6 Number System 6.5 Estimating Answers 1. Express each of the following correct to significant figures: 96.6 16.5 1.90 5 (d) 0.00 681 (e) 50.9 (f) 0.000 60 8 (g) 0.00 71 (h) 5.98 (i) 6.98. Write each of the following correct to the number of significant figures (s.f.) indicated. 08.67 ( s.f.) 0.099 8 (1 s.f.) 0.65 ( s.f.) (d) 0.00 07 ( s.f.). Write 1.00 7 correct to 5 s.f. s.f. s.f.. Nigel, Ali and Sue were given ten calculations to do. The following table shows their answers. For each calculation, only one of the three obtained the correct answer. By working out an estimate for each question, decide who was correct in each calculation. Question Nigel's answer Ali's answer Sue's answer 1.0.9 1.98.958 6.8 0.99 6.7 6..16 5.6.8 10. 6. 9.9 89. (d).6 1. 8. 1.6.5 (e) 5. 8.01. 1. 1. (f) 7.1 7.1 50.1 5.1 6.01 (g) 7.08.66.68 11.08 5.8 (h) 76.16 7.6 1.6 8.6 1. (i) 1.7 8.5 50.85 107.95 0.75 (j) 8. 0.97.7 16.16 8.6 91
5. Without finding the exact answer: Which of the following is nearest in value to 696. + 71. + 71. + 68.? 1.7, 8.09, 90.7 or 1.86 Which of the following is nearest in value to. 1 00.? 57, 87, 870 or 570 Which of the following is nearest in value to 9 7 + 10 1? 18, 67, 1.8 or 6.7 6. Estimate, correct to 1 significant figure, the value of 01. 9.. 7. Express each number correct to 1 significant figure and work out an estimate to 19. 7 9. 75. 1. Use your calculator to evaluate 19. 7 9. 75 1. correct to significant figures. 8. Bottles of mineral water cost 9 p each. Estimate the cost of 1 bottles. Show how you obtained your estimate. Without using a calculator, work out the exact cost of 1 bottles of mineral water at 9 p each. (MEG) 9. Charlie has to work out 5. 9. 1.. He uses a calculator and gets 5.88 for his answer. Saeeda works out an approximate answer for the question. She knows that Charlie's answer must be wrong. (i) Write down approximate values for 5.,.9 and.1. (ii) Use these approximations to find a rough answer to Charlie's calculation. What is the mistake in Charlie's answer? 9
10. The rectangular glass tank shown in the diagram contains 1 litre (1000 cm of water. d cm Not to scale 11.6 cm 9.1 cm Sanjay wanted to find the depth (d cm) of the water. He multiplied 11.6 by 9.1 on his calculator and wrote down the answer. He then divided 1000 by this answer. Explain how you could use your calculator to find the depth without writing down the answer to 11. 6 9. 1. Work out the depth of the water, and write down all the figures on your calculator display. (MEG) 11. Estimate the answer to the following. 8. + 9. 7 5. 8 10. 1 (OCR) 1. Work out the value of. Work out 6 15 + 15. Use approximations to estimate the value of You must show your working. 7. 8 6. 1 195. ( ) 6.6 Using Brackets and Memory on a Calculator 1. Use a calculator to evaluate each of the following: 80 96 + 15 059 15 11 ( ) ( ) ( + ) ( ) ( + ) [( + ) ( )] ( ) + ( ) 5 + 118 7 (d) 51 + 696 15 (e) 18 69 1 19 (f) 1 70 11 19 11 (g) 77175 17 18 78 57 [ ] (h) 50 90 115 798 869 70 9
. For each of the following expressions, evaluate, giving your answer correct to decimal places; express each number correct to the nearest whole number and give an estimate to check your calculations. ( ) (i) 6. + 9.. (ii) 6. + 9.. (iii). 5 0. 615 (iv). 08. 16. 1. (v) 11. 1 18. 1. ( ) (vi) 9. + 6. 19.. Use your calculator to work out the value of 608. 97. 581 + 7 ( ) Write down the full calculator display. (i) Write down a calculation that could be done mentally to check the answer to part using numbers rounded to one significant figure. (ii) Write down the answer to your calculation in part (i). (MEG). Work out: 06. 5. 78 1 11 86 7 5. (MEG) 5. Gabriel buys a packet of 18 biscuits. The packet weighs 85 g. Gabriel wants to calculate the weight of one of these biscuits. He presses the following buttons on his calculator. 1 8 8 5 = Explain what is wrong with his calculation. Calculate the weight of one of these biscuits. Give your answer to the nearest gram. Gabriel checks his answer without using a calculator. Show how you can use approximation to check that his answer is of the right order. You must show all your working. 6. Use your calculator to work out: 95. + 18. 9 5. 17. Write down your full calculator display. Give your answer to significant figures. 9
7. Calculate the following. 57. 76. (d) 9. + 05. 9. 05. 9. 065. 0. 01 Give your answer to the nearest integer. ( ) (e) + 5cos 0 (OCR) 6.7 Upper and Lower Bounds 1. Write down the upper and lower bounds for each of the following measurements. 56 g.0 litres.5 metres (d) 5.6 km (e) 17.8 metres (f) 8.5 kg (g) 17. seconds (h) 0.5 mm (i) 1.9 cm. Find the upper and lower bounds for each of the calculations shown below, assuming the dimensions given are subject to rounding errors. (Give a sensible answer to each calculation.) The perimeter of a rectangle 65 cm by 8 cm. The area of a rectangle 65 cm by 8 cm. The perimeter of an octagon of side mm. (d) The volume of a cube of edge length 96 mm. (e) The total weight of 5 objects, each weighing.6 kg.. Angela measures the lengths of some sticks to the nearest centimetre. She arranges them in groups. The length of the sticks in the shortest group is 1 cm, to the nearest centimetre. (i) What is the smallest possible length for a stick in this group? (ii) What is the smallest possible length for a stick which is not in this group? Angela measures the lengths of some other sticks. She records the length of one of these sticks as 5. cm, to the nearest tenth of a centimetre. What is the smallest possible length of this stick?. Sections of a railway line are measured to the nearest metre as either 00 m or 80 m. What are the bounds on the total length of 15 sections, consisting of eight 00 m sections and seven 80 m sections? 95
5. The area of a rectangle is 5. square centimetres, correct to 1 decimal place. The length of this rectangle is 8. centimetres, correct to 1 decimal place. From this information, write down (i) the largest value (ii) the smallest value that the length of the rectangle could have. Use your answers in to calculate the largest possible width of the rectangle. (NEAB) 6. The formula S F = is used in engineering. A F = 819, correct to significant figures A = 9., correct to significant figures. For the value of F, write down (i) the upper bound (ii) the lower bound. For the value of A, write down (i) the upper bound (ii) the lower bound. Calculate (i) the upper bound (ii) the lower bound for the value of S for these values of F and A. Write down all the figures on your calculator display. (d) Write down this value of S correct to an appropriate number of significant figures. (LON) 7. Cases each weigh 0 kg to the nearest kilogram. What is the least that six cases could weigh? 8. I wish to paint the outside walls of my house. A tin of paint covers 5 m, correct to the nearest 5m. The outside walls of my house have an area of 0 m, correct to the nearest 10 m. Calculate the maximum number of tins of paint that I may have to buy. (OCR) 9. The time period T of a simple pendulum of length l is given by the formula T = π l g where g is the acceleration due to gravity. The length of a simple pendulum is given as 0 cm, correct to significant figures. The value of g is given as 9.8, correct to significant figures. Calculate the greatest value of T. Give your answer correct to significant figures. (Edexcel) 96
10. A circle has an area of 100 cm, measured to the nearest square centimetre. What is the lower bound of the radius? 6.8 Number System 1. In the following list, which are irrational numbers? 0.5, 1, 1, 16, π, 06., 7, 6. Express the following fractions as recurring decimals. (e) (i) 17 18 1 7 (f) (j) 5 6 5 7 (g) 5 11 11 6 (d) (h) 9 1. Express each of the following as a decimal and indicate whether it is recurring or non-recurring. (e) (i) 9 9 8 7 11 (f) (j) 1 0 1 18 59 99 (g) 1 6 11 15. Which of the following numbers are rational? (d) (h) 7 61 90 0 1 (i) 1 (ii) π (iii) When p and q are two different irrational numbers, p q can be rational. Write down one example to show this. Write down a fraction which is equal to the recurring decimal 0. 066... 5. Write down a rational number beween 1. and 1.5. Write down an irrational number between 1. and 1.5. 6. State whether the following numbers are rational or irrational. 1 (d) 9 + 99 (e) 100 + 1000 (f) If rational, state the value, expressed as simply as possible. π 8 6 8 97
7. The recurring decimal 0. 07 07 07... is a rational number. The following method is used to convert it to a fraction. Complete the working. (i) Let x = 0. 07 07 07 (A) then 1000 x =? (B) (ii) (iii) Take (A) from (B) to give 999x =? Hence find x as a fraction in its lowest terms. Using a similar method, or otherwise, find the fraction equivalent to the recurring decimal 0. 18 71 871 87... Give your answer in its lowest term. 8. Find the fraction equivalent of 0. 8 571 8... 0. 07 69 0... 0. 01 698 50 1... 9. Express the recurring decimal 0.777..... as a fraction. Give your answer in its simplest form. 6.9 Surds 1. If a = 1+ and b = 1, state whether the following numbers are rational or irrational. a a + b ab ( ) (d) a b (e) b (f) a + b ( ) (h) ( a b) ( a + b) (i) ab (g) a b. Simplify the expression 1 6, leaving your answer in surd form.. What could be added to + to give an answer which is rational? State whether the following are rational or irrational. (i) (ii) π (iii) 7 (iv) 1.... (NEAB) 98
. Using examples or counter examples, decide whether the following statmements are true or false. When you add two irrational numbers, your answer is still irrational. When you multiply two irrational numbers, your answer is still irrational. When you multiply an irrational number by a rational number, your answer is still irrational. 5. Simplify (i) + (ii) Show that simplifies to 7. 75 1 75 + 1 6. Express the following in the form p q where p and q are integers. Simplify the following. Give your answer in the form a + b, where a and b are integers. ( 1+ ) ( ) (OCR) 7. (i) Show that 0 = 5. (ii) Expand and simplify ( + 10) Is this triangle right-angled? Not to scale + 10 You must show your working + 5 99