1.
SkillSHEET
answers
SkillSHEET 1.1
Percentages
To ﬁnd a certain percentage of a quantity, change the percentage into a fraction or a decimal and multiply by that
quantity.
To increase or decrease some amount by a certain percentage is a calculation which is often required. An increase
yields a larger number and a decrease yields a smaller number compared with the original value.
To increase a number by r%, multiply the number by (1 + ); to decrease a number by r% multiply by
(1 – ).
The number (1 + ) or (1 – ) is often referred to as the multiplying factor.
Try these
1 Find:
a 10% of 60 b 5% of 130 c 40% of $800 d 25% of 700 students
e 16% of 75 f 53% of $20 g 4% of 132 h 7.5% of 260 articles.
2 If 95% of students passed their mid-year examination, how many of the 140 students passed?
3 Enrolments this year increased by 12% from last year. If there were 510 enrolments recorded last year, what is
the total recorded enrolment for this year ?
4 a Wages will be increased by r%. What is the multiplying factor if wages increased by:
i 10% ii 20% iii 25% iv 7% v 12 %?
b If a person is earning $260 per week, calculate the new wage if it increased by:
i 10% ii 20% iii 25% iv 7% v 12 %.
5 a The price of an item is reduced by r%. What is the multiplying factor if the reduction was:
i 10% ii 20% iii 25% iv 7% v 12 %?
b If the printer cost $360, calculate its value if the initial cost depreciated by:
i 10% ii 20% iii 25% iv 7% v 12 %.
Find 15% of $270.
Solution
Express 15% as a fraction and multiply by 270.
15% of $270 = ×
=
= $ 40.50
15
100
---------
270
1
---------
4050
100
------------
WORKEDExample 1
r
100
---------
r
100
---------
r
100
---------
r
100
---------
Increase $50 by 15%. Decrease 340 by 5%.
Solution Solution
To increase by 15%, multiply $50 by (1 + ).
50 × (1 + ) = 50 × (1 + 0.15)
= 50 × 1.15
= $57.50
To decrease by 5%, multiply 340 by (1 − ).
340 × (1 − ) = 340 × (1 − 0.95)
= 340 × 0.95
= 323
15
100
---------
15
100
---------
5
100
---------
5
100
---------
2WORKEDExample 3WORKEDExample
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2
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2
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2
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2.
SkillSHEET
answers
SkillSHEET 1.2
Expressing one number as a percentage of
another
To express one number as a percentage of another, form a fraction using these two numbers and multiply it
by 100.
Note that the number that is being expressed as a percentage is placed in the numerator of the fraction. For
example, if we need to express 5 as a percentage of 20, we put 5 in the numerator and 20 in the denominator of a
fraction (that is, ).
Try these
For each of the following, express the ﬁrst number as a percentage of the second number, giving your answer
correct to two decimal places.
1 42, 53 2 13, 75 3 34, 150
4 47, 95 5 3, 21 6 12, 35
7 23, 60 8 256, 780 9 7, 65
10 5, 41 11 12, 28 12 37, 61
13 341, 730 14 11, 25 15 15, 24
5
20
------
Express 18 as a percentage of 34.
Solution
Form a fraction by placing 18 in the numerator and 34 in the denominator.
To change a fraction into a percentage, multiply by 100.
× 100 = × (Write 100 as a fraction by putting it over 1.)
= (Multiply numerators together and denominators together.)
= 52.94% (Divide the numerator by the denominator, giving the answer correct to two
decimal places.)
18
34
------
18
34
------
18
34
------
100
1
---------
1800
34
------------
WORKEDExample
3.
SkillSHEET
answers
SkillSHEET 1.3
Ratio
A ratio is a comparison of two quantities using the same units. A rate is a comparison of two quantities using
different units. For example the concentrate of a pesticide, Xenon, must be diluted before use. The instructions
suggest that 15 mL of pesticide be mixed with 750 mL of water. This is written:
Pesticide : Water
15 mL : 750 mL.
Clearly if one were to mix 5 mL of pesticide with 250 mL of water the strength of the mixture would be the
same. The ratio
5 mL : 250 mL
is equivalent to
15 mL : 750 mL.
The instructions for this pesticide say that 500 mL will cover 20 m2
in area. Because these quantities have dif-
ferent units their comparison is called a rate:
500 mL : 20 m2
is a rate which is equivalent to
250 mL : 10 m2
.
Alternatively one can write 500 mL per 20 m2
and then reduce it to 25 mL per 1 m2
or 25 mL/m2
.
If these ratios are all equivalent to 400 : 600, ﬁnd A car travels 350 km on 24 L of petrol. Simplify this
the missing number. rate and calculate how many kilometres the same car
a 200 : x will travel on 40 L of fuel.
b x : 200
c x : 150
Solution Solution
a 400 ÷ 2 = 200, so x = 600 ÷ 2 = 300
b 600 ÷ 3 = 200, so x = 400 ÷ 3 = 133·3
c 600 ÷ 4 = 150, so x = 400 ÷ 4 = 100
The rate is 350 kM per 24 litres
= 350/24 km per litre
= 14·58 km/L.
On 40 L the car will travel 40 × 14·6 km = 583 km.
1WORKEDExample 2WORKEDExample
Jan and Dean built a bird-bath which they sold at the markets for $140. They agreed that Jan had spent 4
hours building it and that Dean had worked for 3 hours. If the money is split in the same ratio as the time
they spent, how much does each of them receive?
Solution
Jan : Dean
4 : 3
Total number of hours = 7
So Jan gets of $140 = $80
and Dean gets of $140 = $60
4
7
---
3
7
---
3WORKEDExample
4.
SkillSHEET
answers
Try these
1 If these ratios are all equivalent to 30 : 48, ﬁnd the missing number.
a 120 : x b x : 240 c x : 36
2 If these ratios are all equivalent to 25 : 200, ﬁnd the missing number.
a 200 : x b x : 2 c x : 150
3 If a car travels 250 km on 22 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 600 km.
4 If a car travels 150 km on 18 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 500 km.
5 A weed-killer concentrate should be mixed with water in the ratio of 25 mL to 2000 mL (2 L). If there is only
15 mL of the concentrate remaining, how much water should be mixed with it?
6 Patsy and Crystal set up a screen printing business, with Patsy contributing $1200 and Crystal $700. How
should the ﬁrst month’s proﬁt of $400 be distributed?
7 The Green Pines resort charges $624 per person for a 7-night stay. At the same rate what would you expect to
pay for a 2-night stay?
8 To stay 5 nights at the Cool Breeze Caravan Park costs $47·50. What would you pay for 2 nights at the same
rate?
9 A car travels at 80 km/h. Express this rate in these units.
a metres/hour b metres/minute c metres/second
d hours/100 kilometres
10 Hewey Lewis can run 100 m in 12 s. How fast is this in km/h?
11 The speed of sound is 340 m/s.
a What is this speed in km/h?
b In the 200-metre event the starting gun is 170 m from the timers. If we assume that the smoke from the gun
is seen as soon as the gun is ﬁred, how much of a time lag is there between when the ﬁring is seen and
heard by the timers? If a timer were to start his watch on the sound of the gun, would the athletes’ recorded
time be faster (lower) or slower (higher)?
c For investigation: The speed of sound given here is its speed in air at sea level. How does it change as you
go higher?
5.
SkillSHEET
answers
SkillSHEET 2.1 — Finding a percentage of a
quantity
The most common application of percentages is to ﬁnd a percentage of a quantity. In many cases this will be a
percentage of a money amount.
To ﬁnd the percentage of an amount, write the percentage as a fraction or a decimal and then multiply by the
amount. The simplest way to do this is to divide the percentage by 100, which converts the percentage to a
decimal before multiplying.
Try these
1 Calculate each of the following.
a 25% of $32 b 50% of $460 c 75% of $84
d 4% of $1400 e 8% of $520 f 3% of $624
g 33 % of $540 h 66 % of $360 i 12 % of $400
j % of $640 k % of $585 l 6.5% of $734
2 Ricky works as a clerk and receives $315.00 per week. He receives a 4% pay rise. How much extra does Ricky
receive each week?
3 Sally is an accountant with an annual salary of $42 000. She receives a Christmas bonus of 3% of her salary.
Calculate the size of her Christmas bonus.
4 Mr and Mrs Hamilton sell their house for $192 000. They must pay a commission to the real estate agent of
2.5%. Calculate the amount of commission received by the real estate agent.
5 Dagmar has shares in a company to the value of $23 500. The dividend paid to the shareholders is 2.5% of the
value of shares held. Calculate the amount Dagmar receives as her dividend.
6 The value of the Australian economy is $2560 billion. The government announces that the economy will grow
by 2.1% over the next year. Calculate the amount by which the government predicts the economy to grow.
Find 48% of $380.
THINK WRITE
Divide 48 by 100 and then multiply by $380. 48% of $380
= 48 ÷ 100 × $380
= $182.40
WORKEDExample
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3
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3
4
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6.
SkillSHEET
answers
SkillSHEET 2.2 — Increase or decrease by a
percentage
When asked to increase by a percentage, add the percentage to 100% before ﬁnding the total percentage of the
quantity.
If decreasing by a percentage the method is similar, except subtract the percentage from 100%.
Try these
1 Calculate each of the following.
a Increase $380 by 5%.
b Increase $850 by 12%.
c Increase $85 by 65%.
d Increase $750 by 2.5%.
2 Calculate each of the following.
a Decrease $150 by 5%.
b Decrease $386 by 40%.
c Decrease $1250 by 90%.
d Decrease $75 by 1%.
3 A drill set sells at the hardware store for $136.00. The price is then increased by 8%. Calculate the new price
of the drill set.
4 A motel room is advertised at $120 per night. A 12% surcharge is added on weekends. Calculate the cost per
night of the motel on weekends.
5 Kristan receives $430 per week in her part-time job as a journalist. She pays 25% of this in tax. Calculate the
amount of money that Kristan has left in her pay packet after tax has been deducted.
6 The cost of a restaurant bill has been reduced by 7.5% due to poor service. If the original bill was $98.50, cal-
culate the amount to be paid.
A sports store buys softball bats for $35 each and the mark-up on each bat is 40%. Calculate the
selling price of the softball bats.
THINK WRITE
Add 40% to 100% to ﬁnd the percentage that
we are ﬁnding of $35.
140% of $35
Divide 140 by 100 and then multiply by $35. = 140 ÷ 100 × $35
= $49
Give a written answer. The softball bats sell for $49 each.
1
2
3
WORKEDExample
7.
SkillSHEET
answers
SkillSHEET 2.3
Ratio
A ratio is a comparison of two quantities using the same units. A rate is a comparison of two quantities using
different units. For example the concentrate of a pesticide, Xenon, must be diluted before use. The instructions
suggest that 15 mL of pesticide be mixed with 750 mL of water. This is written:
Pesticide : Water
15 mL : 750 mL.
Clearly if one were to mix 5 mL of pesticide with 250 mL of water the strength of the mixture would be the
same. The ratio
5 mL : 250 mL
is equivalent to
15 mL : 750 mL.
The instructions for this pesticide say that 500 mL will cover 20 m2
in area. Because these quantities have dif-
ferent units their comparison is called a rate:
500 mL : 20 m2
is a rate which is equivalent to
250 mL : 10 m2
.
Alternatively one can write 500 mL per 20 m2
and then reduce it to 25 mL per 1 m2
or 25 mL/m2
.
If these ratios are all equivalent to 400 : 600, ﬁnd A car travels 350 km on 24 L of petrol. Simplify this
the missing number. rate and calculate how many kilometres the same car
a 200 : x will travel on 40 L of fuel.
b x : 200
c x : 150
Solution Solution
a 400 ÷ 2 = 200, so x = 600 ÷ 2 = 300
b 600 ÷ 3 = 200, so x = 400 ÷ 3 = 133·3
c 600 ÷ 4 = 150, so x = 400 ÷ 4 = 100
The rate is 350 kM per 24 litres
= 350/24 km per litre
= 14·58 km/L.
On 40 L the car will travel 40 × 14·6 km = 583 km.
1WORKEDExample 2WORKEDExample
Jan and Dean built a bird-bath which they sold at the markets for $140. They agreed that Jan had spent 4
hours building it and that Dean had worked for 3 hours. If the money is split in the same ratio as the time
they spent, how much does each of them receive?
Solution
Jan : Dean
4 : 3
Total number of hours = 7
So Jan gets of $140 = $80
and Dean gets of $140 = $60
4
7
---
3
7
---
3WORKEDExample
8.
SkillSHEET
answers
Try these
1 If these ratios are all equivalent to 30 : 48, ﬁnd the missing number.
a 120 : x b x : 240 c x : 36
2 If these ratios are all equivalent to 25 : 200, ﬁnd the missing number.
a 200 : x b x : 2 c x : 150
3 If a car travels 250 km on 22 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 600 km.
4 If a car travels 150 km on 18 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 500 km.
5 A weed-killer concentrate should be mixed with water in the ratio of 25 mL to 2000 mL (2 L). If there is only
15 mL of the concentrate remaining, how much water should be mixed with it?
6 Patsy and Crystal set up a screen printing business, with Patsy contributing $1200 and Crystal $700. How
should the ﬁrst month’s proﬁt of $400 be distributed?
7 The Green Pines resort charges $624 per person for a 7-night stay. At the same rate what would you expect to
pay for a 2-night stay?
8 To stay 5 nights at the Cool Breeze Caravan Park costs $47·50. What would you pay for 2 nights at the same
rate?
9 A car travels at 80 km/h. Express this rate in these units.
a metres/hour b metres/minute c metres/second
d hours/100 kilometres
10 Hewey Lewis can run 100 m in 12 s. How fast is this in km/h?
11 The speed of sound is 340 m/s.
a What is this speed in km/h?
b In the 200-metre event the starting gun is 170 m from the timers. If we assume that the smoke from the gun
is seen as soon as the gun is ﬁred, how much of a time lag is there between when the ﬁring is seen and
heard by the timers? If a timer were to start his watch on the sound of the gun, would the athletes’ recorded
time be faster (lower) or slower (higher)?
c For investigation: The speed of sound given here is its speed in air at sea level. How does it change as you
go higher?
9.
SkillSHEET
answers
SkillSHEET 3.1 — Finding a percentage of a
quantity
The most common application of percentages is to ﬁnd a percentage of a quantity. In many cases this will be a
percentage of a money amount.
To ﬁnd the percentage of an amount, write the percentage as a fraction or a decimal and then multiply by the
amount. The simplest way to do this is to divide the percentage by 100, which converts the percentage to a
decimal before multiplying.
Try these
1 Calculate each of the following.
a 25% of $32 b 50% of $460 c 75% of $84
d 4% of $1400 e 8% of $520 f 3% of $624
g 33 % of $540 h 66 % of $360 i 12 % of $400
j % of $640 k % of $585 l 6.5% of $734
2 Ricky works as a clerk and receives $315.00 per week. He receives a 4% pay rise. How much extra does Ricky
receive each week?
3 Sally is an accountant with an annual salary of $42 000. She receives a Christmas bonus of 3% of her salary.
Calculate the size of her Christmas bonus.
4 Mr and Mrs Hamilton sell their house for $192 000. They must pay a commission to the real estate agent of
2.5%. Calculate the amount of commission received by the real estate agent.
5 Dagmar has shares in a company to the value of $23 500. The dividend paid to the shareholders is 2.5% of the
value of shares held. Calculate the amount Dagmar receives as her dividend.
6 The value of the Australian economy is $2560 billion. The government announces that the economy will grow
by 2.1% over the next year. Calculate the amount by which the government predicts the economy to grow.
Find 48% of $380.
THINK WRITE
Divide 48 by 100 and then multiply by $380. 48% of $380
= 48 ÷ 100 × $380
= $182.40
WORKEDExample
1
3
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2
3
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2
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1
2
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3
4
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10.
SkillSHEET
answers
SkillSHEET 3.2 — Increase or decrease by a
percentage
When asked to increase by a percentage, add the percentage to 100% before ﬁnding the total percentage of the
quantity.
If decreasing by a percentage the method is similar, except subtract the percentage from 100%.
Try these
1 Calculate each of the following.
a Increase $380 by 5%.
b Increase $850 by 12%.
c Increase $85 by 65%.
d Increase $750 by 2.5%.
2 Calculate each of the following.
a Decrease $150 by 5%.
b Decrease $386 by 40%.
c Decrease $1250 by 90%.
d Decrease $75 by 1%.
3 A drill set sells at the hardware store for $136.00. The price is then increased by 8%. Calculate the new price
of the drill set.
4 A motel room is advertised at $120 per night. A 12% surcharge is added on weekends. Calculate the cost per
night of the motel on weekends.
5 Kristan receives $430 per week in her part-time job as a journalist. She pays 25% of this in tax. Calculate the
amount of money that Kristan has left in her pay packet after tax has been deducted.
6 The cost of a restaurant bill has been reduced by 7.5% due to poor service. If the original bill was $98.50, cal-
culate the amount to be paid.
A sports store buys softball bats for $35 each and the mark-up on each bat is 40%. Calculate the
selling price of the softball bats.
THINK WRITE
Add 40% to 100% to ﬁnd the percentage that
we are ﬁnding of $35.
140% of $35
Divide 140 by 100 and then multiply by $35. = 140 ÷ 100 × $35
= $49
Give a written answer. The softball bats sell for $49 each.
1
2
3
WORKEDExample
11.
SkillSHEET
answers
SkillSHEET 3.3
Ratio
A ratio is a comparison of two quantities using the same units. A rate is a comparison of two quantities using
different units. For example the concentrate of a pesticide, Xenon, must be diluted before use. The instructions
suggest that 15 mL of pesticide be mixed with 750 mL of water. This is written:
Pesticide : Water
15 mL : 750 mL.
Clearly if one were to mix 5 mL of pesticide with 250 mL of water the strength of the mixture would be the
same. The ratio
5 mL : 250 mL
is equivalent to
15 mL : 750 mL.
The instructions for this pesticide say that 500 mL will cover 20 m2
in area. Because these quantities have dif-
ferent units their comparison is called a rate:
500 mL : 20 m2
is a rate which is equivalent to
250 mL : 10 m2
.
Alternatively one can write 500 mL per 20 m2
and then reduce it to 25 mL per 1 m2
or 25 mL/m2
.
If these ratios are all equivalent to 400 : 600, ﬁnd A car travels 350 km on 24 L of petrol. Simplify this
the missing number. rate and calculate how many kilometres the same car
a 200 : x will travel on 40 L of fuel.
b x : 200
c x : 150
Solution Solution
a 400 ÷ 2 = 200, so x = 600 ÷ 2 = 300
b 600 ÷ 3 = 200, so x = 400 ÷ 3 = 133·3
c 600 ÷ 4 = 150, so x = 400 ÷ 4 = 100
The rate is 350 kM per 24 litres
= 350/24 km per litre
= 14·58 km/L.
On 40 L the car will travel 40 × 14·6 km = 583 km.
1WORKEDExample 2WORKEDExample
Jan and Dean built a bird-bath which they sold at the markets for $140. They agreed that Jan had spent 4
hours building it and that Dean had worked for 3 hours. If the money is split in the same ratio as the time
they spent, how much does each of them receive?
Solution
Jan : Dean
4 : 3
Total number of hours = 7
So Jan gets of $140 = $80
and Dean gets of $140 = $60
4
7
---
3
7
---
3WORKEDExample
12.
SkillSHEET
answers
Try these
1 If these ratios are all equivalent to 30 : 48, ﬁnd the missing number.
a 120 : x b x : 240 c x : 36
2 If these ratios are all equivalent to 25 : 200, ﬁnd the missing number.
a 200 : x b x : 2 c x : 150
3 If a car travels 250 km on 22 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 600 km.
4 If a car travels 150 km on 18 L of petrol, express this rate in kilometres per litre and calculate the amount of
petrol needed by this car for a journey of 500 km.
5 A weed-killer concentrate should be mixed with water in the ratio of 25 mL to 2000 mL (2 L). If there is only
15 mL of the concentrate remaining, how much water should be mixed with it?
6 Patsy and Crystal set up a screen printing business, with Patsy contributing $1200 and Crystal $700. How
should the ﬁrst month’s proﬁt of $400 be distributed?
7 The Green Pines resort charges $624 per person for a 7-night stay. At the same rate what would you expect to
pay for a 2-night stay?
8 To stay 5 nights at the Cool Breeze Caravan Park costs $47·50. What would you pay for 2 nights at the same
rate?
9 A car travels at 80 km/h. Express this rate in these units.
a metres/hour b metres/minute c metres/second
d hours/100 kilometres
10 Hewey Lewis can run 100 m in 12 s. How fast is this in km/h?
11 The speed of sound is 340 m/s.
a What is this speed in km/h?
b In the 200-metre event the starting gun is 170 m from the timers. If we assume that the smoke from the gun
is seen as soon as the gun is ﬁred, how much of a time lag is there between when the ﬁring is seen and
heard by the timers? If a timer were to start his watch on the sound of the gun, would the athletes’ recorded
time be faster (lower) or slower (higher)?
c For investigation: The speed of sound given here is its speed in air at sea level. How does it change as you
go higher?
13.
SkillSHEET
answers
SkillSHEET 4.1
Rounding decimal numbers to 2 decimal
places
To round a decimal number correct to 2 decimal places, follow these steps:
1. Consider the digit in the third decimal place (that is, the thousandth’s place).
2. If it is less than 5, simply omit this digit and all digits that follow. (That is, omit all digits beginning from the
third decimal place.)
3. If it is 5 or greater than 5, add 1 to the preceding digit (that is, the one in the hundredth’s place) and omit all
digits beginning from the third decimal place.
Note that the sign ≈ is read as ‘is approximately equal to’.
Try these
Round each of the following numbers correct to 2 decimal places.
1 0.322 2 0.257 3 1.723 4 2.555 5 4.308
6 12.195 7 8.4678 8 25.033 78 9 18.333 333 10 0.166 666 6
Round each of the following numbers correct to 2 decimal places.
a 0.239 b 4.5842
THINK WRITE
a The digit in the third decimal place is 9, which is greater than 5. So add 1 to
the preceding digit (that is, to 3) and omit 9.
a 0.239 ≈ 0.24
b The digit in the third decimal place is 4. Since it is less than 5, simply omit all
digits beginning from the third decimal place (that is, omit 4 and 2).
b 4.5842 ≈ 4.58
WORKEDExample
14.
SkillSHEET
answers
SkillSHEET 4.2
Multiplying decimal numbers by powers of
10
To multiply a decimal number by powers of 10, move the decimal point to the right one space for each zero in the
power of 10. For example, to multiply by 10, move the decimal point one place to the right, while to multiply by
1000 move it three places to the right. Note that if there are not enough digits after the decimal point, we can
always add extra zeros.
Try these
Calculate each of the following.
1 2.56 × 10 2 7.6 × 10 3 0.98 × 10 4 3.49 × 100
5 2.6 × 100 6 70.1 × 100 7 0.2 × 100 8 5.321 × 1000
9 10.2 × 1000 10 0.758 × 1000 11 2.5 × 10 000 12 3.576 × 10 000
13 0.003 × 1000 14 0.000 6 × 10 000 15 0.000 08 × 100 000 16 0.04 × 10 000
Calculate each of the following.
a 5.67 × 10 b 0.7 × 100
THINK WRITE
a To multiply a decimal by 10, move the decimal
point one place to the right (as there is one
zero in 10).
a 5.67 × 10 = 56.7
b To multiply a decimal by 100, we need to
move the decimal point two places to the right.
However, there is only one digit after the
decimal (7). So add a zero ﬁrst (to create two
decimal places), then move the decimal point
two places to the right. Note that we write the
answer as 70, rather than 070.
b 0.7 × 100
= 0.70 × 100
= 70
WORKEDExample
15.
SkillSHEET
answers
SkillSHEET 4.3
Multiplying decimal numbers by 100
To multiply a decimal number by 100, move the decimal point two places to the right. Note that if there are not
enough digits after the decimal point, we can always add extra zeros.
Try these
Calculate each of the following.
1 0.56 × 100 2 0.76 × 100 3 0.98 × 100 4 0.49 × 100
5 2.6 × 100 6 1.1 × 100 7 0.2 × 100 8 5.321 × 100
9 10.2 × 100 10 0.758 × 100 11 0.5 × 100 12 0.0006 × 100
Calculate each of the following.
a 0.67 × 100 b 0.7 × 100
THINK WRITE
a To multiply a decimal number by 100, move
the decimal point two places to the right. Note
that we do not write the zero in front of the
number and the decimal point at the end of the
number (that is, we write 67, rather than 067.).
a 0.67 × 100 = 67
b We need to move the decimal point two places
to the right; however, there is only one digit
after the decimal point (7). So add a zero ﬁrst
(to create two decimal places), then move the
decimal point.
b 0.7 × 100
= 0.70 × 100
= 70
WORKEDExample
16.
SkillSHEET
answers
SkillSHEET 4.4
Dividing whole numbers and decimals by
100
To divide a whole or a decimal number by 100, move the decimal point two places to the left. Note that although
a whole number does not have a decimal point, we can always add it at the end of the number. (For example, 35
and 35. are the same numbers.) Also note that if there are not enough digits to move the decimal point the required
number of places, we can always add extra zeros.
Try these
1 Calculate each of the following.
a 28 ÷ 100 b 60 ÷ 100 c 34 ÷ 100 d 2 ÷ 100
e 15 ÷ 100 f 7 ÷ 100 g 560 ÷ 100 h 721 ÷ 100
i 3 ÷ 100 j 75 ÷ 100 k 600 ÷ 100 l 250 ÷ 100
2 Calculate each of the following.
a 9.2 ÷ 100 b 52.3 ÷ 100 c 0.5 ÷ 100 d 8.19 ÷ 100
e 4.9 ÷ 100 f 123.4 ÷ 100 g 0.3 ÷ 100 h 71.1 ÷ 100
i 155.6 ÷ 100 j 4.25 ÷ 100 k 75.3 ÷ 100 l 100.5 ÷ 100
Calculate each of the following.
a 34 ÷ 100 b 350 ÷ 100 c 75.6 ÷ 100 d 4.1 ÷ 100
THINK WRITE
a Put a decimal point at the end of the whole
number.
a 34 ÷ 100 = 34. ÷ 100
To divide by 100, move the decimal point 2
places to the left. Add a zero in front of the
decimal point.
34 ÷ 100 = 0.34
b Put a decimal point at the end of the whole
number.
b 350 ÷ 100 = 350. ÷ 100
Move the decimal point 2 places to the left.
You may wish to rewrite your answer,
omitting the zero at the end of the resulting
decimal (as 3.50 = 3.5).
350 ÷ 100 = 3.50
350 ÷ 100 = 3.5
c Move the decimal point 2 places to the left and
add a zero in front of it.
c 75.6 ÷ 100 = 0.756
d Move the decimal point two places to the left.
(Add some extra zeros in front of the number
as you go.)
d 4.1 ÷ 100 = 0.041
1
2
1
2
WORKEDExample
17.
SkillSHEET
answers
SkillSHEET 4.5
Finding the size of a sector in degrees, given
its size as a fraction of a circle
There are 360° in a full circle. So to ﬁnd the size of a sector in degrees, given its size as a fraction of a full circle,
multiply the fraction by 360°.
To multiply a fraction by 360°:
1. change 360 into a fraction by writing it over 1
2. simplify as much as possible
3. multiply the numerators together and the denominators together
4. if the answer is an improper fraction, convert it to a mixed number.
Try these
Find the size of each of the following sectors in degrees, given that their size as a fraction of a circle is:
1 2 3 4
5 6 7 8
9 10 11 12
Finding the size of a sector in degrees, if the sector is of a circle.
THINK WRITE
To express a fraction of a circle in degrees, multiply
by 360.
× 360
Convert 360 into a fraction by writing it over 1. = ×
Cross-cancel 5 and 360 by dividing each by 5 (that is,
360 ÷ 5 = 72; 5 ÷ 5 = 1).
= ×
Multiply the numerators together and the
denominators together.
=
Convert the improper fraction into a mixed number
(which in this case is actually a whole number) and
include the degree sign.
= 216°
3
5
---
1
3
5
---
2
3
5
---
360
1
---------
3
3
1
---
72
1
------
4
216
1
---------
5
WORKEDExample
2
5
---
1
3
---
3
10
------
3
4
---
5
12
------
2
9
---
7
8
---
4
15
------
4
5
---
1
6
---
23
36
------
29
60
------
18.
SkillSHEET
answers
SkillSHEET 6.1
Trigonometry review 1
A trigonometric function is a function of any angle (θ) inside a right angled triangle and can be deﬁned as the
ratio of two sides of a right angled triangle.
Sine (abbreviated as sin), cosine (abbreviated as cos) and tangent (abbreviated as tan) are used to deﬁne the
ratios of the sides.
Try these
1 For each of the following right angled triangles, identify the side labelled x in respect to the given angle θ.
Where appropriate, also identify the side labelled y.
a b c
d e f
g h i
2 For each of the triangles in question 1, identify which trigonometric function (sine, cosine or tangent) could be
used to write an equation to ﬁnd x.
sin θ =
or
cos θ =
or
tan θ =
or
opposite
hypotenuse
---------------------------
Hypotenuse
Opposite
side
θ
Hypotenuse
Opposite side
θ
adjacent
hypotenuse
---------------------------
Hypotenuse
Adjacent side
θ
Hypotenuse
Adjacent
side
θ
opposite
adjacent
--------------------
Opposite
side
Adjacent side
θ
Opposite side
Adjacent
side
θ
θ
10 cm
x
θ
10 cm
x
θ
10 cm
x
θ
10 cm
8 cm
x
θ
10 cm
x
θ
6 cm
x
θ
15 mm
xy
θ
6 cm
x
y
θ
8 cmx
y
19.
SkillSHEET
answers
SkillSHEET 6.2
Trigonometry review 2
The sine, cosine or tangent of an angle can be obtained simply from a scientiﬁc or graphics calculator.
For example, to calculate sin 30°:
1. with a scientiﬁc calculator: Enter 30 then press .
2. with a graphics calculator: Press then enter 30 and press .
Caution: Before entering the angle, check that the calculator is in DEGree mode.
Try these
1 Find the value of the following, giving answers correct to 4 decimal places:
a sin 15° b sin 34° c sin 79° d sin 20°46′ e sin 37°25′ f sin 83°17′
g cos 15° h cos 34° i cos 79° j cos 20°46′ k cos 37°25′ l cos 83°17′
m tan 15° n tan 34° o tan 79° p tan 20°46′ q tan 37°25′ r tan 83°17′
2 Find the value of the following, giving answers correct to 3 decimal places:
a sin 90° b cos 90° c tan 180° d sin 220°50′ e cos 139°25′ f tan 283°30′
g sin 115° h cos 234° i tan 370° j sin 180° k cos 180° l tan 90°
SIN
SIN ENTER
Find: a sin 35∞ (to 4 decimal places) b sin 35∞30¢ (to 4 decimal places).
Solution Solution
a sin 35° = 0.5736 b 30′ means 30 minutes or 0.5° so sin 35°30′ is the
same as sin 35.5°.
sin 35°30′ = 0.5807
1WORKEDExample
Find: a cos 45∞ (to 3 decimal places) b cos 45∞20¢ (to 3 decimal places).
Solution Solution
a cos 45° = 0.707 b cos 45°20′ = 0.703
(Note: cos 45°20′ is the same as cos 45.333°)
2WORKEDExample
Find: a tan 60∞ (to 2 decimal places) b tan 60∞52¢ (to 2 decimal places).
Solution Solution
a tan 60° = 1.73 b tan 60°52′ = 1.79
(Note: tan 60°52′ is the same as tan 60.867°)
3WORKEDExample
20.
SkillSHEET
answers
To ﬁnd the angle when the sine, cosine or tangent value is given, use the inverse or 2nd function keys.
Try these
3 Find the value of θ in each of the following. Where appropriate, give your answer in degrees and minutes.
a sin θ° = 0.5 b sin θ° = 0.3145 c sin θ° = 0.7974 d sin θ° = 0.866
e cos θ° = 0.5 f cos θ° = 0.3145 g cos θ° = 0.7974 h cos θ° = 0.866
i tan θ° = 0.5 j tan θ° = 0.3145 k tan θ° = 0.7974 l tan θ° = 0.866
m tan θ° = 0.95 n tan θ° = 3.145 o tan θ° = 1.79 p tan θ° = 2.567
Find the value of angle θ, if sin θ∞ = 0.55, giving your answer in degrees and minutes.
Solution
θ° = sin-1
0.55
θ° = 33.3670° (degrees only)
or 33°22' (degrees and minutes)
4WORKEDExample
Find the value of angle θ, if cos θ∞ = 0.55, giving your answer in degrees and minutes.
Solution
θ° = cos−1
0.55
θ° = 56.63299°
or 56°38′
5WORKEDExample
Find the value of angle θ, if tan θ∞ = 55, giving your answer in degrees and minutes.
Solution
θ° = tan−1
0.55
θ° = 28.8108°
or 28°49′
6WORKEDExample
21.
SkillSHEET
answers
SkillSHEET 7.1
Conversion of units — length
Common units are: millimetre (mm), centimetre (cm), metre (m), kilometre (km)
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
÷ 10 ÷ 100 ÷ 1000
mm cm m km
× 10 × 100 × 1000
Remember: Converting to a smaller unit → Multiply
Converting to a larger unit → Divide
Try these
Convert each of the following measurements to the unit shown in the brackets.
1 2 m (cm) 2 53 mm (cm)
3 610 km (m) 4 0.0003 km (cm)
5 5600 mm (m) 6 11.3 cm (mm)
7 12 304 m (km) 8 0.0007 m (mm)
9 6300 mm (m) 10 0.8 km (m)
11 0.011 km (cm) 12 0.0042 km (mm)
13 9000 mm (m) 14 765 m (km)
15 0.000 089 km (mm) 16 14 683 mm (km)
Convert 5 centimetres to millimetres. Convert 80 000 centimetres to kilometres.
Solution Solution
Converting to a smaller unit so we need to multiply.
To convert cm to mm, multiply by 10.
5 cm × 10 = 50 mm
Converting to a larger unit so we need to divide.
To convert cm to km, divide ﬁrst by 100 (to convert
to m) then by 1000 (to convert to km).
80 000 cm ÷ 100 = 800 m
800 m ÷ 1000 = 0.8 km
1WORKEDExample 2WORKEDExample
22.
SkillSHEET
answers
SkillSHEET 7.2
Reading scales (How much is each interval
worth?)
When reading scales it is important to remember that the intervals between the adjacent marks are equal. To ﬁnd
the value of each interval, ﬁnd the value of the section of the scale whose end points are known (that is, the length
between the adjacent major marks) and then divide by the number of intervals along this section.
Try these
For each of the following scales, ﬁnd how much each interval is worth.
1 2
3 4
5 6
7 8
9 10
For each of the following scales, ﬁnd how much each interval is worth.
a b c
THINK WRITE
a Find the value of the section of the scale
between the major marks by calculating
the difference between the end points.
a 30 − 20 = 10
There are 10 intervals between the major
marks. So, to ﬁnd the value of each
interval, divide the value of the section of
the scale by 10.
10 ÷ 10 = 1
Write the answer in words. Each interval is worth one unit.
b Repeat steps 1–3 as in part a. b 6 − 5 = 1
1 ÷ 10 = 0.1
Each interval is worth 0.1 of a unit.
c Repeat steps 1–3 as in part a. c 200 − 100 = 100
100 ÷ 5 = 20
Each interval is worth 20 units.
20 30 5 6 100 200
1
2
3
WORKEDExample
40 50 70 80
11 12 23 24
10 20 300 400
8 9 20 30
400 500 4000 5000
23.
SkillSHEET
answers
SkillSHEET 7.3
Pythagoras’ theorem
In any right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This known as Pythagoras’ theorem.
For the triangle shown, Pythagoras’ theorem can be written as c2
= a2
+ b2
.
b
c a
Find the length of x to 1 decimal place. Find the length of y to 1 decimal place.
Solution Solution
The hypotenuse is x (which relates to c in the
formula) and the other side lengths are 5 and 6
(which relate to a and b in the formula).
c2
= a2
+ b2
x2
= 52
+ 62
= 25 + 36
= 61
x = (Take the square root of both sides)
=7.8
The hypotenuse is 15 (which relates to c in the
formula) and the other side lengths are y and 11
(which relate to a and b in the formula).
c2
= a2
+ b2
152
= y2
+ 112
225 = y2
+ 121
y2
+ 121 = 225 (Swap sides to bring the y
term to the left side)
y2
= 225 − 121 (Subtract 121 from both sides)
= 104
y = (Take the square root of both
sides)
= 10.2
6
5
x
1115
y
61
104
1WORKEDExample 2WORKEDExample
24.
SkillSHEET
answers
Try these
For each of the following triangles, ﬁnd the value of the pronumeral correct to 1 decimal place.
1 2
3 4
5 6
7 8
11
x
7
19
y
28
10
13
a
25
14
x
17
3
k
135
73
m
19.8
8.5
x
0.6
0.7
x
25.
SkillSHEET
answers
SkillSHEET 7.4
Using trigonometric ratios
Trigonometric ratios deal with right-angled triangles. The sides of the triangle are named according to their pos-
ition with respect to a speciﬁc angle. The hypotenuse is the longest side length of the right-angled triangle.
Trigonometric ratios: SOH CAH TOA means
sin θ = , cos θ = and tan θ =
Hypotenuse
Opposite side with
respect to the angle θ
θ
Adjacent side
with respect
to the angle θ
opposite
hypotenuse
----------------------------
adjacent
hypotenuse
----------------------------
opposite
adjacent
--------------------
Find the value of x. Express your answer Find the value
correct to 3 decimal places. of x.
Express your
answer correct
to 2 decimal
places.
Solution Solution
Given information is:
Angle is 47°, hypotenuse length is 18 cm.
The side marked x is the opposite side.
Use the sine ratio:
sin θ =
sin 47° =
= sin 47°
x = 18 × sin 47°
x = 13.164 cm
Given information is:
Angle = 30°, hence the adjacent side = 45 mm.
The side marked x is the hypotenuse.
Use the cosine ratio:
cos θ =
cos 30° =
x cos 30° = 45
x =
x = 51.96 mm
45 mm
30°
x
18 cm
x
47°
opposite
hypotenuse
---------------------------
x
18
------
x
18
------
adjacent
hypotenuse
---------------------------
45
x
------
45
cos 30°
-----------------
1WORKEDExample 2WORKEDExample
26.
SkillSHEET
answers
Try these
1 For the following right-angled triangles state which trigonometric ratio would be appropriate in ﬁnding the side
length marked x.
a b c
d e
2 Find the side length marked x for each of the triangles in question 1. Express your answers correct to 3 decimal
places.
Find the value of the angle θ, to the nearest degree.
Solution
Given information is:
Opposite side is 24 mm and adjacent side is 37 mm.
Use tangent ratio:
tan θ =
tan θ =
tan θ = 33°
24 mm
37 mm
θ
opposite
adjacent
--------------------
24
37
------
3WORKEDExample
6 cmx
45°
7 cmx
15°
4 cm
x
20°
13 cm
x
73°
10 cm
x
27°
27.
SkillSHEET
answers
3 Find the value of the angle in each of the following triangles. Express your answer to the nearest degree.
a b
c d
e
22 cm
15 cm
θ
31 mm
39 mm
θ
10 cm
12 cm
θ
8.3 cm
5.9 cm
θ
0.52 m
1.34 m
θ
28.
SkillSHEET
answers
SkillSHEET 7.5
Identifying sides of a right-angled triangle
with respect to the given angle.
In a right-angled triangle, the longest side (which is opposite the right angle) is called the hypotenuse. The other
two sides are named depending on their position with respect to the given angle. The side opposite to the given
angle is called the opposite side and the side next to the given angle is called the adjacent side.
Observe the two triangles below.
In the triangle at left, side a is opposite and side b is adjacent, while in the triangle at right side a is adjacent and
side b is opposite. This is because sides are named according to their position in relation to a given angle and in
the two triangles above the given angle is not the same.
Try these
For each of the following right-angled triangles identify the side labelled x with respect to the given angle θ.
Where necessary, identify also the side labelled y.
1 2 3
4 5 6
7 8 9
Side a
Side b
θ
Opposite
side
Hypotenuse
Adjacent side
Side a
Side b
θ
Adjacent
side
Hypotenuse
Opposite side
For each of the following right-angled triangles, identify the side labelled x with respect to the given
angle θ. Where necessary, also identify the side labelled y.
a b
Solution Solution
a The side labelled x is directly opposite angle θ.
So x is the opposite side.
b The side labelled x is next to given angle θ, and the
side labelled y is opposite the right angle.
So x is the adjacent side and y is the hypotenuse.
x
θ
12 cm
x
y
θ
WORKEDExample
x
10 cm
θ
x
10 cm
θ
x
10 cm
θ
x
10 cm
θ
8 cm
x
10 cm
θ
x
6
θ
x
y
θ
15 mm
xy
θ
6 cm
x
y
θ
8 mm
29.
SkillSHEET
answers
SkillSHEET 7.6
Finding trigonometric values and angles
The sine, cosine and tangent of any angle can be obtained from a scientiﬁc or graphics calculator by simply
pressing the button for the appropriate function.
Note that the calculator must be in the same mode as the given angle. That is, if the angle is in degrees, the
calculator must also be in the DEG (i.e. degree) mode.
If the value of sin or cos or tan of a certain angle is known, the size of that angle can be found by using an appro-
priate inverse function of your calculator.
Use a calculator to ﬁnd the value of each of the following, giving answers correct to 4 decimal places.
a sin 45° b cos 8° c tan 53°
Solution
a Ensure that your calculator is in the degree mode. To ﬁnd the sine of a given angle, press the following
sequence of buttons on your scientiﬁc calculator: (or press instead of ,
if using a graphics calculator).
Record the number shown on the display and round off to 4 decimal places.
sin 45° = 0.707 106 781
sin 45° ≈ 0.7071
Note that the sign ≈ is read as ‘approximately equal to’.)
b To ﬁnd the cosine of the given angle, press the following sequence of buttons on your calculator:
. Copy the number from the display and round off to 4 decimal places.
cos 8° = 0.990 268 068
cos 8° ≈ 0.9903
c To ﬁnd tangent of the given angle, press the following sequence of buttons on your calculator:
. Record the number shown on the display and round off to 4 decimal places.
tan 53° = 1.327 044 822
tan 53° ≈ 1.3270
SIN 4 5 = ENTER =
COS 8 =
TAN 5 3 =
1WORKEDExample
For each of the following, ﬁnd the value of θ, giving answers correct to 1 decimal place.
a sin θ° = 0.5505 b cos θ° = 0.5505 c tan θ° = 0.5505
Continued over page
Solution
a To ﬁnd the value of an angle when given the value of its sine, press [SIN–1
], enter 0.5505 and press
(or press in place of , if you are using a graphics calculator).
Record the number shown on the display and round it off to 1 decimal place. (Your answer is in degrees,
provided that the calculator was in the degree mode prior to your making the calculations.)
sin θ° = 0.5505
sin° θ = sin−1
0.5505
sin° θ = 33.401 321 82
sin° θ ≈ 33.4°
2nd
= ENTER =
2WORKEDExample
30.
SkillSHEET
answers
Try these
1 Find the value of each of the following, giving answers correct to 4 decimal places.
a sin 15° b sin 34° c sin 79° d sin 20° e sin 37° f sin 83°
g cos 15° h cos 34° i cos 79° j cos 20° k cos 37° l cos 83°
m tan 15° n tan 34° o tan 79° p tan 20° q tan 37° r tan 83°
2 For each of the following, ﬁnd the value of θ, giving answers correct to 1 decimal place.
a sin θ° = 0.5 b sin θ° = 0.3145 c sin θ° = 0.7974 d sin θ° = 0.866
e cos θ° = 0.5 f cos θ° = 0.3145 g cos θ° = 0.7974 h cos θ° = 0.866
i tan θ° = 0.5 j tan θ° = 0.3145 k tan θ° = 0.7974 l tan θ° = 0.866
m tan θ° = 0.95 n tan θ° = 3.145 o tan θ° = 1.79 p tan θ° = 2.567
Solution
b To ﬁnd the value of an angle when given the value of its cosine, press , enter 0.5505 and
press .
Record the number shown on the display and round it off to 1 decimal place.
cos θ° = 0.5505
cos° θ = cos−1
0.5505
cos° θ = 56.598 678 18
cos° θ ≈ 56.6°
c To ﬁnd the value of an angle when given the value of its tangent, press , enter 0.5505 and
press .
Record the number shown on the display and round it off to 1 decimal place.
tan θ° = 0.5505
tan° θ = tan−1
0.5505
tan° θ = 28.832 783 64
tan° θ ≈ 28.8°
2nd [COS-1
]
=
2nd [TAN-1
]
=
31.
SkillSHEET
answers
SkillSHEET 8.1
Area and perimeter of composite ﬁgures
Composite ﬁgures are ﬁgures made up of a number of distinct shapes.
Area of a composite ﬁgure = sum of the areas of each individual shape
Perimeter of a composite ﬁgure = length around the outside edge of the composite ﬁgure
Calculate a the area and b the perimeter of Calculate a the area and b the perimeter of the
the following shape. following shape. State your answers correct to 2
decimal places.
Solution Solution
The composite shape is made up of a rectangle
and a triangle.
a Area = area of rectangle + area of triangle
Area = length × width + × base × height
For the rectangle:
Length = 16 cm, width = 12 cm
For the triangle:
Base = 12 cm (same as width of rectangle),
Height = (24 − 16) = 8 cm
So area = 16 × 12 + × 12 × 8
So area = 192 + 48
So area = 240 cm2
b Perimeter = sum of the lengths of 5 sides
Perimeter = 16 + 10 + 10 + 16 + 12
Perimeter = 64 cm
The composite shape is made up of a semicircle and a
trapezium.
a Area = area of semicircle + area of trapezium
Area = πr2
+ (a + b)h
For the semi circle:
r = × 30 = 15 mm
For the trapezium:
a = 30 mm, b = 48 mm and h = 15 mm.
So area = × π × 152
+ × (30 + 48) × 15
So area = 353.43 + 585
So area = 938.43 mm2
b Perimeter = (circumference of circle) +
Perimeter = 3 side lengths of trapezium
To ﬁnd missing side length of trapezium:
Let c = hypotenuse of right-angled triangle with
side lengths of 15 and 18
c2
= 152
+ 182
(Using Pythagoras’ theorem)
= 225 + 324
= 549
c =
c = 23.43
So perimeter = × 2πr + 15 + 48 + 23.43
So perimeter = π × 15 + 86.43
So perimeter = 47.12 + 86.43
So perimeter = 133.55 mm
12 cm
16 cm
24 cm
10 cm
15 m
48 m
30 m
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
549
1
2
---
1WORKEDExample 2WORKEDExample
32.
SkillSHEET
answers
Try these
Calculate a the area and b the perimeter of each of the following shapes. State your answers correct to 2 decimal
places where appropriate.
1 2
3 4
5 6
10 m
18 m
30 m
13 m
20 cm
10 cm
6 cm
18 cm
49 cm
23 cm
35 cm
12 m
9 m
7 m
3 mm 3 mm
3 mm 3 mm
5 mm
15 mm
5 cm 7 cm
33.
SkillSHEET
answers
SkillSHEET 8.2
Volume
The volume of an object is the amount of space that the object occupies. Some examples of volume units are:
mm3
, cm3
and m3
.
Prisms: V = cross-sectional area × height of prism
Tapered objects: V = × area of base × height of object
Sphere: V = πr3
Try these
Find the volume of each of the following shapes. State your answers correct to 2 decimal places (where
appropriate).
1 2 3
4 5 6
1
3
---
4
3
---
Find the volume of the following shape. Find the volume of the following shape. Express
your answer correct to 2 decimal places.
Solution Solution
The shape is a rectangular prism.
V = cross-sectional area × height
V = area of rectangle × height
V = 12 × 5 × 7
V = 420 cm3
The shape is a cone (tapered object).
V = × area of base × height of object
V = × area of circle × height of object
V = × πr2
× H
V = × π × 32
× 10
V = 94.25 m3
7 cm
5 cm
12 cm
6 m
10 m
1
3
---
1
3
---
1
3
---
1
3
---
1WORKEDExample 2WORKEDExample
19 m
8 m
26 m
4 cm
9 cm
8 mm
6 mm
16 mm
2 cm
8 cm
13 cm
9 cm
5 cm
27.2 m
11.5 m
34.
SkillSHEET
answers
7 8 9
10 11 12
16 cm
21 cm
14 cm
17 m38 m
5.7 cm
12.4 cm
6 mm
15 mm
7 mm
4 mm
42 cm
65 cm
35.
SkillSHEET
answers
SkillSHEET 8.3
Conversion of units — volume
Common units for volume are: cubic millimetre (mm3
), cubic centimetre (cm3
) and cubic metre (m3
).
10 mm = 1 cm → 103
mm3
= 13
cm3
so 1000 mm3
= 1 cm3
100 cm = 1 m → 1003
cm3
= 13
m3
so 1 000 000 cm3
= 1 m3
÷ 103
÷ 1003
mm3
cm3
m3
× 103
×1003
Remember: converting to a smaller unit → multiply
converting to a larger unit → divide
Try these
Convert each of the following measurements to the unit shown in brackets.
1 268 mm3
(cm3
) 2 0.035 m3
(cm3
)
3 0.000 610 cm3
(mm3
) 4 37.569 cm3
(m3
)
5 5.03 cm3
(mm3
) 6 68 005 624 cm3
(m3
)
7 3.1 m3
(mm3
) 8 15 000 000 000 mm3
(m3
)
9 86 mm3
(m3
) 10 0.0704 m3
(mm3
)
Convert 2300 mm3
to cm3
. Convert 3.057 m3
to cm3
.
Solution Solution
Converting to a larger unit so we need to divide.
To convert mm3
to cm3
, divide by 1000.
2300 mm3
÷ 1000 = 2.3 cm3
Converting to a smaller unit so we need to multiply.
To convert m3
to cm3
, multiply by 1 000 000.
3.057 m3
× 1 000 000 = 3 057 000 cm3
1WORKEDExample 2WORKEDExample
36.
SkillSHEET
answers
SkillSHEET 8.4
Labelling right-angled triangles
In trigonometry, the sides of right-angled triangles are named according to their position relative to a speciﬁc
angle (not the right angle). The symbol θ (theta) is one of many Greek letters of the alphabet used to represent this
speciﬁc angle.
If the speciﬁc angle is θ, then
the side of the triangle opposite θ is called the opposite side
the longest side of the triangle, opposite the right angle, is called the hypotenuse
the third side, next to the angle θ, is called the adjacent side
Try these
1 Label the sides of each of the following triangles using the words Opposite, Adjacent, and Hypotenuse.
a b
c d
e f
g h
Hypotenuse
Opposite side with
respect to the angle θ
θ
Adjacent side
with respect
to the angle θ
45°
x
6 cm
22 cm
15 cm
θ
18 cm
x
47°
10 cm
12 cm
θ
73°
x
13 cm 27°
10 cm
x
8.3 cm
5.9 cm
θ
24 mm
37 mm
θ
37.
SkillSHEET
answers
i j
2 For each triangle in question 1, list the two sides which provide information. That is, which two of Opposite,
Adjacent and Hypotenuse have either a known value or a pronumeral.
x
7 cm
15°
20°
x
4 cm
38.
SkillSHEET
answers
SkillSHEET 8.5
Using trigonometric ratios
Trigonometric ratios deal with right-angled triangles. The sides of the triangle are named according to their pos-
ition with respect to a speciﬁc angle. The hypotenuse is the longest side length of the right-angled triangle.
Trigonometric ratios: SOH CAH TOA means
sin θ = , cos θ = and tan θ =
Hypotenuse
Opposite side with
respect to the angle θ
θ
Adjacent side
with respect
to the angle θ
opposite
hypotenuse
----------------------------
adjacent
hypotenuse
----------------------------
opposite
adjacent
--------------------
Find the value of x. Express your answer Find the value
correct to 3 decimal places. of x.
Express your answer
correct to 2 decimal
places.
Solution Solution
Given information is:
Angle is 47°, hypotenuse length is 18 cm.
The side marked x is the opposite side.
Use the sine ratio:
sin θ =
sin 47° =
= sin 47°
x = 18 × sin 47°
x = 13.164 cm
Given information is:
Angle = 30°, hence the adjacent side = 45 mm.
The side marked x is the hypotenuse.
Use the cosine ratio:
cos θ =
cos 30° =
x cos 30° = 45
x =
x = 51.96 mm
30°
x
45 mm18 cm
x
47°
opposite
hypotenuse
---------------------------
x
18
------
x
18
------
adjacent
hypotenuse
---------------------------
45
x
------
45
cos 30°
-----------------
1WORKEDExample 2WORKEDExample
39.
SkillSHEET
answers
Try these
1 For the following right-angled triangles state which trigonometric ratio would be appropriate in ﬁnding the side
length marked x.
a b c
d e
2 Find the side length marked x for each of the triangles in question 1. Express your answers correct to 3 decimal
places.
Find the value of the angle θ, to the nearest degree.
Solution
Given information is:
Opposite side is 24 mm and adjacent side is 37 mm.
Use tangent ratio:
tan θ =
tan θ =
tan θ = 33°
24 mm
37 mm
θ
opposite
adjacent
--------------------
24
37
------
3WORKEDExample
45°
6 cm
x
15°
7 cmx
20°
4 cm
x
73°
13 m
x
27°
10 cm
x
40.
SkillSHEET
answers
3 Find the value of the angle in each of the following triangles. Express your answer to the nearest degree.
a b
c d
e
22 cm
15 cm
θ
31 mm
39 mm
θ
10 cm
12 cm
θ
8.3 cm
5.9 cm
θ
0.52 m
1.34 m
θ
41.
SkillSHEET
answers
SkillSHEET 9.1
Finding the size of a sector in degrees, given
its size as a fraction of a circle
There are 360° in a full circle. So to ﬁnd the size of a sector in degrees, given its size as a fraction of a full circle,
multiply the fraction by 360°.
To multiply a fraction by 360°:
1. change 360 into a fraction by writing it over 1
2. simplify as much as possible
3. multiply the numerators together and the denominators together
4. if the answer is an improper fraction, convert it to a mixed number.
Try these
Find the size of each of the following sectors in degrees, given that their size as a fraction of a circle is:
1 2 3 4
5 6 7 8
9 10 11 12
Finding the size of a sector in degrees, if the sector is of a circle.
THINK WRITE
To express a fraction of a circle in degrees, multiply
by 360.
× 360
Convert 360 into a fraction by writing it over 1. = ×
Cross-cancel 5 and 360 by dividing each by 5 (that is,
360 ÷ 5 = 72; 5 ÷ 5 = 1).
= ×
Multiply the numerators together and the
denominators together.
=
Convert the improper fraction into a mixed number
(which in this case is actually a whole number) and
include the degree sign.
= 216°
3
5
---
1
3
5
---
2
3
5
---
360
1
---------
3
3
1
---
72
1
------
4
216
1
---------
5
WORKEDExample
2
5
---
1
3
---
3
10
------
3
4
---
5
12
------
2
9
---
7
8
---
4
15
------
4
5
---
1
6
---
23
36
------
29
60
------
42.
SkillSHEET
answers
SkillSHEET 10.1
Finding a mean
To ﬁnd the mean of two or more numbers, add the numbers together and then divide the sum by the number of
numbers.
Try these
Find the mean of each of the following sets of numbers.
1 3, 5 2 2, 9 3 2, 4, 8 4 5, 6, 9 5 4, 7, 8, 12
6 7, x 7 x, 19 8 x, 5, 12 9 2, 4, x 10 17, x, 25
Find the mean of each of the following sets of numbers.
a 2, 3, 7 b 5, x
THINK WRITE
a To ﬁnd the mean of three numbers, add
them together and then divide by 3.
a Average =
Evaluate the numerator of the fraction ﬁrst
and then divide.
Average =
Average = 4
b To ﬁnd the mean of two numbers, add them
together and divide the total by 2. (Since the
value of x is not given, the expression can not
be evaluated and should be left as is.)
b Average =
1
2 3 7+ +
3
---------------------
2
12
3
------
5 x+
2
------------
WORKEDExample
43.
SkillSHEET
answers
SkillSHEET 10.2
Finding the median
The median of a set of scores is the middle score when the data are arranged in order of size.
The position of the median is found using the formula:
Median position = score
where n is the number of scores.
For example, if there are 13 scores, the median position is the score; that is, the 7th score. If there are
16 scores, the median position is the score; that is, the 8 th score. The 8 th score is halfway between
and is equal to the average of the 8th and 9th scores.
Try these
Find the median of each of the following sets of data.
1 7, 5, 8, 9, 2, 4, 5, 1, 8, 2, 5 2 21, 19, 28, 25, 17, 24, 22, 25, 19
3 32, 45, 58, 21, 57, 84, 26 4 52, 59, 48, 53, 49, 56, 58, 42, 60
5 12, 5, 7, 9, 5, 14, 7, 2, 9, 5, 4, 1 6 27, 40, 33, 37, 46, 32, 19, 21
7 9, 7, 8, 12, 6, 13, 14, 11, 5, 10 8 20, 16, 26, 21, 15, 20
n 1+( )th
2
----------------------
13 1+( )th
2
-------------------------
16 1+( )th
2
------------------------- 1
2
---
1
2
---
Find the median of the following set of data:
8, 3, 7, 4, 9, 1, 5, 8, 6, 13, 12, 2, 6.
Find the median of the following set of data:
28, 33, 27, 29, 29, 29, 32, 28, 34, 31, 32, 33.
Solution Solution
There are 13 scores.
∴ n = 13
Median score =
=
= the 7th score.
Arrange the data in order of size and locate the
seventh score.
1, 2, 3, 4, 5, 6, , 7, 8, 8, 9, 12, 13.
So, median = 6.
There are 12 scores.
∴ n = 12
Median score =
=
= the 6 th score.
Rewrite the data in order of size. The median is
located halfway between the 6th and 7th scores.
27, 28, 28, 29, 29, , 32, 32, 33, 33, 34.
To ﬁnd the median, calculate the average of the 6th
and 7th scores.
Median =
=
= 30
n 1+( )th
2
----------------------
13 1+( )th
2
-------------------------
6
n 1+( )th
2
----------------------
12 1+( )th
2
-------------------------
1
2
---
29, 31
29 31+
2
------------------
60
2
------
1WORKEDExample 2WORKEDExample
44.
SkillSHEET
answers
SkillSHEET 11.1
Measuring the rise and the run
To measure the rise and the run for a straight line, follow these steps:
1. Select two points on the line. If the line goes through the origin, it is best to select the origin and any other
point. If the line cuts both axes, select the x-intercept and the y-intercept.
2. Construct the gradient triangle, so that the two points are the vertices.
3. Measure the horizontal distance (that is, the distance along the horizontal side of a triangle) between the two
points. This distance represents the run. Note that the run is always positive.
4. Measure the vertical distance (that is, the distance along the vertical side of a triangle) between the two points.
This distance represents the rise. Note that if the line slopes upward from left to right, the rise is positive, while
if the line slopes downward, the rise is negative.
State the rise and the run for each of the following straight lines.
a b
THINK WRITE/DRAW
a Since the line goes through the origin,
select the origin and some other point.
Draw the gradient triangle so that the
selected points are the vertices.
a
Measure the distance along the horizontal
side of the triangle (that is, how much it is
from 0 to 2). Hence state the value of the
run.
Run = 2
Measure the distance along the vertical
side of the triangle (that is, how much it is
from 0 to 3) to ﬁnd the value of the rise.
Since the line slopes upward from left to
right, the rise is positive.
Rise = 3
y
x
4
3
2
1
–1
–2
–3
–4
–1 1 2 3 4–2–3–4
y
x
4
2
1
3
–2
–3
–1
–4
2 31 4 5 6–2–3 –1
1 y
4
3
2
1
–1
–2
–3
–4
–1 1 2 3 4–2–3–4
2
3
2
3
WORKEDExample
45.
SkillSHEET
answers
Try these
State the rise and the run for each of the following straight lines.
1 2
3 4
5 6
THINK WRITE/DRAW
b Since the line cuts both axes, select the
x- and y-intercepts. Draw the gradient
triangle so that the selected points are the
vertices.
b
Measure the distance along the horizontal
side of the triangle (it is from 0 to 6) and
hence state the value of the run.
Run = 6
Measure the distance along the vertical
side of the triangle (that is, from 0 to 2) to
ﬁnd the value of the rise. Since the line
slopes downward from left to right, the rise
is negative.
Rise = −2
1 y
x
4
2
–2
6
1
3
–2
–3
–1
–4
2 31 4 5 6–2–3 –1
2
3
y
x
4
2
1
3
–2
–3
–1
2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1 2 31 4 5 6–2–3 –1
y
x
4
5
2
1
3
–2
–3
–1
2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1 2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1
2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1 2 31 4 5 6–2–3 –1
46.
SkillSHEET
answers
7 8
9 10
y
x
4
2
1
3
–2
–3
–1
2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1
2 31 4 5 6–2–3 –1
y
x
4
2
1
3
–2
–3
–1 2 31–2–3 –1
y
x
2
1
3
–2
–4
–3
–1 21–2–4 –3 –1
47.
SkillSHEET
answers
SkillSHEET 11.2
Substitution into a rule
To substitute a given value of the pronumeral into an algebraic sentence or rule means to replace the pronumeral
with that value.
When all pronumerals have been replaced with numbers, the expression can be evaluated. Order of operations
must be observed at all times when evaluating.
Try these
1 Substitute 5 for x in each of the following rules and then ﬁnd the value of y.
a y = x + 9 b y = x − 3 c y = 12 + x d y = 25 − x e y = 4x
f y = 7x g y = 3x − 4 h y = 2x + 6 i y = 25 − 3x j y = 11 + 6x
2 Substitute 3 for x in each of the rules in question 1 and hence ﬁnd the value of y.
Substitute 5 for x in each of the following rules and then ﬁnd the value of y.
a y = x + 7 b y = 2x − 3
THINK WRITE
a Replace x with the given value (i.e. 5). The
rest of the expression remains unchanged.
a y = 5 + 7
Add 5 and 7 to ﬁnd the value of y. y = 12
b Substitute 5 for x, remembering that in
algebra 2x means 2 × x.
b y = 2 × 5 − 3
To ﬁnd the value of y, perform
multiplication ﬁrst, followed by
subtraction.
y = 10 − 3
y = 7
1
2
1
2
WORKEDExample
48.
SkillSHEET
answers
SkillSHEET 11.3
Average rate of change
If a rate is variable, it is sometimes useful to know the average rate of change over a speciﬁc interval from x1 to x2.
Average rate of change = = =
Try these
1 a For the function f(t) = 5t − 4t2
− 6, ﬁnd f(5) and f(7).
b For the function h(t) = 8t − t2
− 3, ﬁnd h(1) and h(4).
c For the function y = 7x − 4x2
− 1, ﬁnd the values of y when x = 2 and x = 4.
2 a Consider the function f(t) = 5t − 4t2
− 6 which deﬁnes the position of a particle at time t (seconds) from a
ﬁxed point. Find the average rate of change (velocity) between t = 5 s to t = 7 s.
b A discus is tossed so that its height, h metres above the ground, is given by the rule h(t) = 8t − t2
− 3, where
t represents time in seconds. Find the average rate of change between t = 1 s to t = 4 s.
c If y = 7x − 4x2
− 1, ﬁnd the average rate of change of y as x changes from 2 to 4.
3 The position of a ball, x cm from the start at time t s is given by x(t) = t + 6t3
. Find the average rate of change
of position with respect to time (velocity) between t = 0 to t = 3.
4 The volume, V m3
, of dripping tap water into a bucket is increasing according to the rule V(t) = ,
time t (minutes) from the beginning. Find the average rate of change of volume of water between t = 2 to t = 5
minutes.
5 If y = x 3
− 4x2
− 1, ﬁnd the average rate of change of y as x changes from −3 to −1.
rise
run
--------
change in f x( )
change in x
------------------------------------
f x2( ) f x1( )–
x2 x1–
---------------------------------
For f(x) = x2
- 4x + 1, ﬁnd the average rate of A cricket ball was hit so that its height (h) metres
change between x = 1 and x = 2. above the ground followed the rule: h(t) = 5t - t2
,
where t represents the time in seconds. Find the
average rate of change over the ﬁrst 3 seconds.
Solution Solution
Average rate of change
= where x1 = 1 and x2 = 2
=
Now f(2) = (2)2
− 4(2) + 1
= 4 − 8 + 1
= −3
and f(1) = (1)2
− 4(1) + 1
= 1 − 4 + 1
= −2
So average rate of change =
So average rate of change =
So average rate of change = −1
Average rate of change
=
=
=
= 2
Average rate of change is 2 m/s.
f x2( ) f x1( )–
x2 x1–
---------------------------------
f 2( ) f 1( )–
2 1–
-----------------------------
−3 −2–
2 1–
-------------------
1–
1
------
f 3( ) f 0( )–
3 0–
-----------------------------
5 3 3
2
–×( ) 5 0 0
2
–×( )–
3 0–
---------------------------------------------------------------
6 0–
3
------------
1WORKEDExample 2WORKEDExample
t 1–( )
3
1+
2
---------------------------
49.
SkillSHEET
answers
SkillSHEET 11.4
Using the regression equation to make
predictions
The regression equation can be used to predict a value of the dependent variable (y) given a value of the indepen-
dent variable (x) and vice versa.
Try these
1 For the regression equation y = 2x + 11 ﬁnd:
a y if x = 5 b y if x = 23 c x if y = 93 d x if y = 214
2 For the regression equation y = 11x − 2 ﬁnd:
a y if x = 4 b y if x = 60 c x if y = 75 d x if y = 125
3 For the regression equation y = 1.2x + 35.5 ﬁnd:
a y if x = 20 b y if x = 56 c x if y = 120 d x if y = 65
4 For the regression equation y = 1.05x − 56 ﬁnd:
a y if x = 65 b y if x = 80 c x if y = 45 d x if y = 38
5 For the regression equation y = 2.35x + 45 ﬁnd:
a y if x = 30 b y if x = 65 c x if y = 80 d x if y = 480.
For the regression equation y = 3x + 9 ﬁnd:
a y if x = 7 b x if y = 72.
Solution
a Substitute 7 for x into the given equation and evaluate.
y = 3x + 9, x = 7
y = 3 × 7 + 9
y = 21 + 9 (Perform multiplication ﬁrst, followed by addition.)
y = 30
b Substitute 72 for y into the equation and solve for x.
y = 3x + 9, y = 72
72 = 3x + 9
72 − 9 = 3x (Subtract 9 from both sides.)
63 = 3x
63 ÷ 3 = x (Divide both sides by 3.)
x = 21
WORKEDExample
50.
SkillSHEET
answers
SkillSHEET 12.1
Understanding a deck of playing cards
A standard deck of playing cards consists of 52 cards.
All cards are divided into 4 suits. There are two black
suits — spades (m) and clubs (p) and two red suits —
hearts (n) and diamonds (o). In each suit there are
13 cards including a 2, 3, 4, 5, 6, 7, 8, 9, 10, a jack,
a queen, a king and an ace. (Note that there is no 1.)
A jack, a queen and a king are called picture cards.
Try these
For a standard deck of playing cards, state the number of:
1 black cards 2 aces
3 picture cards 4 queens of hearts
5 kings 6 clubs
7 not spades 8 red cards
9 tens 10 red jacks
11 black threes 12 red nines
13 number cards greater than 6 14 red picture cards.
For a standard deck of playing cards, state the number of:
a diamonds
b black queens.
THINK WRITE
a Diamonds is one of the four suits and there are
13 cards in any suit.
a There are 13 diamonds.
b There is one queen in each suit and there are
two black suits (clubs and spades).
b There are 2 black queens.
WORKEDExample
51.
SkillSHEET
answers
SkillSHEET 12.2
Converting fractions to percentages
To convert a fraction into a percentage, multiply the fraction by 100%.
Try these
Convert each of the following fractions into percentages.
1 2 3 4 5
6 7 8 9 10
Convert each of the following fractions into percentages.
a b
THINK WRITE
a To change the fraction to a percentage,
multiply by 100%.
a × 100%
Write 100 as a fraction by putting it over 1. = ×
Multiply numerators together and
denominators together.
=
Simplify by dividing 300 by 5. = 60%
b To change the fraction to a percentage,
multiply by 100%.
b × 100%
Write 100 as a fraction by putting it over 1. = ×
Multiply numerators together and
denominators together.
=
Convert the improper fraction into a mixed
number.
= 66 %
3
5
---
2
3
---
1
3
5
---
2
3
5
---
100
1
---------
3
300
5
---------
4
1
2
3
---
2
2
3
---
100
1
---------
3
200
3
---------
4
2
3
---
WORKEDExample
3
4
---
4
5
---
3
8
---
2
5
---
7
8
---
1
3
---
5
6
---
3
7
---
5
9
---
7
12
------
52.
SkillSHEET
answers
SkillSHEET 12.3
Converting a fraction to a decimal number
To convert any fraction into a decimal number, divide its numerator by the denominator. This can be easily done
using a calculator.
Try these
Convert each of the following fractions into decimals.
1 2 3 4 5
6 7 8 9 10
Convert into a decimal.
THINK WRITE
To convert the given fraction into a decimal, divide
15 by 80. This can be done with the help of a
scientiﬁc calculator by pressing the following
sequence of buttons and recording the result:
With a graphics calculator, press the keys for
then ﬁnish by pressing .
= 0.1875
15
80
------
15 ÷ 80 =
15 ÷ 80 ENTER
15
80
------
WORKEDExample
15
40
------
5
16
------
7
8
---
25
80
------
41
100
---------
45
300
---------
91
128
---------
15
64
------
138
200
---------
159
160
---------
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