. / % 0% + / %, so /. + 0. Non Calculator Finding Percentage Increase or Decrease The Glossary contains vocabulary associated with increase and decrease. Example Increase by %. From example Finding a Percentage find % of ( 0). Increased amount is + 0 Example Decrease by / % From example Finding a Percentage find / % of () Decreased amount is -
Calculator Method for Finding the Percentage Of, and Increasing or Decreasing a Value Method uses decimal multipliers. To change a percentage to decimal see page on changing fractions/decimals and percentages. Finding a Percentage of a Value Change percentage into decimal by dividing by 00 Multiply decimal by the value. Find 7% of 8 7% 0.7 7% of 8 0.7 x 8. Example Example Find 8.% of.7 8.% 0.8 8.% of.7 0.8 x.7 8.8 Finding a Percentage Increase or Decrease In percentage questions, the first event in time is 00% If the cost of downloading songs from itunes was p what will it be if it increases by %? First event, is 00% Second event is % more, so 0% The decimal multiplier for 0% is.0 (divide by 00) Calculation is x.0. because this is money we must round to the nearest penny, p
The method works for decrease as well. If a pair of trainers is 0% off in the sale what will be the sale price? First event, is 00% Second event is 0% less, so 0% The decimal multiplier for 0% is 0.0 (divide by 00) Calculation is x 0. 7 7
Changing between Fractions, Decimals and Percentages Fraction Example put t he percent age as numerat or and 00 as denominat or and simplify The percentage is the numerator value of the equivalent fraction with a denominator of 00 Example Example divide the numerator by the denominator the digits are the numerator and the final column heading is the denominator Example Example Percentage divide by 00 Decimal multiply by 00 Example Example a. 7 % simplifies to 00 0 b.. 8.% simplifies to and this simplifies to 00 000 8
Example x 8 a. The fraction makes which is 8% x 00 7 x. 87. b. The fraction makes which is 87.% x. 00 8 / 0 / 00 / 000 Example a. 0. the final column is thousandths so b. 0. 0 the final column is hundredths so 00 000 Example a. the line in a fraction means divide so means 8. Use bus-stop or a 8 calculator to get 0. b. when the denominator is 0, 00 or 000 etc it is easy to do the division in 7 your head. So 0 0., 7 00 0.7 and 0.0 0 00 000 Example and Multiplying by 00 the digits move two places to the left and dividing by 00 the digits move two places to the right a. so 0.7 becomes 7% and.7 becomes 7% b. Harder examples are 0. becomes 0% and 0. becomes.% 9
Prime Factors The Prime Numbers up to 0:,,, 7,,, 7 and 9 Q. Find the Prime Factors of 0 First draw a Prime Factor Tree 0 Find numbers which multiply to make 0 0 Then 0 7 Then Then The Prime Factors of 0 are,,,, 7 They can be expressed in index form as ² x x x 7 Finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using Prime Factor trees Q. Find the LCM and HCF of and 0 0 0 Index form ² x Index form ² x 0
Put these figures into a Venn Diagram 0 HCF x (Multiply the numbers in the overlapping segment) LCM x x x 0 (Multiply all the numbers) The Ladder Method Q. Find the LCM and HCF of and 0. We write a common factor of both numbers beside them. 0 0. The numbers that we are using go at the top of the ladder. We keep writing common factors down as long as possible To find the HCF multiply the numbers on the left side of the ladder: - HCF To find the LCM multiply the numbers on the left side and the numbers beneath the ladder: - LCM 0
Rounding Numbers Rounding to the nearest whole number: Draw a vertical line to the right of the number that will be rounded.. 7 8 Ignore these numbers This number will either stay as a or round up to. The number directly to the right of the vertical line is the decider. If the decider is less than the number left of the line stays the same. If the decider is or more the number left of the line is rounded up. So.78 rounded to the nearest whole number is. Rounding to the nearest 0 The answer will either be 80 or 90 8 7 The number 7 is the decider. As it is larger than the number to the left of the vertical line will round up. So 87 to the nearest 0 is 90 Rounding to the nearest 00 The answer will either be 00 or 00 8 7 The number 8 is the decider. It is greater than so 87 rounded to the nearest hundred is 00
Rounding to decimal places 8 9. 8 rd decimal place st decimal place nd decimal place Round to decimal place 8 9. 8 st decimal place The decider The decider is greater than so the number to the left of the vertical line rounds up 89. Rounding to significant figures 8 st th nd rd th Significant figures (S.F) Round to significant figure 8 st S.F The decider The decider is greater than so 8 rounded to significant figure 0000
Round to significant figures 8 rd S.F The decider The decider is less than so the number to the left of the vertical line stays the same. To round to significant figures we round 8 down to 800. The first significant figure is always the first non zero number so in the number 0. 0 0 8 7 the 8 is the first significant figure. A zero counts as a significant figure if it is between two non zero numbers. Rounding decimals to significant figures 0. 0 0 8 7 nd S.F The decider The decider is greater than so 0.0087 is rounded up to 0.008. When you write out a decimal to significant figures you never add extra zeros to the end of the number. Always stop after the last significant figure. Remember Less than stays the same, or more round up!
Ratio Ratio is the share of something in proportion to other shares. For example, if Andy and Graham share sweets in the ratio :, for every sweet Andy gets, Graham gets. Ratio can be simplified in the same way fractions can - by finding common factors between all parts. Example :: Method :: All parts have as a common factor, so divide each part by :: All parts still have as a common factor so divide each part by :: All parts only have as a common factor and therefore this is the simplest form. Method :: All parts have as a common factor, so divide each part by :: All parts only have as a common factor and therefore this is the simplest form. It is O.K. to take more than step if you cant find the highest common factor.
Example Divide 0 into this ratio: Danielle : Emmanuel : Freddie : : In total there are parts ( + + ). First calculate one part of the ratio: 0 0 Danielle gets parts, so she gets 0 0 Emmanuel gets parts, so he gets 0 0 Freddie gets parts so he gets 0 90 So the money is shared out as follows: D : E : F : : 0 : 0 : 90 If we divide 0 into equal parts of 0, here we see that Danielle takes parts, Emmanuel takes parts and Freddie takes parts. Danielle 0 0 0 0 0 0 7 0 8 0 9 0 0 0 0 0 Emmanuel Freddie
Example Danielle, Emmanuel and Freddie share some money in the following ratio. Danielle : Emmanuel : Freddie : : If Freddie gets 8, how much do Danielle and Emmanuel get and how much is there altogether? Note: Notice the difference between Example and Example. In Example we are given the total amount, but in Example we are given one persons share. Freddie Gets parts worth 8 in total. We first need to find the value of on share. To find share we would do 8. This equals. If one share is then to find out Danielle s share we need to do share multiplied by - so. This equals 0. If one share is then to find out Emmanuel s share we need to do share multiplied by - so. This equals. So Danielle gets 0 Emmanuel gets Freddie gets 8 In total that is 0++8. This equals 7. Danielle 7 8 9 0 Emmanuel Freddie 7