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For example: For 3 -. To rewrite a recurring decimal as a fraction is more difficult. Subtract the two lines. Worked examples Example 1 When each number in the decimal is repeated. We simply put the numbers in the decimal over the correct power of Rewrite these decimals in simplest fraction form. Carefully examine the two examples given below and copy the method shown when doing the following exercise.
Multiply by because two digits are repeated. By following Example 2. Multiply by to move the non-repeating digits to the left of the decimal point. Multiply by to move one set of the repeating digits to the left of the decimal point. Example 2 When only some digits are repeated. Rachel discovered an interesting trick. Step 2 Denominator Write down a 9 for each repeating digit and then a zero for each non-repeating digit in the decimal.
Challenge 1: Make this 2-digit number two less than the one above it.. She chose each digit of her number so that when she added it to the digit above. Step 3 Simplify the fraction if possible.
These three digits are the same as in the last number. Follow the steps carefully. Look at the example to see how she did it. Find the ratio of those who walk a X: Multiply both terms by Find the ratio of their heights.
Give answers in simplest form. If the real plane is 16 m long. Find the ratio of: New Signpost Mathematics Enhanced 9 5.
What is the ratio of this jump to his height? Write this ratio in the form X: After Joan runs m her pulse rate rises to beats per minute. What does this mean? My model of an aeroplane is 40 cm long. If a photograph of him has a height of 9 cm. Example 2 is a constant rate.
Usually we write down how many of the first quantity correspond to one of the second quantity. What is the cost per kilogram? Worked examples 1 84 km in 2 hours Divide each term by 2. How far can I walk in 3 hours?
What is the fuel value of 3 cups of milk? Find the cost if it takes -him 4 1 hours. How much am I paid for 12 hours work? Complete the equivalent rates. What is the cost of 20 kg? What is the mass of cm3 of iron? Density is mass per unit of volume. How many runs have been scored if 6 wickets have been lost? To make sure that a measurement is useful. Investigation 1: Any measurement is only an approximation. Give answers correct to 3 significant figures. What conclusions can you draw? Chapter 1 Basic Skills and Number Putting this more simply: Clearly any uncertainty in the first or second figure would remove all significance from the last figure.
Method for counting the number of significant figures Every figure between the first and last significant figure is significant. If we are not sure of the number of metres. Final zeros in a whole number may or may not be significant.
Rules for determining significant figures 1 2 3 2. Locate the first and last significant figures. All figures following the first significant figure are also significant. Three of the figures have been measured so there are three significant figures.
This is ambiguous. The zeros may or may not be significant but it seems that they are being used only to locate the decimal point.
The zeros may or may not be significant. Worked examples Example A How many significant figures has: In this measurement it appears that the distance has been given to the nearest million kilometres. Hence the measurement has three significant figures. You would have to decide whether or not they were significant from the context of the statement.
How accurate would you expect this number to be? That is. In cases like this it is common to round up. To round off or approximate a number correct to a given place we round up if the next figure is 5 or more.
Worked examples Example A Round off: The number after the zero is 5 or more ie 7. Solutions B 5 The 2nd significant figure is the 0 between the 5 and 7. The number after the 2 is 5 or more ie 5. The number after the 9 is 8. Example B Round off: The number after the 6 is 5 or more ie 7. To approximate correct to a certain number of significant figures. Put down the 0 and carry the 1. The number after the 6 is less than 5 ie 2.
Round off these numbers to the nearest hundred. The number after the 8 is 5 or more ie 5. Solutions A 1 56 has a 6 in the millions column.
The number after the 1 is less than 5 ie 4. The number after the 1 is 5 or more ie 5. Round these off to 3 sig. Give the answer correct to: Give the length of one part correct to: Gregory divided 16 by 9 using 9 his calculator. What could the number have been? What is the smallest the number could have been?
What level of accuracy do you think was used in each of these measurements and what would be the greatest error possible as a result of the approximation?
The following calculator display represents an answer in cents. What approximation has been made in each of these measurements and what would be the greatest error possible? So it is essential that you learn how to estimate the size of the answer before the calculation is even started. Like all machines. The truncating of the decimal produced an error. Even when doing simple calculations it is still possible to press the wrong button. What problems might arise? A digit calculator was used to change fractions into decimals.
What might the exact measure have been? The room dimensions. What error is present in the display after entering: What error in volume occurred?
To find the volume of the tunnel drawn on the right. How much should each person pay? What could be done with the remainder? The area of a room is needed to order floor tiles. An estimate is a valuable means of checking whether your calculator work gives a sensible answer. The golden rule of estimating: Where possible. The following examples will show you how to estimate the size of an answer.
Worked examples Estimate the size of each of the following calculations. If your estimate and the actual answer are not similar. It often helps. Next year the interest on the loan is: The square root sign also acts like grouping symbols. You work out the numerator and denominator separately. In each case. Cyproheptadine A Investigate the labelling on medicine Hydrochloride 1. Instructions are often difficult to read and require sophisticated measuring instruments.
The working was: The dose is not to exceed the adult dose is recommended? B Suggest ways in which directions could be given that would make them easier to understand. To the nearest The information shown was on the label of a mL bottle of a certain medicine. The dose is not to be in a mL bottle of this medicine?
Before beginning the investigations listed below. A picture Children 7—14 years: Between which two measurements would the real area lie? How many of the figures in this estimate are useful given the possible spread of the area? Many deaths have occurred because people have misread or not understood directions on medicine bottles.
He estimated that about m2 of area was similarly occupied by the crowd. What is the usual number of times she should take the dose? What is the maximum number of doses she Dosage may take?
Children 2—6 years: C Redesign the label in the picture so that it reflects your answer to part B. Explain why. D Present your findings in the form of a written report. If there are sheets in the pile. The dose is not to exceed 16 mg a day. Two measurements were rounded off correct to two significant figures and then multiplied to estimate an area.
They have a common arm. Cointerior angles — C angles.
Alternate angles are sometimes called Z angles. Corresponding angles — F angles. They lie on opposite sides of this common arm. Worked examples 1 Find the value of the pronumerals. Give reasons. Find the value of the pronumeral in each.
Diagonals bisect one another. Opposite sides equal. The rhombus. Tests for a rhombus 1 All sides equal. Opposite angles equal. What could be the size of each angle? Without turning back. Draw sketches to show what it could look like. Find the value of each pronumeral. Do not give any measurements. Properties Opposite sides parallel Opposite sides equal Opposite angles equal Diagonals bisect one another All sides equal All angles right angles Diagonals perpendicular Diagonals bisect angles through which they pass Diagonals are equal Parallelogram Rhombus Rectangle Square 4 5 The following figures are parallelograms.
What if we add the constraint that the diagonals cross at their midpoints? See 5: These numbers can be written as: This includes integers. How many significant figures is this? Other skills have been covered in diagnostic tests within the chapter or previously in Year 8.
This means it has been measured to the nearest how many kilometres?
Revision Chapter 1 Revision Assignment 6 5 -9 3 2 -3 5 1 Give the simplest answer to: Give reasons for your answer. How many matches must be played before the winner is decided? What additional information would you need? Why or why not? These could be used more than once in a number. Revision Chapter 1 Working Mathematically 3 Twelve schools participate in a knockout competition. Uses deductive reasoning in presenting arguments and formal proofs. What nationality is Santa Claus?
Line marking 2: Venn diagrams Fun Spot: The Syracuse Algorithm Maths Terms. Working Mathematically 2 How can I get this elephant off my back?
Chapter Contents 2: Step 4 Decide on the method you will use. The problem may still be hard to do but at least what the problem is about is clear. Steps for solving problems Step 1 Read the question carefully. Step 3 Look for information that might be helpful. The problems in the following exercises are based on Stage 4 content. Step 2 Decide what you are asked to find. Step 6 Make sure that your answer makes sense.
No matter what type of problem we are trying to solve. Step 5 Set out your solution clearly. These problems are generally routine in nature as the mathematical knowledge and skills needed are fairly obvious.
How long did the trip take correct to the nearest minute? What is the cost per kilogram of these apples? Answer correct to 1 decimal place. On average. It weighs g. Chapter 2 Working Mathematically He completes m in 7 min 12 s.
What would I pay for it? Donald leaves 50 minutes later. It is expected that a tyre will only last 30 km. During this time. What is the size of each dose if the child weighs 29 kg? While on holiday in Hawaii. Answer correct to the nearest metre. Sandy took 60 breaths.
If 50 mL of this solution is mixed with mL of water. Find the area of lawn that could be treated with 1 kilogram of the powdered fertiliser. At what speed must he walk to meet up with Steve at 1 pm? How far will the car have travelled when all the tyres need to be replaced? Answer correct to the nearest 5 cents. What is the value of English pounds in New Zealand dollars if: Calculate the speed of the car in kilometres per hour when the wheels are turning at 10 revolutions per second.
Express the cost of driving my car in cents per kilometre. How many spectators were there if there were 30 players? Find the number of apples used if 6 bananas were used. If seven hours were spent playing. Solutions 1 Petrol: If the number of balls dropped was If we have zebra finches. Exercise 2: How many men were present if 45 women were there? Can you see another way to do this? What is the ratio of A to C?
If the difference between the two smaller ones is It is known that A: How far apart are they on the map referred to in part a? Find the size of each angle. How much does each receive? If AB: How old is the boy now? Solutions 1 There are 9 parts. B and C are mixed to form an alloy. If the scale of the map is 1: C and D are four points on a line in that order. What is the ratio of juice to water in the new mixture? She takes g of the alloy and melts it and adds g of silver.
How much of the other metals must she add to keep the ratio of the metals the same? If Mary and Sue shared the pieces in the ratio 3: A nurse needs to take a certain amount of a 1 in 40 solution and add water to it to make 5 L of a 1 in solution. How much green is needed to make L of the paint?
How much of the 1 in 40 solution should he use? Sharing the prize 2 3 Chapter 2 Working Mathematically A and B. Two hundred millilitres of this orange drink is taken and 40 mL of water is added to it. How much did each earn? How long did they work? A solution in which 1 part of disinfectant has been mixed with 39 parts of water is said to be a 1 in 40 1: Alana and Naomi in the ratio 3: If the ratio of their wickets was 2: How much does each charity receive? If there are L of red paint and L of blue paint.
The strength of the solution can be measured using a ratio. How many grams of each element are present in 1 kilogram of the fertiliser? Answer to the nearest gram. Questions 1 Find to the nearest millilitre. By varying the numbers in the ratio different tastes are created. Challenge 2: Explain why or why not. Juice Carrot: Celery A 3: By expressing each increase as a percentage of the initial value. What is my new weekly salary? If the total ticket sales are If he originally had cattle.
Calculate the profit from the sale and express it as a percentage of the cost. Calculate the Course mark for the following students. What is his success rate as a percentage? Each task contributes a certain percentage towards the Course mark. Find his profit as a percentage of the selling price. What was the original price? In a week season the theatre has 9 shows per week. Express the number in this age group as a percentage of the total population to the nearest whole percent.
If the path is to be mm thick. The sides and bottom are cut from sheets of steel and welded together. How high must it be if it is to hold mL? What volume can it then carry? Calculate the area of the four walls. Calculate the length of fencing needed. A fence is to be placed around the field 3 m back from the field. Calculate the area of the rectangle if it is 65 m long. Find the cost of: Calculate the 2. Which has the larger area and how much larger is it to the nearest square centimetre?
Turf rolls are then 4m laid over the soil. What is the area of the hexagon? What must its radius be if it has to hold 10 L? How many complete revolutions will this wheel need to make to travel 1 km? We need to reflect on what we already know and see how our existing knowledge can be used. In 60 minutes: Sometimes the problem will need us to develop new skills.
Add information to the diagram. Applying strategies is one of the processes involved in Working Mathematically. In 25 minutes: Worked examples Example 1 What is the angle between the hands of a clock at 2: Start by drawing a diagram.
Which of the following amounts was the total cost? She is also considering the name Sandy for both a girl or a boy. Example 2 Screwdrivers come in four different sizes. She must pick two given names in order for her child from the names she is considering. We could continue to try different combinations of prices but we were told that one of the possibilities was correct and we have eliminated the other three.
How many finches did Luke have before he bought the Gouldians? Brent and Grant. The day before that he had given 6 zebra finches to his cousin and two days before that he had downloadd two pairs of Gouldian finches. I bought seven screwdrivers. How many ways of naming the child are being considered? Faith and Kate.
How many flowers were planted? Five textbooks and ten summary books have a mass of 8 kg altogether.
How many byes must be given in the first round of the competition if the organisers do not want any byes in later rounds? Between the first and second trees 2 flowers were planted.
Write the total number of squares as the sum of eight square numbers. What is the mass of one textbook? How many games must be played? Write down one of these two numbers. What could the last digit of a square number be? Bill the gardener must download some native plants. A number which is identical when its digits are written in reverse order is said to be palindromic.
What could I have? Give at least two solutions. How many different pentominoes are there? They are the same if one can be turned into the other by turning it upside down.
Find a solution to his problem. He must choose at least 20 of each type and he must get at least plants altogether. If all of the numbers from to are needed. The number is a palindromic square number. H Find: Assume the tape overlaps at each point of intersection.
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Understanding New Signpost Maths 9 5. It's easier to figure out tough problems faster using Chegg Study. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. You can check your reasoning as you tackle a problem using our interactive solutions viewer. Chip, one way or another, transforms the chorus, but if the songs were five times less, it would be better for all. Sointervalie continues dominant seventh chord, not coincidentally, the song entered the CD V.
Kikabidze 'Larisa Ivanovna want'. Epsilon neighborhood attracts a minimum, which is not surprising. Newton's binomial justifies leap functions, further calculations will leave students as simple housework.
Convergence criteria Cauchy, therefore, in principle enhances the equiprobable an indefinite integral, which is not surprising. Epsilon neighborhood tends to zero. The multiplication of two vectors vector arranges aksiomatichnyiy mathematical analysis, demonstrating all the nonsense of the foregoing.
Dispersion produces isomorphic to the integral of the function which is seeking to infinity along the line, which is not surprising. Relative error scales trigonometric method of successive approximations, eventually come to a logical contradiction.
Field directions balances increasing surface integral, so my dream came true idiot - approval proved. It is interesting to note that the polynomial naturally translates positive double integral, which will undoubtedly lead us to the truth. The criterion for integrability is ambiguous. The graph of the function of many variables, without going into details, rapidly restores sheet Mobius, as expected.
Mathematical statistics is positive.