Units of measurement and applications -...

73

3.1 Units of measurementMeasurement 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 defi ne their units of measurement. We use the SI metric system.

3.1

C H A P T E R

3Units 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

ISBN: 9781107627291 Photocopying is restricted under law and this material must not be transferred to another party

© The Powers Family Trust 2013 Cambridge University Press

74 Preliminary Mathematics General

SI unitsThe 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 MetreMillimetreCentimetreKilometreNautical mile

mmmcmkmnm

Base unit1000 mm = 1 m100 cm = 1 m1 km = 1000 m1 nm = 1852 m

Area Square metreSquare centimetreHectare

m2

cm2

ha

Base unit10 000 cm2 = 1 m2

1 ha = 10 000 m2

Volume Cubic metreCubic centimetreLitreMillilitreKilolitre

m3

cm3

LmLkL

Base unit1 000 000 cm3 = 1 m3

1L = 1000 cm3

1000 mL = 1 L1 kL = 1000 L

Mass KilogramGramTonne

kggt

Base unit1000 g = 1 kg1 t = 1000 kg

Time SecondMinuteHourDay

sminhd

Base unit1 min = 60 s1 h = 60 min1 d = 24 h

Converting between unitsA prefi x is a simple way to convert between units. It indicates a multiple of 10. Some common

prefi xes are mega (1 000 000), kilo (1000), centi 1100( ) and milli 1

1000( ). Length, mass and volume Time

unit

kilo× 1000

× 100

× 10

÷ 1000

÷ 100

÷ 10centi

milli

× 1000 ÷ 1000mega

× 24

× 60

× 60

÷ 24

÷ 60

÷ 60

hours

days

minutes

seconds



75Chapter 3 — Units of measurement and applications

Example 1 Converting units of length

Complete the following.a =35 cm mm b 4500 m km km k=m k= m

Solution1 To change cm to mm multiply by 10.

2 To change m to km divide by 1000.

a 35 cm = 35 × 10 mm = 350 mmb 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 3 h 15 min = 3 × 60 + 15 min = 195 minb 10 080 min = 10 080 ÷ 60 h = 168 h = 168 ÷ 24 d = 7 d

Converting area and volume unitsTo convert area units, change the side length units and compare the values for area.

1 m

1 m 100 cm

100 cm=1 m2 = 100 × 100 = 10 000 cm2

1 m2 = 10 000 cm2

or 1 cm0

m2 2m2 2m12 212 2

10 00=2 2=2 2

To convert volume units, change the side length units and compare the values for volume.

1 m

1 m 100 cm100 cm

100 cm=1 m

1 m3 = 100 × 100 × 100 = 1 000 000 cm3

1 m3 = 1000 000 cm3

or 1 cm0

m3 3m3 3m13 313 3

1000 00=3 3=3 3




Exercise 3A 1 Complete the following.

a 5 cm mm mmm m=m m b 78 m cm cmm c=m c c 2 km mm mm m=m m

d 890 m cm cmm c=m c e 57 cm mm= f 6 km cm cmm c=m c

g 9400 m km kmm k=m k h 600 mm cm= i 8100 cm m=

j 49 000 cm km= k 22 000 m km kmm k=m k l 51 mm cm=

2 Complete the following.

a 3 t kg= b 45 kg g= c 76 t kt kgt k=t k

d 8100 kg g= e 4 t g= f 0 52. t0 5. t0 52. t2 kg=

g 6800 g kg kgg k=g k h 9 300 000 g tg tg t=g t i 45 000 000 g tg tg t=g t

j 300 kg t= k 2300 g kg kgg k=g k l 60 000 g kg kgg k=g k


a 2 L mL= b 12 kL L= c 9 kL mL mLL m=L m

d 7800 kL L= e 50 L mL mLL m=L m f 300 kL mL=

g 6100 L kL kLL k=L k h 400 mL L= i 210 000 mL kL=

j 80 mL L= k 79 000 mL kL= l 8 000 000 mL kL=


a 2 5. h2 5. h2 5 min= b 2 min s= c 20 d hd hd h=d h

d 40 min s= e 4 5. d4 5. d4 5 h= f 10 h min=

g 720 min h= h 48 000 s min= i 96 h dh dh d=h d

j 1080 h dh dh d=h d k 390 min h= l 780 s min=




5 What unit of length is most appropriate to measure each of the following?a Length of a pen b Height of a buildingc Thickness of a credit card d Distance from Sydney to Newcastlee 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 mugc Bag of onions d Weight of a babye 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 microwavec 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.




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 5 000 000 mm in:a Ascending order (smallest to largest)b Descending order (largest to smallest)


a 1 2 22 2km m2 2m2 2=2 2=2 2 b 1 2 22 2m mm m2 2m m2 22 2m m2 2m2 2m2 2m m=m m2 2m m2 2=2 2m m2 2

c 1 2 22 2cm mm2 2mm2 2=2 2=2 2 d 1000 cm m2 22 2m2 2m=2 2=2 2

e 2000 mm cm2 22 2cm2 2cm=2 2=2 2 f 5000 m km km2 22 2m k2 2m km k2 2m km2 2mm k=m km k2 2m k=m k2 2m k

g 3 9 2 22 2. m3 9. m3 9 cm2 2cm2 2=2 2=2 2 h 310 2 22 2km m2 2m2 2=2 2=2 2

i 4 7 2 22 2. m4 7. m4 7 mm2 2mm2 2=2 2=2 2 j 74300 m km km2 22 2m k2 2m km k2 2m km2 2mm k=m km k2 2m k=m k2 2m k

k 6500 mm cm2 22 2cm2 2cm=2 2=2 2 l 4000 cm m2 22 2m2 2m=2 2=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.




3.2 Measurement errorsThere 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 measurementsThe 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

2 of the limit of reading. For example,

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.

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

2 of the limit of reading.

Limit of reading Absolute error

Smallest unit on measuring instrument Measured value – Actual value

Maximum value is 12

× limit of reading

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

measurement is 0 510

0 050 5.0 5 0 0.0 0( ) = . The relative error is often expressed as a percentage and

called the ‘percentage error’. For example, the percentage error for the above measurement is 0 5

10100 50 5.0 5 %( ) × =100× =100 .

1 cm 2 3 4 5




Relative error Percentage error

±

Absolute error

Measurement±

×Absolute error

Measurement100%

Example 3 Finding the measurement errors

a What is the length indicated by the arrow on the above ruler?

b What is the limit of reading?c What is the upper and lower limit for each measurement?d Find the relative error. Answer correct to three decimal places.e 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

2 the limit of

reading. 5 Upper limit is the measured

value plus 12

the limit of 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 Length is 38 mm.

b Limit of reading is 1 mm.

c 12

× limit of reading = 12

× 1

= 0.5 mm

Lower limit = 38 − 0.5 = 37.5 mm

Upper limit = 38 + 0.5 = 38.5 mm

d Relative error = ± Absolute error

Measurement

= ± 0 5

38

0 5.0 5

= ± 0.013

e Percentage error = ± Absolute error

Measurement

× 100%

= ± 0.5

38 × 100%

= ± 1.3%

0 10 20 30 40 50 60 70 80




Exercise 3B 1 Four measurements of length are shown on the ruler below.

0 10 20 30 40 50 60 70 80 90 100

A B C D

a What length is indicated by each letter? Answer to the nearest millimetre.b What is the limit of reading?c What the largest possible absolute error?d What is the upper and lower limit for each measurement?e Calculate the relative error, correct to three decimal places, for each measurement.f Calculate the percentage error, correct to one decimal place, for each measurement.

2 Two measurements of mass are shown on the scales below.

a What mass is indicated by each letter? Use the outer scale.b What is the limit of reading?c What the largest possible absolute error?d What is the upper and lower limit for each measurement?e Calculate the relative error, correct to three decimal places, for each measurement.f Calculate the percentage error, correct to one decimal place, for each measurement.

0

0 8

8

8

8

88

8

8

8

8

8

0.5

1kg1lb

2lb

3lb

4lb

5lb6lb

7lb

8lb

9lb

10lb

1.5

2kg2.5

3kg

3.5

4kg

4.5

A

0

0 8

8

8

8

88

8

8

8

8

8

0.5

1kg1lb

2lb

3lb

4lb

5lb6lb

7lb

8lb

9lb

10lb

1.5

2kg2.5

3kg

3.5

4kg

4.5

B




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.




3.3 Scientific notation and significant figures

Scientific notationScientifi c 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 scientifi c notation as 4.1 × 106. The power of ten indicates the number of tens multiplied together. For example:

4.1 × 106 = 4.1 × (10 × 10 × 10 × 10 × 10 × 10) = 4 100 000

When writing numbers in scientifi c 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 scientifi c notation

1 Find the fi rst 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 scientifi c notation.

Solution1 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 8153 400 000 = 1.534 × 108

3.3




Significant figuresSignifi cant fi gures are used to specify the accuracy of a number. It is often used to round a number. Signifi cant fi gures 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 fi nish 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 signifi cant fi gures: 5, 1, 3 and 4.• 0.00871 has three signifi cant fi gures: 8, 7 and 1.• 56091 has fi ve signifi cant fi gures: 5, 6, 0, 9 and 1.

The signifi cant fi gures 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 signifi cant fi gures. To prevent this problem, the last signifi cant fi gure of a number is underlined. For example, the number 8000 has two signifi cant fi gures. If the digit is not underlined the context of the problem is a guide to the accuracy of the number.

Writing numbers to signifi cant fi gures

1 Write the number in scientifi c notation.2 Count the digits in the number to determine its accuracy (ignore zeros at the end).3 Round the number to the required signifi cant fi gures.

Example 5 Writing numbers to significant figures

Write these numbers correct to the signifi cant fi gures indicated.a 153 400 000 (3 signifi cant fi gures)b 0.000 657 (2 signifi cant fi gures)

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.

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.

a 153 400 000 = 1.534 × 108

1.534 has 4 digits 1.53 rounded to 3 sig. fig. 153 400 000 = 1.53 × 108

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




Exercise 3C 1 Write these numbers in scientific notation.

a 7600 b 1 700 000 000 c 590 000 d 6 800 000 e 35 000 f 310 000 000g 77 100 000 h 523 000 000 000 i 95 400 000 000

2 Write these numbers in scientific notation.a 0.000 56 b 0.000 068 7 c 0.000 000 812 d 0.0043 e 0.000 058 f 0.000 003 12g 0.26 h 0.092 i 0.000 000 000 167

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 × 105 b 5.34 × 108

c 5.2 × 103 d 8.678 × 107

e 2.4 × 102 f 7.8 × 109

g 3.9 × 106 h 2.8 × 101

i 6.4 × 104

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




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 × 103) × (5.9 × 106) b (4.7 × 105) × (6.3 × 102)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 9.1 105×× −2 8 10 22 8.2 8

b 7.2 107×× −4 8 10 34 8.4 8

c 4.8 10 4××

−

−3 2 10 53 2.3 2

10 Write these numbers correct to significant figures indicated.a 1 561 231 (2 sig. fi g.) b 3 677 720 (4 sig. fi g.) c 789 001 (5 sig. fi g.)d 3 300 000 (1 sig. fi g.) e 777 777 (3 sig. fi g.) f 3 194 729 (5 sig. fi g.)g 821 076 (4 sig. fi g.) h 7091 (1 sig. fi g.) i 49 172 (2 sig. fi g.)

11 Write these numbers correct to significant figures indicated.a 0.0035 (1 sig. fi g.) b 0.191 785 (4 sig. fi g.) c 0.001 592 (3 sig. fi g.)d 0.111 222 33 (6 sig. fi g.) e 0.000 0271 (1 sig. fi g.) f 0.019 832 6 (5 sig. fi g.)g 0.008 12 (2 sig. fi g.) h 0.092 71 (3 sig. fi g.) i 0.000 419 (2 sig. fi g.)

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.




Development

15 If y xy xy x=y x1y x1y x2

y x2

y x2, find the value of y when:

a x = 2.4 × 103 b x = 9.8 × 10−3

16 The arc length of a circle is θ

°πl rl r

θl r

θ=l r= πl rπ360

l r2l 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 × 108. Answer in scientific notation correct to one significant figure.

17 Given that Vr

h= find the value of r in scientific notation when:

a V = 5 × 104 and h = 9 × 106 b V = 6 × 10−7 and h = 4 × 102

18 Use the formula E = md2 to find d correct to three significant figures given that:a m = 0.08 and E = 5.5 × 109

b m = 2.7 × 103 and E = 1.6 × 104

19 Find x given x3 = 2.7 × 1012. 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 × 105 and q = 2.5 × 104. Answer in scientific notation correct to one significant figure.

22 The volume of a cylinder is V = πr2h 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 × 104 and h = 2.8 × 103. Answer in scientific notation correct to three significant figures.

23 The Earth is 1.496 × 108 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.

13.3




3.4 Calculations with ratiosA ratio is used to compare amounts of the same units in a defi nite order. For example, the ratio 3:4 represents 3 parts to 4 parts or 34

or 0.75 or 75%.

A ratio is a fraction and can be simplifi ed in the same way as a fraction. For example, the ratio 15:20 can be simplifi ed 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.

15 : 12 = 5 : 4÷3 ÷3

5 : 4 = 15 : 12×3 ×3

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

8: 3

4 multiply both sides by 8. This results in the

equivalent ratio of 1:6.

Ratio

A ratio is used to compare amounts of the same units in a defi nite order.Equivalent ratios are obtained by multiplying or dividing by the same number.

Dividing a quantity in a given ratioRatio 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.

3.4




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?

Solution1 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 parts9 parts $450

1 part$450

= + ==

= == =

4 5= +4 5= + 9

9$550$550$5

4 5 2005 5 250

4 parts5 parts

= ×4 5= ×4 5= ×5 5= ×5 5

$ $4 5$ $4 50$ $0 =$ $=$ $5 5$ $5 50$ $0 =$ $=

($200 + $250 = $450)

Mikhail receives $200 and Ilya receives $250.

The unitary methodThe unitary method involves fi nding 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?

Solution1 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

30 by a 7 to solve the problem.

4 Evaluate.5 Write answer to an appropriate degree of accuracy.6 Write the answer in words.

30 L km

1 L km

7 L km

84 km

L k=L k

=

= ×= ×

=

360L k360L k360

30360

307

The car travels 84 km.




Exercise 3D 1 Express each ratio in simplest form.

a 15:3 b 10:40 c 24:16d 14:30 e 8:12 f 49:14

g 9:18:9 h 5:10:20 i 11

3:

j 1

2

1

5: k

2

3

3

7: l

3

41:

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 3:2c 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




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 35

of the profits and Mathilde 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

2 kilograms of sugar in making the jam?

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?

Mixed lollies

Jelly babies $5.95 per kg

Licorice $6.95 per kg

Skittles $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?




3.5 Rates and concentrations

RatesA 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 specifi ed.

The order of a rate is important. A rate is written as the fi rst 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 fi rst 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/minb $6.50/kg to c/g

Solution1 Write the rate as a fraction. 2 The numerator is 55 200 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.

a 55 20055 200 m

h55 200 m

minm/min

=

=

=

1

1 6×1 6× 0920

b 6.50$6.50

kg6.50 c

gc/g

=

= ××

=

1100

1 10000 650 6.0 6

3.5




ConcentrationsA 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?

Solution1 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/10 kg2.5 mL

kg2.5 mL

kg

=

= ×= ×

=

10

100 2. Lm. Lm0 2. L0 25. L5 L/. LL/. 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.

Solution1 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 g6.2 g

50 g6.2

=

= ×= ×

=50

100

12

%

. %4. %4

Percentage composition is 12.4%.




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 m2?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

2 kg?

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 39 240 m/min [m/s] b 2 m/s [cm/s] c 88 cm/h [mm/h] d 55 200 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 12 600 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?




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 12 560 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 12 027 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.




3.6 Percentage changePercentage change involves increasing or decreasing a quantity as a percentage of the original amount of the quantity.

Percentage increase Percentage decrease

1 Add the % increase to 100%.2 Multiply the above percentage by the

amount.

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?

Solution1 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 of $36

$37.80

== ×=

1051 0= ×1 0= ×5 3= ×5 3= × 6

%1 0.1 0= ×1 0= ×.= ×1 0= ×

New price is $37.80.

Example 12 Calculating repeated percentage changes

Increase $75 by 20% and then decrease the result by 20%.

Solution1 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 of $75

$90

== ×=

1201 2= ×1 2= ×0 7= ×0 7= × 5

%.1 2.1 2= ×1 2= ×.= ×1 2= ×

100% − 20% = 80%New price of $90

$ 2

== ×=

800 8= ×0 8= ×0 9= ×0 9= × 0

%0 8.0 8= ×0 8= ×.= ×0 8= ×$ 27$ 2

New price is $72.

3.6




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 $120c Increase of 15% on $66 d Increase of 25% on $88e Increase of 40% on $1340 f Increase of 36% on $196

g Increase of 4.5% on $150 h Increase of 12

% on $24

2 What is the amount of the decrease in each of the following?a Decrease of 20% on $110 b Decrease of 60% on $260c Decrease of 35% on $320 d Decrease of 75% on $1096e Decrease of 6% on $50 f Decrease of 32% on $36

g Decrease of 12.5% on $640 h Decrease of 1 14

% on $56

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 $80b Pair of jeans with a retail price of $66c Ladies jacket with a retail price of $450d Boy’s shorts with a retail price of $22e Jumper with a retail price of $124f Girl’s skirt with a retail price of $50

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 240b Salary of $94 860c Salary of $124 280d 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 $190b Cutlery marked at $240c Painting marked at $560d Pot marked at $70

30% OFForiginal price




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?



99Chapter 3 — Units of measurement and applicationsReview

Chapter summary – Units of measurement and applications Study guide 3

Units of measurement

unit

kilo× 1000

× 1000

× 100

× 10

÷ 1000

÷ 1000

÷ 100

÷ 10centi

milli

mega× 24

× 60

× 60

÷ 24

÷ 60

÷ 60

hours

days

minutes

seconds

10 000 cm2 = 1 m2

1 ha = 10 000 m2

1 000 000 cm3 = 1 m3

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 GeneralRe

view

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 × 105 D 4.5 × 106

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 10 000 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



101Chapter 3 — Units of measurement and applicationsReview

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?


a 500 2 22 2cm m2 2m2 2=2 2=2 2 b 4000 2 22 2cm mm2 2mm2 2=2 2=2 2

c 3 2 22 2km m2 2m2 2=2 2=2 2

3 A fi eld has a perimeter of exactly 400 m. Lily measured the fi eld 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 × 106 b 6.25 × 10−4 c 1.9 × 102

5 Write these numbers in scientifi c notation.a 50 800 b 0.0036 c 381 000 000

6 Evaluate the following and express your answer in scientifi c notation.

a (7.2 × 105) × (2.1 × 104) b 4 6

2 3

4 6.4 6

2 3.2 3

×10×10

4

−2

7 Convert a measurement of 3580 tonnes into milligrams. Express your answer in scientifi c notation correct to two signifi cant fi gures.

8 Find the value of 45 × 154 and express your answer in scientifi c notation correct to two signifi cant fi gures.



102 Preliminary Mathematics GeneralRe

view 9 Simplify the following ratios.

a 500:100 b 20:30 c 28:7d 10:15:30 e 12:9 f 56:88

g 4.8:1.6 h 3

4

1

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 kgd 7 kg e 500 g f 250 g

11 Convert each rate to the units shown.a $15/kg to $/g b 14 400 m/h to m/minc 120 cm/h to mm/min d 4800 kg/g to kg/mge 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 profi ts in the ratio 3:4.a The profi t last week was $3437. How much does Daniel receive?b The profi t 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 $4.90c Coffee at $14.20 d Cheese at $8.40

15 An electrician is buying a light fi tting 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 fi tting?

Challenge questions 3



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