Units of measurement and applications

C H A P T E R 3 Units of measurement and applications Syllabus topic MM1 Units of measurement and applications Determine and convert appropriate units of measurement Convert units of area and volume Calculate the percentage error in a measurement Use numbers in scientifi c notation Express numbers to a certain number of signifi cant fi gures Calculate and convert rates Find ratios of two quantities and use the unitary method Calculate repeated percentage changes 3.1 Units of measurement 3.1 Measurement is used to determine the size of a quantity. It usually involves using a measuring instrument. For example, to measure length, instruments that can be used include the rule, tape measure, caliper, micrometer, odometer and GPS. There are a number of systems of measurement that define their units of measurement. We use the SI metric system. 73

74 Preliminary Mathematics General SI units The SI is an international system of units of measurement based on multiples of ten. It is a version of the metric system which allows easy multiplication when converting between units. Units shown in red (below) are non-si units approved for everyday or specialised use alongside SI units. Quantity Name of unit Symbol Value Length Area Volume Mass Time Metre Millimetre Centimetre Kilometre Nautical mile Square metre Square centimetre Hectare Cubic metre Cubic centimetre Litre Millilitre Kilolitre Kilogram Gram Tonne Second Minute Hour Day Converting between units m mm cm km nm m 2 cm 2 ha m 3 cm 3 L ml kl kg g t s min h d Base unit 1000 mm = 1 m 100 cm = 1 m 1 km = 1000 m 1 nm = 1852 m Base unit 10 000 cm 2 = 1 m 2 1 ha = 10 000 m 2 Base unit 1 000 000 cm 3 = 1 m 3 1L = 1000 cm 3 1000 ml = 1 L 1 kl = 1000 L Base unit 1000 g = 1 kg 1 t = 1000 kg Base unit 1 min = 60 s 1 h = 60 min 1 d = 24 h A prefix is a simple way to convert between units. It indicates a multiple of 10. Some common prefixes are mega (1000000), kilo (1000), centi 1 ( 100 ) and milli 1 ( 1000 ). Length, mass and volume Time mega 1000 1000 kilo 1000 1000 unit 100 100 centi 10 10 milli 24 60 60 days hours minutes seconds 24 60 60

Chapter 3 Units of measurement and applications 75 Example 1 Converting units of length Complete the following. a 35 cm = mm b 4500 m = km Solution 1 To change cm to mm multiply by 10. 2 To change m to km divide by 1000. a b 35 cm = 35 10 mm = 350 mm 4500 = 4500 1000 km = 4.5 km Example 2 Converting units of time Complete the following. a 3 h and 15 min = min b 10 080 min = d Solution 1 To change hours to minutes, multiply by 60. 2 To change minutes to hours, divide by 60. 3 To change hours to days, divide by 24. a b 3 h 15 min = 3 60 + 15 min = 195 min 10080 min = 10080 60 h = 168 h = 168 24 d = 7 d Converting area and volume units To convert area units, change the side length units and compare the values for area. 1 m = 1 m 100 cm 100 cm 1 m 2 = 100 100 = 10000 cm 2 1 m 2 = 10000 cm 2 or 1 cm2 1 = m2 10 000 To convert volume units, change the side length units and compare the values for volume. 1 m 1 m = 1 m 100 cm 100 cm 100 cm 1 m 3 = 100 100 100 = 1000000 cm 3 1 m 3 = 1000000 cm 3 3 1 3 or 1 cm = m 1000 000

76 Preliminary Mathematics General Exercise 3A 1 Complete the following. a 5 cm = mm b 78 m = cm c 2 km = m d 890 m = cm e 57 cm = mm f 6 km = cm g 9400 m = km h 600 mm = cm i 8100 cm = m j 49 000 cm = km k 22 000 m = km l 51 mm = cm 2 Complete the following. a 3 t = kg b 45 kg = g c 76 t = kg d 8100 kg = g e 4 t = g f 0.52 t = kg g 6800 g = kg h 9 300 000 g = t i 45 000 000 g = t j 300 kg = t k 2300 g = kg l 60 000 g = kg 3 Complete the following. a 2 L = ml b 12 kl = L c 9 kl = ml d 7800 kl = L e 50 L = ml f 300 kl = ml g 6100 L = kl h 400 ml = L i 210 000 ml = kl j 80 ml = L k 79 000 ml = kl l 8 000 000 ml = kl 4 Complete the following. a 2.5 h = min b 2 min = s c 20 d = h d 40 min = s e 4.5 d = h f 10 h = min g 720 min = h h 48 000 s = min i 96 h = d j 1080 h = d k 390 min = h l 780 s = min

Chapter 3 Units of measurement and applications 77 5 What unit of length is most appropriate to measure each of the following? a Length of a pen b Height of a building c Thickness of a credit card d Distance from Sydney to Newcastle e Height of a person f Length of a football field 6 What unit of mass is most appropriate to measure each of the following? a Weight of an elephant b Mass of a mug c Bag of onions d Weight of a baby e Mass of a truck f Mass of a teaspoon of sugar 7 What unit of time is most appropriate to measure each of the following? a Lesson at school b Reheating a meal in a microwave c Age of a person d School holidays e Accessing the internet f Movie 8 There are 20 litres of a chemical stored in a container. a What amount of chemical remains if 750 ml is removed from the container? Answer in litres. b How many containers are required to make a kilolitre of the chemical? 9 Christopher bought 3 kg of sultanas. What mass of sultanas remains if he ate 800 grams? Answer in kilograms. 10 The length of the Murray River is 2575 km. The length of the Hawkesbury River is 80 000 m. What is the difference in their lengths? Answer in metres. 11 There are three tonnes of grain in a truck. What is the mass if another 68 kg of grain is added to the truck? Answer in kilograms.

78 Preliminary Mathematics General Development 12 A cyclist travels to and from work over a 1200-metre long bridge. Calculate the distance travelled in a week if the cyclist works for 5 days. Answer in kilometres. 13 Madison travels 32 km to work each day. Her car uses 1 litre of petrol to travel 8 km. a How many litres of petrol will she use to get to work? b How many litres of petrol will she use for 5 days of work, including return travel? 14 Arrange 500 m, 0.005 km, 5000 cm and 5000000 mm in: a Ascending order (smallest to largest) b Descending order (largest to smallest) 15 Complete the following. km a 1 2 2 cm c 1 2 2 = m b 1 m 2 = mm 2 = mm d 1000 cm = m 2 2 e 2000 mm 2 = cm 2 f 5000 m2 = km 2 2 g 3.9 m2 = cm2 h 310 km2 = m2 i 4.7 m2 = mm2 j 74300 m2 = km 2 2 k 6500 mm 2 = cm 2 l 4000 cm = m 2 2 16 The area of a field is 80 000 square metres. Convert the area units to the following. a Square kilometres b Hectares 17 Jackson swims 30 lengths of a 50-metre pool. a How many kilometres does he cover? b If his goal is 4 kilometres, how many more lengths must he swim? 18 Eliza worked from 10.30 a.m. until 4.00 p.m. on Friday, from 7.30 a.m. until 2.00 p.m. on Saturday, and from 12 noon until 5.00 p.m. on Sunday. a How many hours did Eliza work during the week? b Express the time worked on Friday as a percentage of the total time worked during the week. Answer correct to the nearest whole number.

Chapter 3 Units of measurement and applications 79 3.2 Measurement errors There are varying degrees of instrument error and measurement uncertainty when measuring. Every time a measurement is repeated, with a sensitive instrument, a slightly different result will be obtained. The possible sources of errors include mistakes in reading the scale, parallax error and calibration error. The accuracy of a measurement is improved by making multiple measurements of the same quantity with the same instrument. Accuracy in measurements The smallest unit on the measuring instrument is called the limit of reading. For example, a 30 cm rule with a scale for millimetres has a limit of reading of 1 mm. The accuracy of a measurement is restricted to ± 1 of the limit of reading. For example, 2 if the measurement on the ruler is 10 mm then the range of errors is 10 ± 0.5 mm. Here the upper limit is 10 + 0.5 mm or 10.5 mm and the lower limit is 10 0.5 mm or 9.5 mm. 1 cm 2 3 4 5 Every measurement is an approximation and has an error. The absolute error is the difference between the actual value and the measured value indicated by the instrument. The maximum value for an absolute error is 1 of the limit of reading. 2 Limit of reading Smallest unit on measuring instrument Absolute error Measured value Actual value Maximum value is 1 limit of reading 2 Relative error gives an indication of how good a measurement is relative to the size of the quantity being measured. The relative error of a measurement is calculated by dividing the limit of reading by the actual measurement. For example, the relative error for the above 0. 5 measurement is ( 0. 05 10 ) =. The relative error is often expressed as a percentage and called the percentage error. For example, the percentage error for the above measurement 0 5 is. 100 5 %. ( ) = 10

80 Preliminary Mathematics General Relative error ± Absolute error Measurement ± Percentage error Absolute error Measurement 100% Example 3 Finding the measurement errors a b c d e What is the length indicated by the arrow on the above ruler? What is the limit of reading? 0 10 20 30 40 50 60 70 80 What is the upper and lower limit for each measurement? Find the relative error. Answer correct to three decimal places. Find the percentage error. Answer correct to one decimal place. Solution 1 The arrow is pointing to 38 mm. 2 Limit of reading is the smallest unit on the ruler (millimetre). 3 Calculate half the limit of reading. 4 Lower limit is the measured value minus 1 the limit of 2 reading. 5 Upper limit is the measured value plus 1 the limit of 2 reading. 6 Write the formula for relative error. 7 Substitute the values for absolute error and the measurement. 8 Evaluate correct to three decimal places. 9 Write the formula for percentage error. 10 Substitute the values for absolute error and the measurement. 11 Evaluate. a b Length is 38 mm. Limit of reading is 1 mm. c 1 limit of reading = 1 1 2 2 = 0.5 mm Lower limit = 38 0.5 = 37.5 mm Upper limit = 38 + 0.5 = 38.5 mm d Relative error = ± = ± 0. 5 38 = ± 0.013 e Percentage error = ± Absolute error Measurement Absolute error Measurement 100% = ± 0.5 38 100% = ± 1.3%

Chapter 3 Units of measurement and applications 81 Exercise 3B 1 Four measurements of length are shown on the ruler below. A B C D 0 10 20 30 40 50 60 70 80 90 100 a b c d e f What length is indicated by each letter? Answer to the nearest millimetre. What is the limit of reading? What the largest possible absolute error? What is the upper and lower limit for each measurement? Calculate the relative error, correct to three decimal places, for each measurement. Calculate the percentage error, correct to one decimal place, for each measurement. 2 Two measurements of mass are shown on the scales below. 4kg 3.5 0 4.5 0.5 8 0 8 10lb 1lb 8 8 9lb 2lb 8 A 8 8lb 3lb 8 8 7lb 4lb 8 8 6lb 8 5lb 3kg 2kg 2.5 1kg 1.5 4kg 3.5 0 4.5 0.5 8 0 8 10lb 1lb 8 8 9lb 2lb 8 B 8 8lb 3lb 8 8 7lb 4lb 8 8 6lb 8 5lb 3kg 2kg 2.5 1kg 1.5 a b c d e f What mass is indicated by each letter? Use the outer scale. What is the limit of reading? What the largest possible absolute error? What is the upper and lower limit for each measurement? Calculate the relative error, correct to three decimal places, for each measurement. Calculate the percentage error, correct to one decimal place, for each measurement.

82 Preliminary Mathematics General Development 3 A dishwasher has a mass of exactly 49.6 kg. Abbey measured the mass of the dishwasher as 50 kg to the nearest kilogram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error to the nearest whole number. 4 An ipod has a mass of exactly 251 g. Jake measured the mass of the ipod as 235 g to the nearest gram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error correct to two decimal places. 5 An LCD screen has a mass of exactly 2.71 kg. Saliha measured the mass of the screen as 3 kg to the nearest kilogram. a Find the absolute error. b Find the relative error. Answer correct to three decimal places. c Find the percentage error correct to three decimal places. 6 A measurement was taken of a skid mark at the scene of a car accident. The actual length of the skid mark was 25.15 metres, however it was measured as 25 metres. a What is the absolute error? b Find the relative error. Answer correct to three decimal places. c Find the percentage error. Answer correct to one decimal place. 7 The length of a building at school is exactly 56 m. Cooper measured the length of the building to be 56.3 m and Filip measured the building at 55.8 m. a What is the absolute error for Cooper s measurement? b What is the absolute error for Filip s measurement? c Compare the relative error for both measurements. Answer correct to four decimal places. d Compare the percentage error for both measurements. Answer correct to three decimal places.

Chapter 3 Units of measurement and applications 83 3.3 Scientific notation and significant figures 3.3 Scientific notation Scientific notation is used to write very large or very small numbers more conveniently. It consists of a number between 1 and 10 multiplied by a power of ten. For example, the number 4 100 000 is expressed in scientific notation as 4.1 10 6. The power of ten indicates the number of tens multiplied together. For example: 4.1 10 6 = 4.1 (10 10 10 10 10 10) = 4 100 000 When writing numbers in scientific notation, it is useful to remember that large numbers have a positive power of ten and small numbers have a negative the power of ten. Writing numbers in scientific notation 1 Find the first two non-zero digits. 2 Place a decimal point between these two digits. This is the number between 1 and 10. 3 Count the digits between the new and old decimal point. This is the power of ten. 4 Power of ten is positive for larger numbers and negative for small numbers. Example 4 Expressing a number in scientific notation The land surface of the earth is approximately 153 400 000 square kilometres. Express this area more conveniently by using scientific notation. Solution 1 The first two non-zero digits are 1 and 5. 2 Place the decimal point between these numbers. 3 Count the digits from the old decimal point (end of the number) to position of the new decimal point. 4 Large number indicates the power of 10 is positive. 5 Write in scientific notation. 1.534 1.53 400 000 eight digits Power of 10 is +8 or 8 153 400 000 = 1.534 10 8

84 Preliminary Mathematics General Significant figures Significant figures are used to specify the accuracy of a number. It is often used to round a number. Significant figures are the digits that carry meaning and contribute to the accuracy of the number. This includes all the digits except the zeros at the start of a number and zeros at the finish of a number without a decimal point. These zeros are regarded as placeholders and only indicate the size of the number. Consider the following examples. 51.340 has four significant figures: 5, 1, 3 and 4. 0.00871 has three significant figures: 8, 7 and 1. 56091 has five significant figures: 5, 6, 0, 9 and 1. The significant figures in a number not containing a decimal point can sometimes be unclear. For example, the number 8000 may be correct to 1 or 2 or 3 or 4 significant figures. To prevent this problem, the last significant figure of a number is underlined. For example, the number 8000 has two significant figures. If the digit is not underlined the context of the problem is a guide to the accuracy of the number. Writing numbers to significant figures 1 Write the number in scientific notation. 2 Count the digits in the number to determine its accuracy (ignore zeros at the end). 3 Round the number to the required significant figures. Example 5 Writing numbers to significant figures Write these numbers correct to the significant figures indicated. a 153400000 (3 significant figures) b 0.000657 (2 significant figures) Solution 1 Write in scientific notation. 2 Count the digits in the number. 3 Round the number to 3 significant figures. 4 Write answer in scientific notation correct to 3 significant figures. a 153 400 000 = 1.534 10 8 1.534 has 4 digits 1.53 rounded to 3 sig. fig. 153 400 000 = 1.53 10 8 5 Write in scientific notation. 6 Count the digits in the number. 7 Round the number to 2 significant figures. 8 Write answer in scientific notation correct to 2 significant figures. b 0.000 657 = 6.57 10 4 6.57 has 3 digits 6.6 rounded to 2 sig. fig. 0.000 657 = 6.6 10 4

Chapter 3 Units of measurement and applications 85 Exercise 3C 1 Write these numbers in scientific notation. a 7600 b 1700000000 c 590000 d 6800000 e 35000 f 310000000 g 77100000 h 523000000000 i 95400000000 2 Write these numbers in scientific notation. a 0.00056 b 0.0000687 c 0.000000812 d 0.0043 e 0.000058 f 0.000 00312 g 0.26 h 0.092 i 0.000000000167 3 A microsecond is one millionth of a second. Write 5 microseconds in scientific notation. 4 Sharks existed 410 million years ago. a Write this number in scientific notation. b Express this number correct to one significant figure. 5 Write each of the following as a basic numeral. a 1.12 10 5 b 5.34 10 8 c 5.2 10 3 d 8.678 10 7 e 2.4 10 2 f 7.8 10 9 g 3.9 10 6 h 2.8 10 1 i 6.4 10 4 6 Write each of the following as a basic numeral. a 3.5 10 4 b 7.9 10 6 c 1.63 10 7 d 5.81 10 3 e 4.9 10 2 f 9.8 10 1 g 4.12 10 8 h 6.33 10 5 i 3.0 10 9

86 Preliminary Mathematics General 7 Convert a measurement of 5.81 10 3 grams into kilograms. Express your answer in scientific notation. 8 Evaluate the following and express your answer in scientific notation. a (2.5 10 3 ) (5.9 10 6 ) b (4.7 10 5 ) (6.3 10 2 ) c (7.1 10 5 ) (4.2 10 2 ) d (3.0 10 4 ) (6.2 10 5 ) 9 Evaluate the following and express your answer in scientific notation. a 9.1 10 5 2. 8 10 2 b 7.2 10 7 4. 8 10 3 c 4.8 10 4 3. 2 10 5 10 Write these numbers correct to significant figures indicated. a 1561231 (2 sig. fig.) b 3677720 (4 sig. fig.) c 789001 (5 sig. fig.) d 3300000 (1 sig. fig.) e 777 777 (3 sig. fig.) f 3194729 (5 sig. fig.) g 821076 (4 sig. fig.) h 7091 (1 sig. fig.) i 49172 (2 sig. fig.) 11 Write these numbers correct to significant figures indicated. a 0.0035 (1 sig. fig.) b 0.191785 (4 sig. fig.) c 0.001592 (3 sig. fig.) d 0.11122233 (6 sig. fig.) e 0.0000271 (1 sig. fig.) f 0.019832 6 (5 sig. fig.) g 0.00812 (2 sig. fig.) h 0.09271 (3 sig. fig.) i 0.000419 (2 sig. fig.) 12 A bacterium has a radius of 0.000 015 765 m. Express this length correct to two significant figures. 13 Convert a measurement of 2654 kilograms into centigrams. Express your answer correct to two significant figures. 14 Convert a measurement of 4 239 810 milligrams into grams. Express your answer correct to four significant figures.

Chapter 3 Units of measurement and applications 87 Development 15 If y = 1 x 2, find the value of y when: 2 a x = 2.4 10 3 b x = 9.8 10 3 16 The arc length of a circle is l θ = π 360 2 r where θ is the angle at the centre and r is the radius of the circle. Use this formula to calculate the arc length of a circle when θ = 30 and r = 7.4 10 8. Answer in scientific notation correct to one significant figure. r 17 Given that V = find the value of r in scientific notation when: h a V = 5 10 4 and h = 9 10 6 b V = 6 10 7 and h = 4 10 2 18 Use the formula E = md 2 to find d correct to three significant figures given that: a m = 0.08 and E = 5.5 10 9 b m = 2.7 10 3 and E = 1.6 10 4 19 Find x given x 3 = 2.7 10 12. Answer correct to four significant figures. 20 Light travels at 300 000 kilometres per second. Convert this measure to metres per second and express this speed in scientific notation. 21 Use the formula E = 3p q to evaluate E given that p = 7.5 10 5 and q = 2.5 10 4. Answer in scientific notation correct to one significant figure. 22 The volume of a cylinder is V = πr 2 h where r is the radius of the cylinder and h is the height of the cylinder. Use this formula to calculate the volume of the cylinder if r = 5.6 10 4 and h = 2.8 10 3. Answer in scientific notation correct to three significant figures. 13.3 23 The Earth is 1.496 10 8 km from the Sun. Calculate the distance travelled by the Earth in a year using the formula c = 2πr. Answer in scientific notation correct to two significant figures.

88 Preliminary Mathematics General 3.4 Calculations with ratios 3.4 A ratio is used to compare amounts of the same units in a definite order. For example, the ratio 3:4 represents 3 parts to 4 parts or 3 4 or 0.75 or 75%. A ratio is a fraction and can be simplified in the same way as a fraction. For example, the ratio 15:20 can be simplified to 3:4 by dividing each number by 5. Equivalent ratios are obtained by multiplying or dividing each amount in the ratio by the same number. 3 3 15 : 12 = 5 : 4 3 3 5 : 4 = 15 : 12 15:12 and 5:4 are equivalent ratios. When simplifying a ratio with fractions, multiply each of the amounts by the lowest common denominator. For example, to simplify 1 : 3 multiply both sides by 8. This results in the 8 4 equivalent ratio of 1:6. Ratio A ratio is used to compare amounts of the same units in a definite order. Equivalent ratios are obtained by multiplying or dividing by the same number. Dividing a quantity in a given ratio Ratio problems may be solved by dividing a quantity in a given ratio. This method divides each amount in the ratio by the total number of parts. Dividing a quantity in a given ratio 1 Calculate the total number of parts by adding each amount in the ratio. 2 Divide the quantity by the total number of parts to determine the value of one part. 3 Multiply each amount of the ratio by the result in step 2. 4 Check by adding the answers for each part. The result should be the original quantity.

Chapter 3 Units of measurement and applications 89 Example 6 Dividing a quantity in a given ratio Mikhail and Ilya were given $450 by their grandparents to share in the ratio 4:5. How much did each person receive? Solution 1 Calculate the total number of parts by adding each amount in the ratio (4 parts to 5 parts). 2 Divide the quantity ($450) by the total number of parts (9 parts) to determine the value of one part. 3 Multiply each amount of the ratio by the result in step 2 or $50. 4 Check by adding the answers for each part. The result should be the original quantity or $450. 5 Write the answer in words. Total parts = 4 + 5 = 9 9 parts = $450 $450 1 part = = $50 9 4 parts = 4 $50 = $ 200 5 parts = 5 $50 = $ 250 ($200 + $250 = $450) Mikhail receives $200 and Ilya receives $250. The unitary method The unitary method involves finding one unit of an amount by division. This result is then multiplied to solve the problem. Using the unitary method 1 Find one unit of an amount by dividing by the amount. 2 Multiply the result in step 1 by a number to solve the problem. Example 7 Using the unitary method A car travels 360 km on 30 L of petrol. How far does it travel on 7 L? Solution 1 Write a statement using information from the question. 2 Find 1 L of petrol by dividing 360 km by the amount or 30. 3 Multiply the 360 by a 7 to solve the problem. 30 4 Evaluate. 5 Write answer to an appropriate degree of accuracy. 6 Write the answer in words. 30 L = 360 km 360 1 L = km 30 360 7 L = 7 km 30 = 84 km The car travels 84 km.

90 Preliminary Mathematics General Exercise 3D 1 Express each ratio in simplest form. a 15:3 b 10:40 c 24:16 d 14:30 e 8:12 f 49:14 g 9:18:9 h 5:10:20 i 1: 1 3 j 1 1 : k 2 5 2 3 3 : l 7 3 4 : 1 2 A delivery driver delivers 1 parcel on average every 20 minutes. How many hours does it take to drop 18 parcels? 3 Divide 240 into the following ratios. a 2:1 b 3:2 c 1:5 d 7:5 4 A bag of 500 grams of chocolates is divided into the ratio 7:3. What is the mass of the smaller amount? 5 At a concert there were 7 girls for every 5 boys. How many girls were in the audience of 8616? 6 Molly, Patrick and Andrew invest in a business in the ratio 6:5:1. The total amount invested is $240 000. How much was invested by the following people? a Molly b Patrick c Andrew 7 In a boiled fruit cake recipe the ratio of mixed fruit to flour to sugar is 5:3:2. A 250 g packet of mixed fruit is used to make the cake. How much sugar and flour is required? 8 A 5 kg bag of potatoes costs $12.80. Find the cost of: a 1 kg b 10 kg c 14 kg d 6 kg 9 The cost of 3 pens is $42.60. Find the cost of: a 1 pen b 4 pens c 6 pens d 10 pens

Chapter 3 Units of measurement and applications 91 Development 10 A punch is made from pineapple juice, lemonade and passionfruit in the ratio 3:5:2. a How much lemonade is needed if one litre of pineapple juice is used? b How much pineapple juice is required to make 10 litres of punch? 11 Angus, Ruby and Lily share an inheritance of $500 000 in the ratio of 7:5:4. How much will be received by the following people? a Angus b Ruby c Lily 12 Samantha and Mathilde own a restaurant. Samantha gets 3 of the profits and Mathilde 5 receives the remainder. a What is the ratio of profits? b Last week the profit was $2250. How much does Mathilde receive? c This week the profit is $2900. How much does Samantha receive? 13 A jam is made by adding 5 parts fruit to 4 parts of sugar. How much fruit should be added to 2 1 kilograms of sugar in making the jam? 2 14 A local council promises to spend $4 for every $3 raised in public subscriptions for a community hall. The cost of the hall is estimated at $1.75 million. How much does the community need to raise? 15 The ratio of $5 to $10 notes in Stephanie s purse is 3:5. There are 24 notes altogether. What is the total value of Stephanie s $5 notes? 16 Nathan makes a blend of mixed lollies using 5 kg jelly babies, 4 kg licorice and 1 kg skittles. What is the cost of the blend per kilogram to the nearest cent? Jelly babies Licorice Skittles Mixed lollies $5.95 per kg $6.95 per kg $11.90 per kg 17 The three sides of a triangle are in the ratio of 2:3:4. The longest side of the triangle is 12.96 mm. What is the perimeter of the triangle?

92 Preliminary Mathematics General 3.5 Rates and concentrations 3.5 Rates A rate is a comparison of amounts with different units. For example, we may compare the distance travelled with the time taken. In a rate the units are different and must be specified. The order of a rate is important. A rate is written as the first amount per one of the second amount. For example, $2.99/kg represents $2.99 per one kilogram or 80 km/h represents 80 kilometres per one hour. Converting a rate 1 Write the rate as a fraction. First quantity is the numerator and 1 is the denominator. 2 Convert the first amount to the required unit. 3 Convert the second amount to the required unit. 4 Simplify the fraction. Example 8 Converting a rate Convert each rate to the units shown. a 55 200 m/h to m/min b $6.50/kg to c/g Solution 1 Write the rate as a fraction. 2 The numerator is 55200 m and the denominator is 1 h. 3 No conversion required for the numerator. 4 Convert the 1 hour to minutes by multiplying by 60. 5 Simplify the fraction. 6 Write the rate as a fraction. 7 The numerator is $6.50 and the denominator is 1 kg. 8 Convert the $6.50 to cents by multiplying by 100. 9 Convert the 1 kg to g by multiplying by 1000. 10 Simplify the fraction. 55 200 m a 55 200 = 1 h 55 200 m = 1 60 min = 920 m/min $6.50 b 6.50 = 1 kg 6.50 100 c = 1 1000 g = 0. 65 c/g

Chapter 3 Units of measurement and applications 93 Concentrations A concentration is a measure of how much of a given substance is mixed with another substance. Concentrations are a rate that has particular applications in nursing and agriculture. It often involves mixing chemicals. Concentrations may be expressed as: weight per weight such as 10 g/100 g weight per volume such as 5 g/10 ml volume per volume such as 20 ml/10 L. Finding a percentage concentration 1 Write the two quantities as a fraction. 2 Multiply the fraction by 100 to convert it to a percentage. Example 9 Converting a concentration A medicine is given as a concentration of 2.5 ml per 10 kg. What is the dosage rate for this medicine in ml/kg? Solution 1 Write the rate as a fraction. 2 The numerator is 2.5 ml and the denominator is 10 kg. 3 Divide the numerator by the denominator. 4 Evaluate. 5 Write answer to an appropriate degree of accuracy. 6 Write the answer in words. 2.5 ml 2.5 ml/10 kg = 10 kg 2.5 ml = 10 kg = 0.25 m L/ kg The dosage rate is 0.25 ml/kg. Example 10 Expressing as a percentage concentration Express 6.2 g of sugar per 50 g as a percentage concentration. Solution 1 Write as a fraction. The first amount is divided by the second amount. 2 Multiply the fraction by 100 to convert it to a percentage. 3 Evaluate. 4 Write the answer in words. 6.2 g/50 g = 6.2 g 50 g 6.2 = 100% 50 = 12. 4% Percentage composition is 12.4%.

94 Preliminary Mathematics General Exercise 3E 1 Use the rate provided to answer the following questions. a Cost of apples is $2.50/kg. What is the cost of 5 kg? b Tax charge is $28/m². What is the tax for 7 m 2? c Cost savings are $35/day. How much is saved in 5 days? d Cost of a chemical is $65/100 ml. What is the cost of 300 ml? e Cost of mushrooms is $5.80/kg. What is the cost of 1 kg? 2 f Distance travelled is 1.2 km/min. What is the distance travelled in 30 minutes? g Concentration of a chemical is 3 ml/l. How many ml of the chemical is needed for 4 L? h Concentration of a drug is 2 ml/g. How many ml is needed for 10 g? 2 Express each rate in simplest form using the rates shown. a 300 km on 60 L [km per L] b 15 m in 10 s [m per s] c $640 for 5 m [$ per m] d 56 L in 0.5 min [L per min] e 78 mg for 13 g [mg per g] f 196 g for 14 L [g per L] 3 Convert each rate to the units shown. a 39240 m/min [m/s] b 2 m/s [cm/s] c 88 cm/h [mm/h] d 55200 m/h [m/min] e 0.4 km/s [m/s] f 57.5 m/s [km/s] g 6.09 g/ml [mg/ml] h 4800 L/kL [ml/kl] i 12600 mg/g [mg/kg] 4 Mia earns $37.50 per hour working in a cafe. a How much does Mia earn for working a 9-hour day? b How many hours does Mia work to earn $1200? c What is Mia s annual income if she works 40 hours a week? Assume she works 52 weeks in the year. 5 Patrick mixes 35 ml of a pesticide per 20 L as a percentage concentration. a Express this concentration in litres per litre. b What is the percentage concentration?

Chapter 3 Units of measurement and applications 95 Development 6 A tap is dripping water at a rate of 70 drops per minute. Each drop is 0.2 ml. a How many millilitres of water drip from the tap in one minute? b How many litres of water drip from the tap in a day? 7 Natural gas is charged at a rate of 1.4570 cents per MJ. a Find the charge for 12560 MJ of natural gas. Answer to the nearest dollar. b The charge for natural gas was $160.27. How many megajoules were used? 8 Olivia s council rate is $2915 p.a. for land valued at $265 000. Lucy has a council rate of $3186 on land worth $295 000 from another council. a What is Olivia s council charge as a rate of $/$1000 valuation? b What is Lucy s council charge as a rate of $/$1000 valuation? 9 Mira s car uses 9 litres of petrol to travel 100 kilometres. Petrol costs $1.50 per litre. a What is the cost of travelling 100 kilometres? b How far can she drive using $50 worth of petrol? Answer to the nearest kilometre. 10 A motor bike is moving at a steady speed. When the speed is 90 km/h the bike consumes 5 litres of petrol for every 100 kilometres travelled. a The petrol tank holds 30 litres. How many kilometres can the bike travel on a full tank of petrol when its speed is 90 km/h? b When the speed is 110 km/h the bike consumes 30% more petrol per kilometre travelled. Calculate the number of litres per 100 kilometres consumed when the bike travels at 110 km/h. 11 A plane travelled non-stop from Los Angeles to Sydney, a distance of 12027 kilometres in 13 hours and 30 minutes. The plane started with 180 kilolitres of fuel, and on landing had enough fuel to fly another 45 minutes. a What was the plane s average speed in kilometres per hour? Answer to the nearest whole number. b How much fuel was used? Answer to the nearest kilolitre.

96 Preliminary Mathematics General 3.6 Percentage change Percentage change involves increasing or decreasing a quantity as a percentage of the original amount of the quantity. 3.6 Percentage increase 1 Add the % increase to 100%. 2 Multiply the above percentage by the amount. Percentage decrease 1 Subtract the % decrease from 100%. 2 Multiply the above percentage by the amount. Example 11 Calculating the percentage change The retail price of a toaster is $36 and is to be increased by 5%. What is the new price? Solution 1 Add the 5% increase to 100%. 2 Write the quantity (new price) to be found. 3 Multiply the above percentage (105%) by the amount. 4 Evaluate and write using correct units. 5 Write the answer in words. 100% + 5% = 105% New price = 105% of $36 = 1. 05 36 = $37.80 New price is $37.80. Example 12 Calculating repeated percentage changes Increase $75 by 20% and then decrease the result by 20%. Solution 1 Add the 20% increase to 100%. 2 Write the quantity (new price) to be found. 3 Multiply the above percentage (120%) by the amount. 4 Evaluate and write using correct units. 5 Subtract the 20% decrease from 100%. 6 Write the quantity (new price) to be found. 7 Multiply the above percentage (80%) by the amount. 8 Evaluate and write using correct units. 9 Write the answer in words. 100% + 20% = 120% New price = 120% of $75 = 1. 20 75 = $90 100% 20% = 80% New price = 80% of $90 = 0. 80 90 = $ 72 New price is $72.

Chapter 3 Units of measurement and applications 97 Exercise 3F 1 What is the amount of the increase in each of the following? a Increase of 10% on $48 b Increase of 30% on $120 c Increase of 15% on $66 d Increase of 25% on $88 e Increase of 40% on $1340 f Increase of 36% on $196 g Increase of 4.5% on $150 h Increase of 1 % on $24 2 2 What is the amount of the decrease in each of the following? a Decrease of 20% on $110 b Decrease of 60% on $260 c Decrease of 35% on $320 d Decrease of 75% on $1096 e Decrease of 6% on $50 f Decrease of 32% on $36 g Decrease of 12.5% on $640 h Decrease of 1 1 % on $56 4 3 David Jones clearance sale has a discount of 30% off the retail price of all clothing. Find the amount saved on the following items. a Men s shirt with a retail price of $80 b Pair of jeans with a retail price of $66 c Ladies jacket with a retail price of $450 d Boy s shorts with a retail price of $22 e Jumper with a retail price of $124 f Girl s skirt with a retail price of $50 30% OFF original price 4 A manager has decided to award a salary increase of 6% for all employees. Find the new salary awarded on the following amounts. a Salary of $46 240 b Salary of $94 860 c Salary of $124 280 d Salary of $64 980 5 Molly has a card that entitles her to a 2.5% discount at the store where she works. How much will she pay for the following items? a Vase marked at $190 b Cutlery marked at $240 c Painting marked at $560 d Pot marked at $70

98 Preliminary Mathematics General Development 6 A used car is priced at $18 600 and offered for sale at a discount of 15%. a What is the discounted price of the car? b The car dealer decides to reduce the price of this car by another 15%. What is the new price of the car? 7 Find the repeated percentage change on the following. a Increase $100 by 20% and then decrease the result by 20%. b Increase $280 by 10% and then increase the result by 5%. c Decrease $32 by 50% and then increase the result by 25%. d Decrease $1400 by 5% and then decrease the result by 5%. e Increase $960 by 15% and then decrease the result by 10%. f Decrease $72 by 12.5% and then increase the result by 33 1 3 %. 8 An electronic store offered a $30 discount on a piece of software marked at $120. What percentage discount has been offered? 9 The cost price of a sound system is $480. Retail stores have offered a range of successive discounts. Calculate the final price of the sound system at the following stores. a Store A: Increase of 10% and then a decrease of 5% b Store B: Increase of 40% and then a decrease of 50% c Store C: Increase of 25% and then a decrease of 15% d Store D: Increase of 30% and then a decrease of 60% 10 The price of a clock has been reduced from $200 to $180. a What percentage discount has been applied? b Two months later the price of the clock was increased by the same percentage discount. What is new price of the clock?

Chapter 3 Units of measurement and applications 99 Chapter summary Units of measurement and applications Study guide 3 Units of measurement mega 1000 1000 kilo 1000 1000 unit 100 100 centi 10 10 milli 10000 cm 2 = 1 m 2 1 ha = 10000 m 2 1000000 cm 3 = 1 m 3 24 60 60 days hours minutes seconds 24 60 60 Review Writing numbers in scientific notation 1 Find the first two non-zero digits. 2 Place a decimal point between these two digits. 3 Power of ten is number of the digits between the new and the old decimal point. (Small number negative value, Large number positive value) Writing numbers in significant figures 1 Write the number in scientific notation. 2 Count the digits in the number to determine its accuracy. 3 Round the number to the required significant figures. Ratios A ratio is used to compare amounts of the same units in a definite order. Equivalent ratios are obtained by multiplying or dividing by the same number. Unitary method 1 Find one unit of an amount by dividing by the amount. 2 Multiply the result in step 1 by the number. Converting a rate 1 Write the rate as a fraction. First quantity is the numerator and 1 is the denominator. 2 Convert the first amount to the required unit. 3 Convert the second amount to the required unit. 4 Simplify the fraction. Percentage change 1 Add the % increase or subtract the % decrease from 100%. 2 Multiply the above percentage by the amount.

100 Preliminary Mathematics General Review Sample HSC Objective-response questions 1 Convert 7.5 metres to millimetres. A 0.0075 mm B 75 mm C 750 mm D 7500 mm 2 How many square millimetres are in a square centimetre? A 10 B 100 C 1000 D 10 000 3 Write 4 500 000 in scientific notation. A 4.5 10 6 B 4.5 10 5 C 4.5 10 5 D 4.5 10 6 4 Express 0.0655 correct to two significant figures. A 0.06 B 0.07 C 0.065 D 0.066 5 The ratio of adults to children in a park is 5:9. How many adults are in the park if there are 630 children? A 70 B 126 C 280 D 350 6 A 360 gram lolly bag is divided in the ratio 7:5. What is the mass of the smaller amount? A 150 g B 168 g C 192 g D 210 g 7 A hose fills a 10 L bucket in 20 seconds. What is the rate of flow in litres per hour? A 0.0001 B 30 C 1800 D 7200 8 Which of the following is the slowest speed? A 60 km/h B 100 m/s C 10000 m/min D 6000 m/h 9 The concentration of a drug is 3 ml/g. How many ml are required for 30 g? A 0.1 ml B 10 ml C 27 ml D 90 ml 10 What is the new price when $80 is increased by 20% then decreased by 20%? A $51.20 B $76.80 C $80.00 D $115.20

Chapter 3 Units of measurement and applications 101 Sample HSC Short-answer questions 1 There are six tonnes of iron ore in a train. What is the mass (in tonnes) if another 246 kg of iron ore is added to the train? 2 Complete the following. a 500 cm2 = m2 b 4000 cm2 = mm2 c 3 km2 = m2 Review 3 A field has a perimeter of exactly 400 m. Lily measured the field to be 401.2 m using a long tape marked in 0.1 m intervals. a Calculate the limit of reading. b What is the absolute error for Lily s measurement? c What is the percentage error for Lily s measurement? Answer correct to three decimal places. 4 Write each of the following as a basic numeral. a 4.8 10 6 b 6.25 10 4 c 1.9 10 2 5 Write these numbers in scientific notation. a 50800 b 0.0036 c 381000000 6 Evaluate the following and express your answer in scientific notation. a (7.2 10 5 ) (2.1 10 4 ) 4. 6 104 b 2 2. 3 10 7 Convert a measurement of 3580 tonnes into milligrams. Express your answer in scientific notation correct to two significant figures. 8 Find the value of 45 15 4 and express your answer in scientific notation correct to two significant figures.

102 Preliminary Mathematics General Review 9 Simplify the following ratios. a 500:100 b 20:30 c 28:7 d 10:15:30 e 12:9 f 56:88 g 4.8:1.6 3 1 h : 4 2 10 A 5 kg bag of rice costs $9.20. What is the cost of the following amounts? a 10 kg b 40 kg c 3 kg d 7 kg e 500 g f 250 g 11 Convert each rate to the units shown. a $15/kg to $/g b 14400 m/h to m/min c 120 cm/h to mm/min d 4800 kg/g to kg/mg e 14 L/g to ml/kg f $3600/g to c/mg 12 A car travels 960 km on 75 litres of petrol. How far does it travel on 50 litres? 13 Daniel and Ethan own a business and share the profits in the ratio 3:4. a The profit last week was $3437. How much does Daniel receive? b The profit this week is $2464. How much does Ethan receive? 14 Jill has a shareholder card that entitles her to a 5% discount at a supermarket. How much will she pay for the following items? Answer to the nearest cent. a Breakfast cereal at $7.60 b Milk at $4.90 c Coffee at $14.20 d Cheese at $8.40 15 An electrician is buying a light fitting for $144 at a hardware store. He receives a clearance discount of 15% then a trade discount of 10%. How much does the electrician pay for the light fitting? Challenge questions 3