Arithmetic Revision Sheet Questions and of Paper Basics Factors/ Divisors Numbers that divide evenly into a number. Factors of,,,, 6, Factors of 8,,, 6, 9, 8 Highest Common Factor of and 8 is 6 Multiples A multiple of a natural number is a number into which it divides evenly. Multiples of, 6, 9,. Multiples of, 8,, 6.. Lowest Common Multiple of and is Prime Numbers A numbers whose factors are only one and itself.,, 5, 7,,, 7, 9,. Reciprocal This is a fraction turned upside down. To calculate the value of a reciprocal divide the bottom number into the top number and round off to two decimal places. Reciprocal of 7 is 7 or 0.9 Scientific Notation We often write very large or very small numbers in the form and 0 Examples - 0,000 = 5. 0 6. 0 5 60, 000 7 7,000,000 =.7 0.5 0 50 0.0078 = 7.8 0.6 0 0. 006 n a 0 where a is between Example Calculate (5.0 ) (.6 0 ) and give your answer in scientific notation. (5. 0 ) (.6 0 ) write equation 5000 600 put in normal notation 55600 add together = 5.56 0 put back in scientific notation Measurement If asked a question where measurements are in different units it is important to change them all into the same unit before attempting to solve. Length Mass These are the most 0mm = cm 000mg = g common but others can 00cm = m 000g = kg be found in your tables 000m = km 000kg = tonne if asked
Percentages Percentages can come up in a variety of questions (Profit and Loss, VAT, Currency etc). To get a % of a number multiply the number by the percentage over 00. Example Calculate VAT of % on a Fridge costing 60 00,0 00 60.0 We want to calculate % of 60 Total cost of Fridge = 60 +.0 = 76.0 To express one number as a % of another put one over the other and multiply by 00 Example Express 5cm as a percentage of m (convert m to 00cm first) 5,500 00 7.5% 00 00 (Measurements must be in the same unit) To express a % increase or decrease in change in quantity, put the change over the original and multiply by 00 Example Calculate the % increase of a class that went from 0 students to 5 5 0 500 00 5% 0 (change in quantity is 5 0 = 5) To calculate the % Profit/ Loss put the profit/loss over the cost price and multiply by 00 Example A fruit importer buys apples for 0c and sells them for c. What is his percentage profit? Profit = Sales Costs = c 0c = c His profit PROFIT 00 % profit 00 00 0% COSTPRICE 0 0 Answer Example Fred sold his car for 750 making a profit of 5%. How much did he pay for the car. (This can also be asked as a VAT question and would be solved the very same way) Example Fred bought a fridge for 750 including VAT of 5%. What was the price before VAT was added. Cost Price + Profit = 750 750 is made up of Cost price plus the profit he made. 00% + 5% = 750 Cost is obviously00% of cost, Profit is a further 5% 5% = 750 So the sale price was 5% of the cost price 750 Divide by 5 to find % of the cost price 5 % = 6 00% = 6 x 00 = 600 Multiply by 00 to find cost price. The car cost 600 or the fridge was 600 before VAT was added.
Currency The easiest way to solve currency questions is to use cross multiplication. Example A supplier buys 00 parts for $6 each. They will be sold for a total of 8,000. Calculate the percentage profit on the cost price if the exchange rate is = $0.8. 00 x $6 = $00,800 Firstly work out total cost in dollars = $0.8 Write down exchange rate x = $00,800 Underneath put the amount you want to convert 0.8x = (00,800) Cross Multiply 00,800 x 0.8 Divide across by number next to x, 0.8 x = 0,000 This is the cost in euro Profit = Sales Costs = 8,000-0,000 = 8,000 PROFIT 00 8,000 00,800,000 Percentage Profit = 5% COSTPRICE 0,000 0,000
Interest Rates Interest is the money we receive or pay for investing or borrowing. We need to learn interest rate formula PxTxR I where 00 I = Interest P = Principal (the amount you invest/ borrow) T = Time R = Rate Simple Interest Interest only calculated once over the term, T of the investment. Compound Interest Interest is added at the end of every year to create a new principal for the next year. Example Calculate the difference between the simple and compound interest on an investment of 6,000 at 7% p.a (per annum) for years. PxTxR 00 6,000 7 6,000,60 00 00 I Simple interest (put figures into formula) Compound Interest (we calculate one years interest at a time and add this to principal) Year Principal 6000 Interest at 7% 0 Year Principal 60 Interest at 7% 9.0 Year Principal 6869.0 Interest at 7% 80.86 Value of investment 79.0 Total interest = (0 + 9.0 + 80.86) = 9.0 Difference between simple and compound = 9.0 60 = 89.0 Example What some of money invested at 6% p.a over years would amount to 5,056.0. To do this we first see what 00 would amount to at the same rate and time period Year Principal 00 Interest at 6% 6 Year Principal 06 Interest at 6% 6.6.6 00.6 If we invest 00 at 6% for years we get.6 so we can cross x 5056.0 multiply to investigate what we need to invest to get 5056.0.6x = 00(5056.0) Cross Multiply.6x = 505,60 x = 50560 500 Divide across by number next to x,.6..6 So if we invest 500 for years at 6% we will get 5056.0
Income Tax Questions normally ask you to calculate a person s net pay. Net pay (take home pay) = Gross Wages/ Income/ Salary Tax Due Tax Due = Gross Tax Tax Credits To calculate the tax payable multiply the gross pay by the tax rate (sometimes there will be two tax rates). Example James has income of 0,000. Tax is charged on the first,000 at rate of %. The rest is charged at 0%. His Tax credits are,000. Calculate James take home pay. Income is 0,000 0,000, 000 will be charged at % = 000 x % = 080 The remaining 6, 000 will be charged at 0% =6 000 x 0% = 600 + Gross Tax = 980 Tax Credits = 000 - Tax Due = 680 Take home pay ( 0,000 6,80) =,50 Example Joe has income of 0,000. Tax is charged on the first,500 at rate of %. The rest is charged at r%. His Tax credits are,00. Joe s take home pay is,655. In this example we must work backwards. Firstly this is a two-interest rate question.,500 is charged at % The remainder ( 0,000,500) = 5, 500 is charged at r% 0,000,655 = 6,5 Subtract the take home from the income to get the Tax Due,00 + 6,5 = 9,55 Add the Tax Due to the Tax Credits to get the Gross Tax Calculate % of,500, 500,90 This gives us the part of the Gross Tax which 00 relates to the first,500 9,55 -,90 = 6,55 This is the remainder of the Gross Tax which relates to the 5,500 So what we want to know is what % is this of 5,500. r 6,55 5,500 00 % 5
Speed, Time and Distance For questions involving Speed, Time and Distance we need to learn the following formulae. Average Speed Distnce Time Distnce Time Distance = Speed x Time Speed Give your answer in kilometres per hour (km/h) unless otherwise asked. If your answer is in kilometres per minute (km/m) multiply by 60 to get km/ h. Example A journey of 9km took 0 mins. Find the average speed. Distnce Average Speed Time Write down formula 9 = 0.5 km/m 0 Answer here is in kilometres per minute 0.5 x 60 = 7 km/ h Multiply by 60 to get km/ h Example A bus leaves Mullingar at.05 and travels to Dublin 50 kms away, at an average speed of 0km/ h. At what time does the bus arrive in Dublin. Distnce Time Speed Write down formula 50 Time.5 hours 0 Enter figures to give time in hours.5 60 = 50 minutes Multiply by 60 to get minutes = hours 0 mins Turn into Hours and Minutes.05 + Add hrs 0mins to.05.0.5 Bus arrives at.5 Example At what average speed should I drive to cover 66km in hours 0 mins. hours 0 mins = We cannot use combinations of hrs and mins ( x 60) + 0 = 0 + 0 = 0 mins so change hrs 0mins into 0 minutes. Distnce Average Speed Time Write down formula 66.9 km/ minute 0 Answer here is in kilometres per minute.9 x 60 = km/ h Multiply by 60 to get km/ h 6
Sets - A B A intersection B What is common to both A B A union B List all the elements in A and all the elements in B A ' A complement List everything outside of A A/ B A without B List the elements in A but don t include any of B The cardinal number of A How many elements are in A A A B A is a subset of B Everything in A is also in B. Doesn t mean that These symbols can be combined: everything in B is in A also. ( A B)' Everything outside of A intersection B ( A B) \ C All of A and all of B but don t list any elements of C Example Examples of things they may ask U = {,,,,5,6,7,8,9} A = {,,,6,9} A = 5 B = {,,7,9} B = C = {,5,7,9} C = A B = {,9} ( A B)' C = {,5,7} A B C = {9} ( A B) \ C = {,,,6} B C = {7,9} ( A B) C' ={} A ' = {,5,7,8} ( A \ C) ( C \ B) = {} A\B = {,,6} A ' ( B \ C) = {,,5,7,8} Example A survey of 50 households were asked whether they used gas, oil or electricity to heat their homes. 7 have electricity, 7 have gas, 9 have oil, 9 have gas and electricity, have electricity and oil and 6 have gas and oil. X represents all. Find X. X goes in center as it is in all 9 have G and E so (9 x) in region shown have E and O so ( x) in region shown 6 have G and O so (6 x) in region shown Adding all the entries ( x ) (6 x) (9 x) ( x) ( x) ( x) x 50 x 50 x 50 8 7 study Electricity. We already have x in there, (9 x) in there and ( x) in there. The remainder is 7 x ( x) (9 x) = + x which we put in the region shown and do likewise for Gas and Oil 7
Ratio and Proportion Firstly to find what fraction one number is of another we just put that number over the other. Ratios describe the way that we divide things up. If I want to divide some money between two people in the ratio : it means that for every I give one person I give the other. :: means that for every A gets, B gets and C gets We can change a ratio by multiplying or dividing all the terms by the same number. : is the same as : Divide both side by :6:9 is the same as :: Divide all across by : : is the same as :: Multiply across by (always get rid of fractions when using ratios) To split amounts into ratios we do the following:. Add up the ratios to get the total number of shares. Divide the total amount to be split by the sum of the ratios to get the amount per share. Multiply each ratio by the amount per share. Example 50 is to be divided :5 between Tom and Frank. How much does each get? :5 +5 = 7 shares Add the ratios to get the number of shares 50 50 The amount each share will get 7 Tom has shares x 50 = 00 Frank has 5 shares 5 x 50 = 50 Example A sum of money was divided between Tom and Frank in the ratio 8:. Tom received 00 more than Frank. How much did they each receive? Difference between what each got was 00 We are told this 8 shares shares = 00 The difference between the ratios 5 shares = 00 share = 0 5 Divide by 5 to find value of share Tom has 8 shares 8 x 0 = 0 Multiply each ratio by the amount Frank has shares x 0 = 0 per share. Total amount = 0 + 0 = 0 Add to get total amount 8
Indices With questions involving indices we must break down all of the terms in the question to the same base number. We can the use the rules below to simplify. 7 7 (a). 7 (b) (c) ( (e) 9 9 (f) 6 = 6 (g) 8 = 8 = 6 5 (h) and 5 Example Solve for x in the equation x x 5 5 6 x x ( 5 ) Change 5 into 5 6. 6 ) (d) 0 x x 5 5 6 Remove bracket to leave both sides in base 5 x = 6-x Let the indices (powers) equal each other x + x = 6 x s to one side, numbers to the other x = 6 6 x Divide across by Example 8 ) ( 5 the equation and express in the form 8 = 5x = The Equation 5x ( ) = Turn everything into base ( )( ) 5x Multiply the powers 5x = ( )( ) 5 x Let the powers equal each other 5 x Remove mixed fraction = (5 x) Multiply across by = 0 x Multiply to remove bracket x = 0 x s to one side, numbers to the other x = 7 7 x Divide across by 5x 9
Surds Surds are irrational numbers in the form Some important points: ab = a. b and a. b = ab Therefore can be broken down into. 6 = 6 a a 6 6 therefore b b 6 Terms with the same surd part can be added and subtracted Therefore 5 6 + 6-6 = 6 Any surd squared is equal to the term under the root sign Therefore a 5 a 5 Equations involving surds can be solved by squaring both sides of the =. This gets rid of the surd part to leave you with a simple or quadratic equation. Example ( )( ) ( )( ) The expression ( ) ( ) Open up the brackets. Multiply in by number outside brackets 6 6() 8 6 Answer Example ( 6 ) 0( 5 0) and express your answer in the form a b ( 6 ) 0( 5 0) The expression 6 0 5 0 0 Multiply in by number outside brackets 8 9 50 00 using a. b = ab 9 () 5 (0) Change each term into either a number or b () (5) 0 6 5 0 8 9 Answer in the form a b 0
Estimation The estimation question has normally two parts. The first part will ask you to estimate an answer by rounding off numbers. The second part will ask you to find the exact value of the same question. It is important to take a step by step approach using the order of operation rules rather than entering all the information straight into the calculator. Brackets then powers then multiplication/ division then addition/ subtraction Example By rounding to the nearest whole number, estimate the value of.5.7 0.6 5. (5.).5.7 0.6 5. (5.) Our expression 5 (5) Round off each figure to nearest whole number 5 5 Work out Brackets and Powers 5 5 Must work out x before, simplify bottom 0 0 Make sure you go through all the steps.5.7 0.6 Then evaluate correct to two decimal places. 5. (5.).5.7 0.6 5. (5.) Our expression.5.7 0.8 5. 6.0 Work out Brackets and Powers.5.8 5. 6.0 Work out.7 x 0.8 first.5.8 0.6 5. 6.0 9.9 0.6.8568 9.9.86 Round off to two decimals