Numbers

1. Numbers http://teachable.net/res.asp?r=766

2. Contents : Calculator questions Best buy questions Long multiplication Long division Simple fractions questions Negative numbers Rounding Estimating Percentages Types of number Products of primes HCF and LCM Indices Simplifying roots Standard Form Ratio Fractions with the four rules Recurring decimals as fractions Units Distance, Speed, Time questions Density, Mass, Volume questions

3. 9.26.1 7.5 8 – 8.5 x 103 3.4 x 10-1 2.5 3 1.72 + 5.22 6.31 1 1 3.6 2.3 – 1000 4 Calculator questions Which buttons would you press to do these on a calculator ? 1.7 + 2.8 2.3 – 0.2 2.5 + 4.1 3.5 1.56

4. 87p 34p 400g 150g OR 78p 32p OR 2.1L 0.95L Best buy questions 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}

5. Long multiplication Use the method that gives you the correct answer !! Question : 78 x 59 70 8 50 3500 400 9 630 72 Total = 3500 + 400 + 630 + 72 Answer : 4602 Now try 84 x 46 and 137 x 23 and check on your calculator !!

6. 23 times table 23 46 69 92 115 138161 184207230 29 227 68 23 2987 Long division Again use the method that gives you the correct answer !! Question : 2987  23 2 9 1 6 22 20 Answer : 129 r 20 Now try 1254  17 and check on your calculator – Why is the remainder different?

7. 4 5 3 ? 5 9 6 7 5 6 9 . 12 ? ? 12 20 ? 24 5 . 25 ? ? 3. 1. 2. 4. 1. 3. 2. = = = = 9 . 12 = 9  12 = 0.75 1 6 5 6 1 2 5 9 3. 1. 4. 2. of of of of $72 £30 £30 £30 4. Put the following fractions in order of size, smallest to largest: 3 4 2 3 5 8 4 7 Simple fractions questions Equivalent fractions Fractions into decimals Divide top by bottom Fractions of amounts Divide by the bottom then times by the top

8. 20  - 5 = - 4  3 = - 8  - 4 = - 5  - 2 = - 4  4 = - 3 + 5 = - 2 - 4 = - 7 + 2 = - 4 - - 2 = 5 - 12 = (- 7)2 = - 9 - - 11 = 5 + - 7 = - 3 + - 1 = - 6 + 4 = 4. 1. 4. 2. 1. 5. 2. 2. 5. 3. 4. 3. 1. 6. 3. Negative numbers Put these in order - smallest first - 4 , - 3.6 , - 0.4 , - 0.36 , - 1 , 0 , 2 , - 1.4 Up and down the scale Two signs next to each other - 3 - 1 = - 4 + 2 = Multiplication and Division Signs same  +ve answer Signs different  -ve answer

9. 4715.692803 cm Rounding • Round this number off to : • 1 decimal place • (b) 1 significant figure • 2 decimal places • (d) 2 significant figures • the nearest centimetre • (f) the nearest metre • 3 decimal places • (h) 3 significant figures • 4715.7 cm • (b) 5000 cm • 4715.69 cm • (d) 4700 cm • 4716 cm • (f) 47 m • 4715.692 cm • (h) 4720 cm 14.9999 But there is always a trickier one • Round this number off to : • the nearest whole number • (b) 3 significant figures • 2 decimal places • 15 • (b) 15.0 • 15.00

10. 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 !! 4899  46 1906  44 377  19 6. 10. 4. 7. 5. 2. 8. 1. 9. 3. 360 x 87 4.89 x 6.01 1.92 58 x 21 47 x 22 399 x 31 58 x 21 7.12 x 39.2 0.87 Estimating 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

11. £600 Percentages Percentage increase and decrease A woman’s wage increases by 13.7% from £240 a week. What does she now earn ? Increase: New amount: 13.17% of £240 240 + 31.608 = 271.608 Her new wage is £271.61 a week 13.17 100 x 240 = 31.608 30% = 75% = 10% = 1% = Percentages of amounts 25% = 45% = 5% = (Do these without a calculator) 85% = 20% = 50% = 2% =

12. % Frac Dec 83% 56% 0.04 0.17 19 20 9 50 1 2 28% 4 25 0.92 Percentages Fractions, decimals and percentages 50% Copy and complete: 0.5

13. ÷ 1.05 ÷0.83 x 1.05 x 0.83 Original amount Original weight £660 Original weight Original amount 6 ton. £660 6 ton. 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 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

14. x 1.065 x 1.065 x 1.065 x 1.065 x 1.065 £12000 ? x 0.77 x 0.77 x 0.77 x 0.77 £40000 ? Repeated % This is not the correct method: 12000 x 0.065 = 780 780 x 5 = 3900 12000 + 3900 = £15900 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 This is not the correct method: 40000 x 0.23 = 9200 9200 x 4 = 36800 40000 – 36800 = $3200 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 $)

15. 16 25 13 100 3 2 27 20 7 1 12 9 6 11 Types of number • From this set of numbers • list the: • Odd numbers • Even numbers • Multiples of 8 • Factors of 12 • Prime numbers • Square numbers • Cube numbers • Some useful words to know the meaning of: • Sum = add together • Product = multiply together • Difference = subtract one number from another • Reciprocal of a number = 1 divided by the number • (e.g. Reciprocal of 4 = ¼ or 0.25)

16. 40 5 3 2 2 2 2 3 315 105 7 5 10 20 35 Products of primes Express 40 as a product of primes 40 = 2 x 2 x 2 x 5 (or 23 x 5) 630 Express 630 as a product of primes Now do the same for 100 , 30 , 29 , 144 630 = 2 x 3 x 3 x 5 x 7 (or 2 x 32 x 5 x 7)

17. Finding the HCF and LCM of a pair of numbers HCF stands for the Highest Common Factor (the biggest number that will go into both numbers) LCM stands for Lowest Common Multiple (the first number to appear in both numbers times table) e.g. Find the HCF and LCM of the two numbers 140 and 112 Write both numbers as a product of primes 140 = 2 x 2 x 5 x 7 and 112 = 2 x 2 x 2 x 2 x 7 For the HCF write out all the primes that appear in both answers HCF = 2 x 2 x 7 = 28 For the LCM write out the largest number of each prime that exists in either number LCM = 2 x 2 x 2 x 2 x 5 x 7 = 560

18. 104 9 43 Indices 52 91 32 102 75  73 23 190 26 53 171 25

19. Simplifying roots Tip: Always look for square numbered factors (4, 9, 16, 25, 36 etc) e.g. Simplify the following into the form a b 12 4 x 3 2 3 8 4 x 2 2 2 45 9 x 5 3 5 72 36 x 2 6 2 700 100 x 7 10 7

20. Standard form Write in Standard Form Write as an ordinary number 9.6 0.0001 4.7 x 109 8 x 10-3 3600 0.041 1 x 102 5.1 x 104 23600 0.2 7 x 10-2 8.6 x 10-1 8900000000 9.2 x 103 3.5 x 10-3 0.003 46.7 6 x 106 2 x 100 Do 3 x 104 x 7 x 105 with and without a calculator

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

22. Fractions with the four rules + – × ÷ Learn these steps to complete all fractions questions: • 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

23. 14 3 14 3 14 3 3 2 3 2 2 3 = = =  +  28 6 9 6 = + 28 9 37 6 = = 6 3 1 6 1 9 = = Fractions with the four rules 4⅔ + 1½ 4⅔ 1½

24. Recurring decimals as fractions Learn this technique which changes recurring decimals into 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 0.13213213….. as a fraction. Let n = 0.132132132….. so 1000n =132.132132132….. so 999n = 132 so n = 132/999 n = 44/333

25. x 1000 x 1000 x 1000 x 100 x 100 x 100 x 10 x 10 x 10 km kl kg g l m cg cl cm mg ml mm ÷ 1000 ÷ 1000 ÷ 1000 ÷ 10 ÷ 10 ÷ 10 ÷ 100 ÷ 100 ÷ 100 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 Learn these rough conversions between imperial and metric units Learn this pattern for converting between the various metric units Metric length conversions Metric weight conversions Metric capacity conversions

26. D T S = S = S = S = D T D T D T S = T = 65 = 600 . 65 17 . 0.417 600 . T 45 = D . 120 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. • Speed = 45 m/s • Time = 2 minutes • Distance = ? 2. Distance = 17 miles Time = 25 minutes Speed = ? 3. Speed = 65 km/h Distance = 600km Time = ? 45 m/s and 120 secs 17 miles and 0.417 hours 45 x 120 = D S = 40.8 mph T = 9.23 hours D = 5400 m

27. M V D = D = D = D = M V M V M V V = D = 12 = 5000 . 800 564 . 12 564 . V 8 = M . 6000 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. • Density = 8 g/cm3 • Volume = 6 litres • Mass = ? 2. Mass = 5 tonnes Volume = 800 m3 Density = ? 3. Density = 12 kg/m3 Mass = 564 kg Volume = ? 8 g/cm3 and 6000 cm3 800 m3 and 5000 kg 8 x 6000 = M D = 6.25 kg/m3 V = 47 m3 M = 48000 g ( or M = 48 kg)