3. Higher Tier - Number revision Contents :Calculator questions Long multiplication & division Best buy questions Estimation Units Speed, Distance and Time Density, Mass and Volume Percentages Products of primes HCF and LCM Indices Standard Form Ratio Fractions with the four rules Upper and lower bounds Percentage error Surds Rational and irrational numbers Recurring decimals as fractions Direct proportion Inverse proportion Graphical solutions to equations

4. Calculator questions Which buttons would you press to do these on a calculator ? 2.5 + 4.1 3.5 1.7 + 2.8 2.3 – 0.2 1.5 6 2.5 3 1000 4 1.7 2 + 5.2 2 6.31 8.5 x 10 3 3.4 x 10 -1 9.2 6.1 7.5 8 – 1 1 3.6 2.3 –

5. Long multiplication Use the method that gives you the correct answer !! Question : 78 x 59 9 50 708 Total = 3500 + 400 + 630 + 72 Answer : 4602 3500400 63072 Now try 84 x 46 and 137 x 23 and check on your calculator !!

6. Long division Again use the method that gives you the correct answer !! Question : 2987  23 23 46 69 92 115 138 161 184 207 230 23 times table 2 9 8 72 9 8 723 29 1 6 68 2 22 227 9 20 Answer : 129 r 20 Now try 1254  17 and check on your calculator – Why is the remainder different?

7. 0.95L 2.1L 78p 32p OR 34p 87p Always divide by the price to see how much 1 pence will buy you Beans Large  400  87 = 4.598g/p Small  150  34 = 4.412g/p Large is better value (more grams for every penny spent) Milk Large  2.1  78 = 0.0269L/p Small  0.95  32 = 0.0297L/p Small is better value OR (looking at it differently) {Large  78  2.1 = 37.14p/L Small  32  0.95 = 33.68p/L} Best buy questions

8. Estimation If you are asked to estimate an answer to a calculation – Round all the numbers off to 1 s.f. and do the calculation in your head. DO NOT USE A CALCULATOR !! e.g. Estimate the answer to 4.12 x 5.98  4 x 6 = 24 Always remember to write down the numbers you have rounded off Estimate the answer to these calculations 1. 58 x 21 2. 399 x 31 3. 47 x 22 4. 4899  46 5. 7.12 x 39.2 0.87 6. 377  19 7. 1906  44 8. 4.89 x 6.01 1.92 9. 360 x 87 10. 58 x 21

9. Units Metric length conversions kmmcmmm x 1000x 10x 100 ÷ 1000÷ 10 ÷ 100 Metric weight conversions kggcgmg x 1000x 10x 100 ÷ 1000÷ 10 ÷ 100 Metric capacity conversions kllclml x 1000x 10x 100 ÷ 1000÷ 10 ÷ 100 Learn this pattern for converting between the various metric units Learn these rough conversions between imperial and metric units 1 inch  2.5 cm 1 yard  0.9 m 5 miles  8 km 2.2 lbs  1 kg 1 gallon  4.5 litres

10. Speed, Distance, Time questions Speed, Distance and Time are linked by this formula To complete questions check that all units are compatible, substitute your values in and rearrange if necessary. S = DTDT 1.Speed = 45 m/s Time = 2 minutes Distance = ? 2.Distance = 17 miles Time = 25 minutes Speed = ? 3.Speed = 65 km/h Distance = 600km Time = ? S = D T 45 m/s and 120 secs 45 = D. 120 45 x 120 = D D = 5400 m S = D T 65 = 600. T T = 9.23 hours S = D T 17 miles and 0.417 hours S = 17. 0.417 S = 40.8 mph T = 600. 65

11. Density, Mass, Volume questions Density, Mass and Volume are linked by this formula To complete questions check that all units are compatible, substitute your values in and rearrange if necessary. D = MVMV 1.Density = 8 g/cm 3 Volume = 6 litres Mass = ? 2.Mass = 5 tonnes Volume = 800 m 3 Density = ? 3.Density = 12 kg/m 3 Mass = 564 kg Volume = ? D = M V 8 g/cm 3 and 6000 cm 3 8 = M. 6000 8 x 6000 = M M = 48000 g D = M V 12 = 564. V V = 47 m 3 D = M V 800 m 3 and 5000 kg D = 5000. 800 D = 6.25 kg/m 3 V = 564. 12 ( or M = 48 kg)

12. Simple % Percentage increase and decrease A woman’s wage increases by 13.7% from £240 a week. What does she now earn ? 13.17% of £240 Increase: 13.17 100 240 = x New amount: 31.608 Her new wage is £271.61 a week 240 + 31.608 =271.608 Percentages of amounts 25% = 20% = 50% =2% = 45% = 1% = 75% = 30% = 10% = 85% = 5% = £600 (Do these without a calculator)

13. Simple % Fractions, decimals and percentages 83% 9 50 0.04 56%28% 19 20 0.92 4 25 0.17 % Frac Dec 50% 0.5 1212 Copy and complete:

14. Reverse % e.g. A woman’s wage increases by 5% to £660 a week. What was her original wage to the nearest penny? Original amount = 660 ÷ 1.05 = £628.57 Original amount x 1.05 £660 Original amount £660 ÷ 1.05 e.g. A hippo loses 17% of its weight during a diet. She now weighs 6 tonnes. What was her former weight to 3 sig. figs. ? Original weight = 6 ÷ 0.83 = 7.23 tonnes Original weight x 0.83 6 ton. Original weight 6 ton. ÷ 0.83

15. Repeated % e.g. A building society gives 6.5% interest p.a. on all money invested there. If John pays in £12000, how much will he have in his account at the end of 5 years. He will have = 12000 x (1.065) 5 = £16441.04 e.g. A car loses value at a rate of approximately 23% each year. Estimate how much a $40000 car be worth in four years ? The car’s new value = 40000 x (0.77) 4 = $14061 (nearest $) £12000 x 1.065 ? This is not the correct method: 12000 x 0.065 = 780 780 x 5 = 3900 12000 + 3900 = £15900 £40000 x 0.77 ? This is not the correct method: 40000 x 0.23 = 9200 9200 x 4 = 36800 40000 – 36800 = $3200

16. Products of primes Express 40 as a product of primes 40 220 210 25 40 = 2 x 2 x 2 x 5 (or 2 3 x 5) Express 630 as a product of primes 630 2315 3105 335 57 630 = 2 x 3 x 3 x 5 x 7 (or 2 x 3 2 x 5 x 7) Now do the same for 100, 30, 29, 144

17. HCF Expressing 2 numbers as a product of primes can help you calculate their Highest common factor LCM Expressing 2 numbers as a product of primes can also help you calculate their Lowest common multiple e.g. Find the highest common factor of 84 and 120. 84 = 2 x 2 x 3 x 7 120 = 2 x 2 x 2 x 3 x 5 Highest common factor = 2 x 2 x 3 = 12 Pick out all the bits that are common to both. e.g. Find the lowest common multiple of 300 and 504. 300 = 2 2 x 3 x 5 2 504 = 2 3 x 3 2 x 7 Lowest common multiple = 2 3 x 3 2 x 5 2 x 7 = 12600 Pick out the highest valued index for each prime factor. LCM Consider the numbers 16 and 20. Their multiples are: 16, 32, 48, 64, 80, 96 and 20, 40, 60, 80, 100 Their lowest common multiple is 80 HCF Consider the numbers 20 and 30. Their factors are: 1, 2, 4, 5, 10, 20 and 1, 2, 3, 5, 6, 10, 15, 30 Their highest common factor is 10

18. Indices 10 -4 19 0 7 5  7 3 16 3/2 2 -3 9 1/ 2 1 15 3 -2 36 -1/2 2 -1 4343 81 1/4 Evaluate: 2 5 x 2 1

19. Standard form Write in Standard Form 3 6000.041 0.0001 8 900 000 000 56 x 10 3 9.6 0.2 Write as an ordinary number 8.6 x 10 -1 1 x 10 2 7 x 10 -2 5.1 x 10 4 9.2 x 10 3 3.5 x 10 -3 4.7 x 10 9 8 x 10 -3 Calculate 4.6 x 10 4 ÷ 2.5 x 10 8 with a calculator Calculate 3 x 10 4 x 7 x 10 -1 without a calculator Calculate 1.5 x 10 6 ÷ 3 x 10 -2 without a calculator

20. Ratio Equivalent Ratios 1 : ? 0.5 : ? ? : 1 2100 : ? ? : 12 14 : ? ? : 12 ? : 6 ? : 10 21 : ? 49 : ? 7:2 Splitting in a given ratio £600 is split between Anne, Bill and Claire in the ratio 2:7:3. How much does each receive? Total parts = 12 Anne gets 2 of 600 = £100 12 Claire gets 3 of 600 = £150 12 Basil gets 7 of 600 = £350 12

21. Fractions with the four rules + – × ÷ Always convert mixed fractions into top heavy fractions before you start When adding or subtracting the “bottoms” need to be made the same When multiplying two fractions, multiply the “tops” together and the “bottoms” together to get your final fraction When dividing one fraction by another, turn the second fraction on its head and then treat it as a multiplication Learn these steps to complete all fractions questions:

22. Fractions with the four rules 4⅔ + 1½ 14 3 3232 += 37 6 = 9696 28 6 += = 6 1616 4⅔  1½ 14 3 3232  = 14 3 2323  = 28 9 = = 3 1919

23. A journey of 37 km measured to the nearest km could actually be as long as 37.49999999…. km or as short as 36.5 km. It could not be 37.5 as this would round up to 38 but the lower and upper bounds for this measurement are 36.5 and 37.5 defined by: 36.5 < Actual distance < 37.5 e.g. Write down the Upper and lower bounds of each of these values given to the accuracy stated: Upper and lower bounds 9m (1s.f.) 85g (2s.f.) 180 weeks (2s.f.) 2.40m (2d.p.) 4000L (2s.f.) 60g (nearest g) 8.5 to 9.5 84.5 to 85.5 175 to 185 2.395 to 2.405 3950 to 4050 59.5 to 60.5 e.g. A sector of a circle of radius 7cm makes an angle of 32 0 at the centre. Find its minimum possible area if all measurements are given to the nearest unit. (  = 3.14) Area = (  /360) x  x r x r Minimum area = (31.5/360) x 3.14 x 6.5 x 6.5 Minimum area = 11.61cm 2 32 0 7cm

24. % error If a measurement has been rounded off then it is not accurate. There is a an error between the measurement stated and the actual measurement. The exam question that occurs most often is: “Calculate the maximum percentage error between the rounded off measurement and the actual measurement”. e.g. This line has been measured as 9.6cm (to 1d.p.). Calculate the maximum potential error for this measurement. Upper and lower bounds of 9.6 cm (1d.p.)  Maximum potential difference (MPD) between actual and rounded off measurements  Max. pot. % error = (MPD/lower bound) x 100 9.55 to 9.65 = (0.05/9.55) x 100 = 0.52% 0.05

25. Simplifying roots Tip: Always look for square numbered factors (4, 9, 16, 25, 36 etc)  20  4 x  52  5 88  4 x  22  2  45  9 x  53  5  72  36 x  26  2  700  100 x  710  7 e.g. Simplify the following into the form a  b

26. Surds A surd is the name given to a number which has been left in the form of a root. So  5 has been left in surd form. A surd or a combination of surds can be simplified using the rules:  M x  N =  MN and visa versa  M ÷  N =  M/N and visa versa Tips: Deal with a surd as you would an algebraic term and always look for square numbers SIMPLIFYING EXPRESSIONS WITH SURDS IN (  3 – 1) 2  5(  5 +  20)  135 ÷  3  12  4 x  32  3  9 x  53  5 5 +  100 15 (  3 – 1) 3 –  3 –  3 + 1 4 – 2  3  45 LEAVING ANSWERS IN SURD FORM Calculate the length of side x in surd form (non-calculator paper): x 66  14 Pythagoras  (  14) 2 = (  6) 2 + x 2 14 = 6 + x 2 x =  8 Answer: x = 2  2

27. Rational and irrational numbers Rational numbers can be expressed in the form a/b. Terminating decimals (3.17 or 0.022) and recurring decimals (0.3333..or 4.7676..) are rational. Irrational numbers cannot be made into fractions. Non-terminating and non-recurring decimals (3.4526473… or  or  2) are irrational. 16  1/5 320 2.3/5.7 5252 2.7 State whether the following are rational or irrational numbers: What do you need to do to make the following irrational numbers into rational numbers:  320 22 5353 66 33

28. Recurring decimals as fractions Express 0.77777777….. as a fraction. Let n = 0.77777777….. so 10n = 7.77777777….. so 9n = 7 so n = 7/9 Express 2.34343434….. as a fraction. Let n = 2.34343434….. so 100n = 234.34343434….. so 99n = 232 so n = 232/99 Express 0.413213213….. as a fraction. Let n = 0.4132132132….. so 10000n = 4132.132132132….. and 10n = 4.132132132…… so 9990n = 4128 so n = 4128/9990 n = 688/1665 Learn this technique which changes recurring decimals into fractions:

29. Direct proportion If one variable is in direct proportion to another (sometimes called direct variation) their relationship is described by: p  t p = kt Where the “Alpha” can be replaced by an “Equals” and a constant “k” to give : e.g. y is directly proportional to the square of r. If r is 4 when y is 80, find the value of r when y is 2.45. Write out the variation: y  r 2 Change into a formula: y = kr 2 Sub. to work out k: 80 = k x 4 2 k = 5 So: y = 5r 2 And: 2.45 = 5r 2 Working out r: r = 0.7 Possible direct variation questions: x  p t  h 2 s  3  v c   i g  u 3 g = ku 3 c = k  i s = k 3  v t = kh 2 x = kp

30. Inverse proportion If one variable is inversely proportion to another (sometimes called inverse variation) their relationship is described by: p  1/t p = k/t Again “Alpha” can be replaced by a constant “k” to give : e.g. y is inversely proportional to the square root of r. If r is 9 when y is 10, find the value of r when y is 7.5. Write out the variation: y  1/  r Change into a formula: y = k/  r Sub. to work out k: 10 = k/  9 k = 30 So: y = 30/  r And: 7.5 = 30/  r Working out r: r = 16 (not 2) Possible inverse variation questions: x  1/p t  1/h 2 s  1/ 3  v c  1/  i g  1/u 3 g = k/u 3 c = k/  i s = k/ 3  v t = k/h 2 x = k/p

31. Graphical solutions to equations If an equation equals 0 then its solutions lie at the points where the graph of the equation crosses the x-axis. e.g. Solve the following equation graphically: x 2 + x – 6 = 0 All you do is plot the equation y = x 2 + x – 6 and find where it crosses the x-axis (the line y=0) y x 2-3 y = x 2 + x – 6 There are two solutions to x 2 + x – 6 = 0 x = - 3 and x =2

32. Graphical solutions to equations If the equation does not equal zero : Draw the graphs for both sides of the equation and where they cross is where the solutions lie e.g. Solve the following equation graphically: x 2 – 2x – 11 = 9 – x Plot the following equations and find where they cross: y = x 2 – 2x – 20 y = 9 – x There are 2 solutions to x 2 – 2x – 11 = 9 – x x = - 4 and x = 5 y x y = x 2 – 2x – 11 y = 9 – x -45

33. If there is already a graph drawn and you are being asked to solve an equation using it, you must rearrange the equation until one side is the same as the equation of the graph. Then plot the other side of the equation to find the crossing points and solutions. e.g. Solve the following equation using the graph that is given: x 3 – 4x + 5 = 5x + 5 y x y = x 3 – 8x + 7

34. Rearranging the equation x 3 – 4x + 5 = 5x + 5 to get x 3 – 8x + 7 : Add 2 to both sidesx 3 – 4x + 5 = 5x + 5 x 3 – 4x + 7 = 5x + 7 x 3 – 8x + 7 = x + 7 Take 4x from both sides So we plot the equation y = x + 7 onto the graph to find the solutions y x y = x 3 – 8x + 7 -3 0 3 Solutions lie at –3, 0 and 3 y = x + 7

35. State the graphs you need to plot to solve the following equations describing how you will find your solutions: 1. 3x 2 + 4x – 2 = 0 2. 7x + 4 = x 2 – 4x 3. x 4 + 5 = 0 4. 0 = 8x 2 – 5x 5. 2 x = 9 6. 6x 3 = 2x 2 + 5 If you have got the graph of y= 4x 2 + 5x – 6 work out the other graph you need to draw to solve each of the following equations: 1. 4x 2 + 4x – 6 = 0 2. 4x 2 + x - 2 = 7 3. 4x 2 – 3x = 2x 4. 3x 2 = – 5 Solve this equation graphically: x 3 + 8x 2 + 3x = 2x 2 – 2x