Calculate the following. a 3b1 THINK WRITE a Write the expression.a3

VCE

VCE

co

covverage

erage

Area of Study

Units 1 and 2 • Arithmetic

In this

In this

cha

chapter

pter

1A Mental computations

1B Calculator computations

1C Rounding off

1D Estimation and

approximation

1E Ratios

1F Proportion

1G Percentages

1H Percentage change

1

Mental computations

Of all the branches of mathematics, arithmetic, and in particular mental computation, is the one most used in our everyday lives. This chapter revises some of the most important arithmetic skills.

Order of operations

It is often necessary to evaluate an expression containing a number of arithmetic oper-ations. In such cases a specific order should be observed:

1. brackets must be evaluated first, followed by

2. multiplication and division (in order from left to right), and then 3. addition and subtraction (again in order from left to right).

When evaluating the contents of brackets the above rules apply (that is, multi-plication and division should be done before addition and subtraction). The mnemonic BODMAS can be used to help you remember the order of operations:

B stands for brackets, O stands for of, D means division, M means multiplication and A and S stand for addition and subtraction respectively.

Computations with fractions

Addition and subtraction

Remember that only fractions with the same denominator can be added and subtracted. If fractions have different denominators, they should be changed to a common denomi-nator first. It is preferable to change all fractions in a question to the lowest common denominator (LCD) possible. If a question contains mixed numbers, they should be changed to improper fractions prior to addition or subtraction.

Evaluate each of the following.

a 3 + 8 × 7 − 4 b (7 + 2 × 3) ÷ (6 × 2 − 11)

THINK WRITE

a There are 3 operations: addition,

multiplication and subtraction. According to the rules, multiplication should be done first, followed by addition and subtraction.

a 3 + 8 × 7 − 4 = 3 + 56 − 4 = 59 − 4 = 55 b The contents of brackets should be

evaluated first; then the value of the first bracket must be divided by the value of the second bracket.

b (7 + 2 × 3) ÷ (6 × 2 − 11)

(a) First bracket: Do multiplication, followed by addition.

(b) Second bracket: Do multiplication, then subtraction.

(c) Divide.

= (7 + 6) ÷ (6 × 2 − 11) = 13 ÷ (6 × 2 − 11) = 13 ÷ (12 − 11) = 13 ÷ 1

= 13 1

2

1

The final answer should be given in the simplest form; if it is an improper fraction, it should be changed to a mixed number.

Multiplication and division

When multiplying or dividing, mixed numbers should be changed into improper fractions first.

To multiply two (or more) fractions, multiply the numerators together and the denominators together.

To divide by a fraction, turn it upside down (invert it) and multiply.

Always remember to check if anything can be cancelled before proceeding with multiplication.

Evaluate 2 + − 1 .

THINK WRITE

Write the expression. 2 + − 1

Change mixed numbers into improper fractions. = + −

LCD = 60. Change to LCD by multiplying the numerator and denominator of the first fraction by 15, the second by 12 and the third by 20.

=

Simplify the numerator. =

Change into a mixed number. = 1

3 4 --- 1 5 --- 2 3

---1 3₄--- 1

5 --- 2

3

---2 11---₄ 1

5 --- 5

3

---3 165+12–100

60

---4 77₆₀

---5 17

60

---2

WORKED

E

xample

Calculate the following.

a 3 × b ÷ 1

THINK WRITE

a Write the expression. a 3 ×

Change the mixed number into an improper fraction. = × Divide 4 in the numerator and 2 in the denominator by 2

(that is, cancel).

= ×

Multiply numerators together and denominators together. =

Change into a mixed number. = 2

b Write the expression. b ÷ 1

Change the mixed number into an improper fraction. = ÷ Turn the second fraction upside down and exchange the

division sign for a multiplication sign.

= × The 7 in the numerator and denominator can be cancelled, while

4 in the numerator and 8 in the denominator can be divided by 4. = × Multiply the numerators together and the denominators together. ₌

1 2 --- 4 5 --- 7 8 --- 3 4

---1 1₂--- 4

5

---2 7₂--- 4

5

---3 7₁--- 2

5

---4 14---₅

5 4₅

---1 7₈--- 3

4 ---2 7 8 --- 7 4 ---3 7 8 --- 4 7 ---4 1 2 --- 1 1 ---5 1 2

---3

Computations with directed numbers

Multiplication and division

The product (or quotient) of two numbers with the same sign is positive while the product or quotient of two numbers with opposite signs is negative.

That is:

Addition and subtraction

To add numbers with the same sign, add the absolute values (the magnitude) together. The answer will have the same sign as the numbers which were added.

To add numbers with opposite signs, find the difference between the absolute values. The answer will have the same sign as the number with the larger magnitude.

To subtract a directed number is equivalent to adding a number with the same mag-nitude but the opposite sign (for example, subtracting −3 is equivalent to adding +3).

First number Second number Product/quotient

+ + +

− − +

+ − −

− + −

Evaluate

a −18 ÷ −6 × 5 b (−3)3

THINK WRITE

a Write the expression.

The two operations should be done in order from left to right. Therefore, divide −18 by −6. Since both of them are negative, the answer is positive.

a −18 ÷−6 × 5 = 3 × 5

Multiply 3 by 5. Since both numbers are positive, the answer is positive.

= 15

b Write the expression. _{b (−3)}3

Write in expanded notation = (−3) × (−3) × (−3) Multiply −3 by −3. The result is

positive, since they have the same sign.

= 9 × (−3) Multiply 9 by −3. The result is negative,

because their signs are opposite.

=−27 1

2

3

1 2 3

4

4

Evaluate

a 5 − 11 b −3 + (−6) c 6 − (−2) d −5 + 12

THINK WRITE

a Write the expression.

To subtract 11 is the same as adding −11.

a 5 − 11 = 5 + (−11) As signs are opposite, find the difference

between 5 and 11. The answer is negative, as −11 has larger magnitude.

= −6

b Write the expression

As both numbers have the same sign, add them together. The answer is negative, as both numbers in question are negative.

b −3 + (−6) = −9

c Write the expression.

To subtract (−2) is the same as adding 2.

c 6 − (−2) = 6 + 2

Add 6 and 2. The answer is positive. = 8

d Write the expression.

Since the numbers have opposite signs, find the difference in magnitude. The answer is positive, because +12 has larger magnitude.

d −5 + 12 = 7 1

2 3

1 2

1 2 3 1 2

5

WORKED

E

xample

remember

(a) brackets

(b) ÷ and × (from left to right) (c) + and − (from left to right).

Inside the brackets the same order should be observed. 2. Operations with fractions

(a) Change mixed numbers to improper fractions prior to any calculations. (b) Change fractions to the lowest common denominator (LCD) before adding/

subtracting.

(c) To divide by a fraction, turn it upside down and multiply. (d) Check whether cancelling is possible before multiplying.

(e) Give the final answer in simplest form; change improper fractions into mixed numbers.

3. Directed numbers

First number Second number Product/quotient

+ + +

− − +

+ − −

− + −

Mental computations

As the heading suggests, all calculations in this exercise should be done mentally.

1 Evaluate each of the following.

2 Evaluate each of the following:

3 Calculate the following multiplications and divisions.

4 Evaluate the following expressions.

5 Evaluate:

6 Calculate the following.

7 Evaluate each of the following. (Express your answer as a fraction in its simplest form.)

a 16 − 3 × 2 b 11 + 4 × 9 c 2 × 8 − 3 × 5

d 5 + 12 ÷ 4 − 6 e 17 − 4 × 3 + 8 f 12 ÷ (6 − 2)

g 8 × (9 + 3) h (19 − 3) ÷ (5 × 3 − 11) i (12 ÷ 4 − 3) × (5 + 9 × 8)

j (20 − 3 × 6) − (5 − 2 × 2)

a + b − c + + d 1 −

e 4 + f 2 − g 3 + 1 h 5 − 4

i 2 +1 − 2 j 3 − 2 + 1

a × b 2 × c 1 × d ÷

e ÷ 2 f 3 ÷ g × × h × ÷

i 1 × 2 ÷ j 2 ÷ 3 × 1

a 2 × −3 b −4 × −6 c 12 ÷ −2 d −20 ÷ −5 × 4 ÷ −8

e 7 × −4 × −2 f 4 ÷ −2 × 5 g −6 × −5 ÷ −3 h (−2)3

i 6 × (−2)3 j −8 × (−3)2÷ 6

a 5 − 8 b −7 + 12 c −8 + 6 d 6 − (−11)

e −12 − 4 f 10 + (−6) g −3 − (−9) h −8 + (−2)

i −19 − (−4) j 7 + (−13)

a 76 − (−3) × 5 b 4 − 32 ÷ 8 − 7 × (−5)

c −6 + 20 ÷ (1 − 2 × 3) d 11 × (−3) + (−16) ÷ 2 − (−8)

e f

g −(2 + 3 × 6 − 72) h 15 −

a × + ÷ b 3 × −

c 1 ÷ (5 + × ) d 9 − (2 − ÷ )

e ( + ) ÷ (2 − ) f × 3 ÷ − 1

g 1 + 5 ÷ 7 − 3 h 8 × (3 − 2 ÷ ) + 6

1A

SkillS HEET

1.1

3 4 --- 2 5 --- 7 8 --- 1 6 --- 3 5 --- 1 3 --- 2 9 --- 3 7 --- 4 8 ---1 2 --- 5 6 --- 3 8 --- 7 10 --- 2 9 --- 5 6 --- 3 10 --- 3 4 ---4 7 --- 9 10 --- 4 5 --- 5 12 --- 7 8 --- 1 4 ---WORKED Example

3 3₅--- 4 7 --- 5 8 --- 1 2 --- 5 6 --- 7 8 --- 3 4 ---5 9 --- 3 4 --- 9 10 --- 3 8 --- 5 9 --- 4 5 --- 7 8 --- 2 9 --- 1 3 ---3 4 --- 1 6 --- 13 15 --- 1 3 --- 3 5 --- 7 15 ---WORKED Example 4 WORKED Example 5

5+2×( )–8 –12 23 –

--- 17+5×2 3 –

---–8 4–3×8

5 –

---+7(–8–12)

---Calculator computations

Order of operations

A graphics calculator can be used for evaluating lengthy expressions with a number of operations. Calculators with algebraic logic will ‘know’ the order of operations only if you enter the expression in the order in which it is written (that is, from left to right).

If calculations involve fractions, follow these steps on a graphics calculator: 1. type the numerator followed by the key (This evaluates the numerator.) 2. press

3. type a left bracket followed by the denominator and a right bracket 4. press to evaluate the full expression.

Use a calculator to evaluate each of the following.

a 3 − 5 × 6 b (4 + 8) ÷ 3 c (9 − 6) ÷ (7 + 5)

THINK WRITE/CALCULATE

a Write the expression. a 3 − 5 × 6

Enter the expression into your calculator in sequence: press and write down the answer shown on the display.

= −27

b Write the expression. b (4 + 8) ÷ 3

Enter the expression: press

and write down the answer.

= 4

c Write the expression. c (9 − 6) ÷ (7 + 5)

Enter the expression:

press and write down the answer.

= 0.25

1 2

3 − 5 × 6 ENTER

1

2 ( 4 + 8 ) ÷ 3

ENTER

1

2 ( 9 − 6 ) ÷ ( 7

+ 5 ) ENTER

6

WORKED

E

xample

ENTER

÷

ENTER

Use your calculator to evaluate the following fraction.

THINK WRITE/CALCULATE

Write down the fraction.

Evaluate the numerator as follows: .

Divide the numerator (which is on your screen) by the

denominator by pressing

.

Copy the answer from the calculator display into your book. = 2.5

20–3×5 15–3

( )÷(5+1)

---1 20–3×5

15–3

( )÷(5+1)

---2

2 0 − 3 × 5 ENTER

3

÷ ( ( 1 5 − 3 ) ÷ ( 5 + 1 ) ) ENTER

4

7

Calculations with negative numbers

To perform calculations with negative numbers on a graphics calculator, the button is used.

Note that to indicate that the number is negative on a scientific calculator, the button is pressed after that number (although in written expressions we put the negative sign in frontof the number).

Calculations involving scientific notation

When the result of the calculations is very large or very small, it will appear on the cal-culator display in scientific notation. In scientific notation, a number is written as a product of a number between 1 and 10 and a power of 10.

On a graphics calculator display, the power of 10 will appear after the letter E (which stands for exponent). For example, on a display means 2.305 001 9 × 1015.

On a scientific calculator, this number may be displayed as .

Note: Scientific notation is sometimes referred to as standard notation. This must not be confused with normal notation, which is the common way of expressing decimal numbers.

(−) −

Evaluate each of the following using a calculator.

a 26 × (−3.5) b − 12 × (−4) + (−6)

THINK WRITE/CALCULATE

a Write the expression. a 26 × (−3.5)

Enter the expression into your graphics calculator using the following sequence:

and write down the answer.

= −91

b Write the expression. b −12 × (−4) + (−6)

Evaluate as follows:

and copy the answer on the display into your book.

= 42 1

2

2 6 × (−) 3 . 5 ENTER

1 2

(−) 1 2 × (−) 4 + (−) 6 ENTER

8

WORKED

E

xample

2.3050019E15

2.305001915

Write the answer to the following calculations:

i in a scientific form ii as a decimal. a 2359 × 7 600 000 b 0.0091 ÷ 32 600

THINK WRITE/CALCULATE

a Write down the expression. a 2359 × 7 600 000

i Press .

The display shows 1.79284E10 which means 1.792 84 × 1010. Write down the answer in scientific form.

i 1.792 84 × 1010

ii To multiply 1.792 84 by 1010, move the decimal point 10 places to the right. (Since there are only 5 digits after the decimal point, add 5 zeroes to have enough places.)

ii 17 928 400 000

b Write down the expression. b 0.0091 ÷ 32 600

i Press . The display shows

2.791411043E−07. Copy the answer interpreting E−07 as 10−7.

i 2.791 411 043 × 10−7 ii To multiply by 10−7, move the decimal point 7 places

to the left. (Since there is only one digit to the left of the decimal point, put 6 zeroes in front.)

ii 0.000 000 279 141 104 3

2359 × 7600000 ENTER

0.0091 ÷ 32600 ENTER

9

Note that the numbers we are dealing with here are quite long. In the next section, rounding-off will be discussed.

To enter numbers in scientific form, 2nd [EE] is used. For example, to enter 2.5 × 106 press [EE] , and to enter 2.5 × 10−6 press [EE]

.

The following screen view shows the previous worked example on a TI-83 screen.

The second result may be obtained as follows: 1. Press and select Sci.

2. Press [QUIT] to return to the home screen.

3. Press [CALC] to recall the previous calculation, then press .

2.5 2nd 6 2.5 2nd ( – ) 6

Evaluate the following expression using your calculator. 5.78 × 103÷ 2.5 × 10−4 Give your answer in scientific form.

THINK WRITE

Write the expression. _{5.78 × 10}3_{÷ 2.5 × 10}−4

Enter the expression as follows:

[EE] [EE]

and copy down the answer from the display.

= 23120 000 Rewrite the answer using scientific notation. _{= 2.312 × 10}7

1 2

5.78 2nd 3 ÷ 2.5 2nd ( – ) 4

3

10

WORKED

E

xample

MODE 2nd

2nd ENTER

remember

1. A calculator with algebraic logic will use the order of operations. Enter the expression in the same way as it is written and press .

2. In expressions with fractions: (a) evaluate the numerator first

(b) press then type a left bracket, followed by the denominator and a right

bracket and press

3. To indicate that the number is negative, press the button before the number.

4. To enter powers of 10, use 2nd [EE].

ENTER

÷

ENTER

(−)

Calculator computations

1 Use your calculator to evaluate each of the following.

2 Use your calculator to evaluate the following fractions.

3 Evaluate each of the following using a calculator.

4 Evaluate each of the following.

5 Write the answer to each of the following calculations: i in scientific form ii as a decimal:

6

The expression 4.9 × 108 has the same value as:

a 3 + 72 × 7 b 106 − 16 × 3.5

c 12 + 14.1 × 2.3 − 6.8 d 26 × 1.1 − 5 × 4.9

e 7 × (3.9 − 4.1 × 0.6) − 2.8 f (2.7 − 3.6 ÷ 4) ÷ (16 − 4 × 2.5)

g (18.5 + 3.96) × (17.2 − 4.7) h 26.7 − (2.3 + 11.01) × 2

i 7.56 ÷ 0.3 + 12.4 × 5 − 4.8 j (5.2 + 7.1 × 4) ÷ (6.9 ÷ 6 − 0.55)

a b

c d

e f

g h

i j

a −3 × 5.7 b 8.2 × (−4.1) c −6.9 × (−1.5)

d (−17) × 2.88 e 56 ÷ (−7) ÷ (−2) f −4.6 ÷ (−0.02)

g 5.3 × (−7.1) × (−1.1) h −6.3 ÷ (−2.1) × (−30) i (−5) × (−6) ÷ 0.04

j 20 × 8 × (−0.34) k −6.4 ÷ (−0.08) ÷ 2 l (−11) × (−3.5) × (−0.48)

a (−4 + 3) × 8 b −2(−7 − 6) c [4 + (−6)]

d −(−8 − 3) + 2 e 5 − (−2) × 9 − 15 f (−4)2× 5 − (−11)

g −27 ÷ (−3)2+ 14 h i 15 ÷ (−5 + 2) + (−6)

j k l

a 1 275 000 × 235 100 b 4 583 007 × 117 952

c 740 000 × 12 305 d 476 × 1230 × 1512

e 65 ÷ 312 ÷ 79 002 f 0.012 ÷ 953 880

g 5200 × 30 × 761 000 h 0.009 15 ÷ 123 094

i 17 254 × 30 076 ÷ 0.263 j 0.765 ÷ 0.12 × 0.000 976

A 4 900 000 000 B 490 000 000 C 49 000 000

D 0.000 000 049 E 0.000 000 004 9

1B

4.2+9.6

--- 120.16

14.5–7.3 ---4.399+59.87

7.8+12.6

--- 5.2×14.35 4.1×6.5 ---7.3+33.11

5.1+2.6×8.4

--- 6.7+10.548 3.2×4.9 ---27.35

33.2×0.6+4.1×2.2

--- 8.62–6.14 9.2–7.4

( )÷(5.1+3.9) ---15×9–44.53

35–8×3+7×14

--- 33.92–3.1×2.4 1.6 15.3( –12.4×0.7) ---WORKED Example 8a WORKED Example 8b 2

– +5×( )–3 +1 8 –

---2

–

( )2_{–3×( )–8}

7 –

--- (7+( )–2 ×6)–(4×3–16) 5

–

--- 7–(2×13–51) 2×( )–4 –

---SkillS HEET

1.2

WORKED Example 9

m

7

The expression 3.25 × 10−4 can be written as:

8

Using scientific notation, the number 0.009 01 can be written as:

9

When written in scientific form, 220 000 is:

10 Use your calculator to evaluate the following expressions. Give your answer in scientific form.

i 3.16 × 1012÷ 6.4 × 1015× 4 × 108

j 9.6 × 10−5_{× 11.36 ÷ 4.8 × 10}−12_{× 3.52 × 10}−21

11 Use your calculator to evaluate each of the following. Write your answer in scientific form.

a 5.5 (7.6 × 109− 3.1 × 106)

b 2.8 × 105+ 3.1 × 109× 20

c d

e

12 The speed of light is 2.998 × 108 m/s.

a How far will the light travel in:

i 1 minute

ii 1 day (24 h)

iii 1 year (365.25 days)?

b If the distance between the Earth and the Sun is 1.50 × 1011 m, how long will ittake for the light to reach the Earth?

c Use your answer to b to decide whether it takes under or over an hour for the sunlight to reach the Earth.

13 The masses of the Earth, the Sun and the Moon are 5.98 × 1024 kg, 1.99 × 1030 kg and 7.35 × 1022 kg respectively.

a Calculate by how many times the Sun is heavier than

i the Earth

ii the Moon.

b Find the difference in the mass of the Earth and the Moon. Express your answer in tonnes.

A 32 500 B 3 250 000 C 0.000032 5

D 0.000 325 E 0.0325

A 9.01 × 103 B 9.1 × 103 C 9.1 × 10−3 D 0.901 × 10−2 E 9.01 × 10−3

A 2.2 × 104 B 22 × 10−4 C 22 × 104 D 2.2 × 105 E 2.2 × 10−5

a 1.5 × 106× 3.28 b 15 × 104÷ 7 × 1012

c 8.95 × 108× 2.16 × 1011 d 3.7 × 1010÷ 4.5 × 1015

e 51.36 × 12.8 × 10−4 f 6 × 107÷ 3 × 10−8

g 15.91 × 10−6× 3.9 × 10−9 h 7 × 10−5× 4.16 × 1019

m

multiple choiceultiple choice

m

multiple choiceultiple choice

m

multiple choiceultiple choice

4.38×103_+6.19×₁₀7

2_×10–6

---35 3.15_{( ×}106_–4.8×104₎

6×105

---4 7( ×109×₈×₁₀–7_–3×₁₀8)+₉×₁₀3

5_×10–10

---Rounding off

Decimal places and significant figures

While completing exercise 1B you had to copy the complete answers from the calculator display. In many cases the number you had to write contained up to 10 digits. However, in real life the exact answer is not always required. For instance, if you are asked how long it takes you to get to school in the morning, you are not expected to give the answer in hundredths of a second. A reasonable answer to such a question would be in minutes. Similarly, when talking about distances between cities, we usually operate in whole numbers of kilometres. That is, when enquiring about the distance to the nearest town, you are not likely to get the answer, say, 25.003 69 kilometres (which would mean 25 km 3 m and 69 cm), but rather 25 kilometres.

The required degree of accuracy of the answers in this section is usually stated, for example, ‘give the answer correct to 2 decimal places’. If it is not specified, the accuracy should be dictated by common sense.

It should be understood that the smaller the required number of decimal places (or significant figures), the less accurate will be the answer. Confusion between significant figures and decimal places can be avoided with the following rules:

1. Decimal places are counted beginning from the first digit after the decimal point. For example, the number 102.591 has three digits after the decimal point (5, 9 and 1) and hence three decimal places.

2. Significant figures are counted from the first non-zero digit. For example, 0.0092 has two significant figures (9 and 2).

3. Any zeros at the end of the number after the decimal point are considered to be significant. For example, the number 1.20 has three significant figures (1, 2 and 0). 4. The trailing zeros at the end of the number, however, are not considered as

significant. That is, 1200 has only two significant figures (1 and 2).

5. While significant figures are not necessarily the same as decimal places, sometimes they do coincide. For example, 0.025 has three decimal places, but only two significant figures, while 0.25 has the same number (two) of decimal places as significant figures.

Note: When used with numerals, the words ‘decimal places’ will be abbreviated to ‘dp’ and ‘significant figures’ to ‘sf’.

In each of the following numbers, state the number of:

i decimal places ii significant figures.

a 20 500 b 3.097 c 0.5702

THINK WRITE

a Write the number. a 20 500

i State the number of decimal places by counting the digits after the point. (There are none in this case.)

i 0 dp ii State the number of significant figures. Start counting from 2,

as it is the first non-zero digit on the left; do not count the two trailing zeros after the 5.

ii 3 sf

11

Rounding off

Whether the question requires you to round off the number to a certain number of decimal places or significant figures, the procedure is the same.

1. Count out the required number of decimal places or significant figures and visualise a cut-off line. (You do not have to actually draw it; just think where it should be.) 2. Consider the digit immediately after the cut-off line:

(a) if it is under 5, leave the preceding digit unchanged (b) if it is 5 or more, add 1 to the preceding digit (round up). 3. Omit all digits after the cut-off line.

THINK WRITE

b Write the number. b 3.097

i State the number of decimal places by counting the digits after the point.

i 3 dp ii State the number of significant figures by counting the number

of digits beginning with 3 (the first non-zero digit on the left).

ii 4 sf

c Write the number. c 0.5702

i Count the number of digits after the point. Hence state the number of decimal places.

i 4 dp ii Start counting digits from the first non-zero digit on the left

(which is 5) and hence state the number of significant figures.

ii 4 sf

Round off 10.5392 to show the following number of decimal places (dp) or significant figures (sf):

a 1 dp b 2 dp c 1 sf d 5 sf

THINK WRITE

a Write the number. a 10.5392

Count out 1 dp and place a cut-off line: it is after the 5. 10.5|392 The number after the cut-off line (3) is less than 5, so leave the

preceding digit unchanged and omit all digits after it.

= 10.5 (to 1 dp)

b Write the number. b 10.5392

Count out 2 dp and place the cut-off line. 10.53|92

The digit after the line (9) is more than 5, so round up by adding 1 to the preceding digit (3 becomes 4) and omit all digits after it.

= 10.54 (to 2 dp)

c Write the number. c 10.5392

Count out 1 sf and place the cut-off line. (Here it is after the 1.) 1|0.5392 Consider the number after the line (which is 0). It is under 5, so

leave the preceding digit unchanged.

Put 0 in the unit’s place and omit all digits that follow. = 10 (to 1 sf)

d Write the number. d 10.5392

Count out 5 sf; the cut-off line is after the 9. 10.539|2 The digit after the line (2) is under 5, so leave the preceding digit

unchanged and omit all digits that follow.

= 10.539 (to 5 sf) 1

2 3

1 2 3

1 2 3

4 1 2 3

12

Truncating

Numbers are truncated by deleting final digits.

As you can see, truncating is similar to rounding, except it does not take the number after the cut-off line into consideration. For example, 2.372 whether truncated or rounded off to 2 dp will produce 2.37, while 2.376 when truncated to 2 dp will still give 2.37 and when rounded off to 2 dp will give 2.38.

1. When you wish to round to a particular number of decimal places press , select NUM and choose 2:round(, then press

2. Enter the decimal that you wish to round or press ANS to paste the last answer obtained on the home screen, followed by a comma and the number of decimal places required.

Truncate 5.3968:

a to 2 dp b to a whole number.

THINK WRITE

a Write the number. a 5.3968

Count out 2 dp and insert the cut-off line. 5.39|68

Rewrite your number leaving out the digits after the cut-off line. = 5.39

b Write the number. b 5.3968

The cut-off line is placed where the whole part of the number ends (that is, after the 5).

5.|3968

Rewrite the number leaving out its decimal part and the decimal point.

= 5 1

2 3 1 2

3

13

WORKED

E

xample

Graphics Calculator

Graphics Calculator

tip!

tip!

Rounding to a number of

decimal places

MATH

ENTER

2nd

remember

1. Significant figures are counted from the first non-zero digit on the left. 2. Decimal places are counted from the first digit after the decimal point. 3. To round off:

(a) count out the required number of decimal places or significant figures and draw a cut-off line

(b) if the digit after the cut-off line is under 5, simply leave out all digits after the line

(c) if the digit after the cut-off line is 5 or more, add 1 to the preceding digit and omit all digits that follow.

4. To truncate, delete the end digits of the number regardless of the value of the digit after the cut-off line.

Rounding off

1 In each of the following numbers, state the number of:

i decimal places ii significant figures.

2 Round off each of the following to:

i 1 decimal place ii 2 decimal places.

3 Round off each of the following to the number of significant figures indicated in brackets.

4

When the number 17.5362 is written as 17.54, it has been rounded off to:

5 Truncate each of the following to the number of decimal places indicated in brackets.

6 Copy and complete the following statement using the correct word(s) from the brackets.

If the digit after the cut-off line is under 5, truncating and rounding off to any number of decimal places will produce . . . answer(s). (the same/a different)

7

If the digit after the cut-off line is 5 or more, a true statement when rounding or trun-cating to a number of decimal places is:

A Truncating and rounding off will produce the same answer. B The truncated answer will be smaller than the rounded-off answer. C The truncated answer will be larger than the rounded-off answer.

D The difference between the truncated and rounded-off answers will always be equal to 0.1.

E The difference between the truncated and rounded-off answers will always be equal to 1.

a 12.305 b 7.0916 c 0.006 d 4500

e 0.7310 f 10.9 g 200 h 360.00

i 0.0930 j 107.09

a 7.953 b 12.076 c 0.083 d 100.9999

e 33.3333 f 6.120 59 g 10.918 h 0.904

i 250.035 j 98.12

a 4.397 (1) b 20.035 (1) c 758.92 (1) d 0.0937 (1)

e 3.5009 (3) f 1200.56 (2) g 0.573 (1) h 0.0762 (2)

i 19.0583 (4) j 10.503 (3)

A 2 decimal places B 2 significant figures C 4 decimal places D 4 significant figures E both A and D

a 23.586 (1) b 4.156 42 (3) c 29.316 (2) d 188.05 (1)

e 20.359 (0) f 15.425 98 (4) g (1) h (3)

1C

WORKED Example 11

WORKED Example

12a, b EX

CEL Spreadsh e_et

Rounding

WORKED Example 12c, d

m

multiple choiceultiple choice

WORKED Example 13

3.6˙. 100.2˙.

m

8 Copy and complete the following table:

9 Examine the supermarket receipt.

a What is the actual total of the goods on the receipt?

b What is the rounded off total?

c Round off each item separ-ately to the nearest 5c by using the following rules: Round down the amounts ending in 1c and 2c and round up the amounts ending in 8c and 9c to the nearest 10c Round up the amounts ending in 3c and 4c and round down those ending in 6c or 7c to the nearest 5c.

d Calculate the total of

rounded off items in c.

e Find the difference of the totals in b and d.

f Which method of rounding off will leave:

i a customer better off

ii a supermarket owner better off?

10 Alex and three of his friends want to buy a computer magazine valued at $3.95. Calculate:

a the exact amount of each share

b the amount of each share to the nearest cent

c the actual amount of money each person has to pay (assuming they all pay equal amounts)

d the amount of change the four friends will receive if each contributes as much as you have calculated in part c. Is it possible to divide the change evenly?

Number

Rounded off to 2 decimal places

Rounded off to 2 significant figures

Truncated to 2 decimal places

a 12.376

b 1540.25

c 0.0748

d 212.0968

e 6.99

f 0.1026

THANK YOU FOR SHOPPING AT

SUPERMART $. ________ GRAPEFRUIT LOOSE

0.375 KG @ $1.78/KG 0.67

CRISPIX 300G 2.99

SPECIAL K 600G 4.97

LIGHT FRUCHE PK4 3.46 MUSL SLC APR 8 PK 3.09

FRUCHE 4X125G 3.46

R/TIM COF BG 45G 2.25 RAZZA BODY MAGIC 3.71 KNEE HI KICK 2PK 2.08 KINDR SPRISE 3PK 3.23

FRUIT BARS 225G 3.55

EXTRA GUM PK3 1.39

BANANA

0.665 KG @ $1.98/KG 1.32 NESCAFE COF 250G 7.99 BANANA

0.720 KG @ $1.98/KG 1.43

15 BALANCE DUE $45.59

Cash $46.00

ROUNDING $0.01

CHANGE $0.40

Estimation and approximation

When using a calculator, it is easy to accidentally press a wrong key or key entries out of sequence. To make sure that the answer on your calculator display makes sense, the solution should be estimated first.

To find an approximate answer, it is usually sufficient to round off all the values involved in calculations to 1 significant figure.

For each of the following, find:

i an approximate answer by rounding off each number involved in the calculation to

1 significant figure (sf)

ii the answer correct to 1 decimal place, using a calculator.

a 7.6598 × 0.3956 b

THINK WRITE

a Write the expression. a 7.6598 × 0.3956

i Round off each number to 1 sf. i ≈ 7 × 0.4

Simplify. = 2.8

ii Use the calculator to find the exact answer.

ii 7.6598 × 0.3956 = 3.030 216 88

Round off to 1 decimal place. = 3.0 (to 1 dp)

b Write the expression. b

i Round off each number to 1 sf. i ≈

Simplify. =

= 4 ii Use the calculator to find the

exact answer. ii = 3.854 161 882

Round off to 1 decimal place. = 3.9 (to 1 dp)

19.853+3.6412

6.0958

---1 2 1

2

19.853+3.6412 6.0958

---1 20+4

6

---2 24---₆

1 19.853 + 3.6412

6.0958

---2

14

WORKED

E

xample

remember

1. It is a good habit to estimate the answer before making any calculations. 2. To find an approximate answer, round off all the values involved in calculations

to 1 significant figure and then evaluate.

Estimation and approximation

1 For each of the following, find:

i an approximate answer by rounding off each number involved in the calculation to 1 significant figure

ii the answer correct to 1 decimal place, using a calculator.

2 For each of the following:

i estimate the answer by rounding off each number involved in the calculation to 1 significant figure

ii use your calculator to find the answer correct to 3 decimal places.

3 Estimate, and then confirm by calculator the value of each of the following expres-sions containing fractions:

4 To simplify each of the following, first estimate an answer and then calculate correct to 2 significant figures using a calculcator.

5

The best approximation to is:

6

The number 8 definitely cannot be used as an approximation to:

a 12.39 × 4.872 b 17.956 × 2.819

c 10.234 × 2.819 d 15.491 × 3.852

e 126.14 × 18.42 f 6.7598 × 352.87

g 1239 × 0.0091 h 0.0356 × 280.53

i 9735.2 × 12.361 j 0.0085 × 0.003 92

a b

c d

e f

g h

i j

a 7 + 4 − 3 b 5 − 4 + 2 c 16 × 3

d 8 ÷ 2 e 89 ÷ (3 + 6 ) f 12 + 16 × 2

a b c

d e f

A 60 B 80 C 100 D 120 E 130

A B C D E 4.509 × 1.703

1D

4.752+3.98 3.21

--- 16.92–13.104 2.465 ---123.48×7.352

148.52

--- 189.16×2.958 56.8 ---18.9 3.92( +4.698)

2.007

--- 27.95+13.42 58.71–11.98 ---12.98 73.16( –18.24)

5.12×9.936

--- 63.4×12.12–16.9×34.8 16.25+19.3 ---12.47+8.93×3.52–7.68

13.6÷44.5

--- (32.65–9.64)÷(3.52+5.71) 3.2 15.48( –8.92) ---3 8 --- 5 6 --- 1 4 --- 2 9 --- 7 8 --- 1 2 --- 4 5 --- 2 7 ---3 4 --- 7 8 --- 2 3 --- 1 2 --- 5 6 --- 5 7 --- 3 5 --- 5 7

---150.42 (98.51)2

1108

3

4.18

( )3 9×(0.017)2

0.042

--- (7×20.398)2 45.36

---m

multiple choiceultiple choice

37.4×9.96 4.803

---m

multiple choiceultiple choice

67.359

7

Which of the following circles has its circumference close to 36 units?

8 A standard brick is 12.5 cm high.

a Estimate the number of layers of bricks needed to construct a wall which is 5 m tall.

b Use your calculator to find the exact number of layers of bricks that would make a wall which is 5 m high.

9 Anna and Karen plan to attend modelling classes. The classes are conducted once a week for 14 consecutive weeks and each lesson costs $28.70. The only other expense is a make-up kit valued at $119.95 (which they are going to share). Also, if their parents are to attend the graduation dinner, it will cost an extra $55.00 per guest (payable in advance).

a Estimate the total cost of the classes. (Include both tuition fees and the associated expenses; assume that two parents are planning to attend the graduation dinner.)

b Use your calculator to find the exact cost of the classes.

10 Victoria is buying lollypops (35c each) for her son’s birthday party. She estimates that she can buy 3 lollypops with $1 and hence her $10 note will be just enough for her to buy 30 lollypops.

a Show that Victoria’s estimation is wrong.

b Explain the fault in her estimation.

11 According to a description in a craft magazine, Tanya needs 8 m of lace to decorate a cradle for her baby daughter. If the lace costs $2.75 per metre:

a estimate the cost of the required length of lace and hence decide whether she will be able to purchase it using a $20 note.

b use your calculator to find the total length of $20 worth of lace to the nearest centimetre and state how that compares with the required amount.

m

multiple choiceultiple choice

15 A

12

B

18

C

6

D

6

E

Estimation errors

For each shape you will need to do the following.

i Estimate the area of the shape by rounding off each measurement to 1 signifi-cant figure and record it in the second column (C2).

ii Find an exact area and record it in the third column (C3).

iii Find the error of estimation by subtracting the estimated area from the actual area (C3 − C2) and record it in C4.

iv Express the error as a percentage of the actual area by dividing the estimation error by the actual area and multiplying by 100% .

The following formulas may be of some assistance to you. Area of a circle (A) =πr2 Area of a square (A) =s2 Area of a triangle (A) = bh

where r is the radius; s is the length of the side of the square; b is the base and h is the height of the triangle.

Use your completed table to answer these questions.

a Did your approximations underestimate or overestimate the actual area? Explain why.

b Were your approximations good or not? Explain your answer. C4 C3

---×100%

 

 

1 2

---Shape

Approximate area

Exact area (to 4 dp)

Estimation error

% Esti-mation error a

b

c

12.12 mm 18.15 mm

10 cm 3.2 cm 7.3 cm

16.9 cm

Ratios

A ratio is a comparison of two or more quantities measured in the same units. Consider the following extract from a microwave recipe book:

‘. . . Place rice and water in the ratio 1:2 in a microwave-proof dish, cover and cook on high for 10 minutes. . .’

The ratio 1:2 (which is read as ‘one to two’) of rice and water means that for each cup of rice we should add two cups of water (that is, the amount of water is twice that of rice). The numbers 1 and 2 are called the terms of the ratio. Naturally, if we expect a large number of guests for dinner, 1 cup of rice won’t be enough. If we use 2 (or 3 or even more) cups of rice, we should then increase the amount of water accordingly. Here are some possible ratios of rice and water:

1 :3 2:4 2.5:5 3:6

All of these ratios are equivalent and, of all of them, the ratio 1:2 (the one from the recipe book) is said to be in its simplest form. It is the most convenient to work with, as the numbers are small and easy to interpret.

Generally the ratio a:b (or ) is in its simplest form if both a and b are whole numbers and it has been reduced to its lowest terms.

Finding the ratio of two quantities

As was discussed before, a ratio compares quantities that are measured in the same units. Therefore to find the ratio of two quantities, we first have to make sure that their units are the same. We then write the two numbers as a ratio and omit the units altogether, as they are not relevant anymore.

Note that the order of the numbers in the ratio is important. Remember the example of the ratio of rice to water? When written as 1:2, it means ‘1 cup of rice for 2 cups of water’; when written as 2:1 however, it would mean ‘2 cups of rice for each 1 cup of water’. (The rice will most likely burn in this case!)

1 2

---a b

---Express each of the following ratios in simplest form.

a 24:8 b 3.6:8.4 c 1 :1

THINK WRITE

a Write the question. a 24:8

Divide both terms by the highest common factor (HCF) of 8.

= 3:1

b Write the question. b 3.6:8.4

Multiply both terms by 10 to obtain whole numbers. = 36:84

Divide both terms by the HCF of 12. = 3:7

c Write the question. c 1 :1

Change both terms into improper fractions. = :

Multiply both terms by the lowest common denominator (LCD) of 9 to obtain whole numbers.

= 13:15

4 9 --- 2

3

---1 2

1 2 3

1 4₉--- 2

3

---2 13---₉ 5

3

---3

15

Keeping this in mind, always write the numbers in the same order as they were given to you in the question when solving problems involving ratios.

Dividing a quantity in a given ratio

Ingredients

cup tomato sauce 1 red chilli, chopped 2 gloves garlic, crushed

1 teaspoon finely chopped ginger 6 shallots, chopped

500 g prawns 360 g cooked rice

This recipe poses a problem, as it does not tell us how many cups of uncooked rice to use, but instead tells us the amount of the ready-made product, containing two ingredients (rice and water) in the ratio 1: 2.

Find the ratio of 2 hours to 112 minutes. Write your answer in simplest form.

THINK WRITE

Convert 2 hours into minutes to make both units the same.

2 hours= 2 × 60 minutes = 120 minutes Omit the units and write the two

quantities as a ratio. (Keep the same order as in the question.)

The ratio of 2 h to 112 min is 120:112

Simplify by dividing both terms by 8. = 15:14 1

2

3

16

WORKED

E

xample

1 4

---Divide 360 g in the ratio 1: 2; hence state the amount of rice and water needed to prepare 360 g of cooked rice.

THINK WRITE

Write the ingredients as a ratio. Rice:water = 1: 2 Count the number of parts. Number of parts= 1 + 2

= 3

Find the size of each part. _{Each part= g}

= 120 g

Find the amount of rice needed. We need 1 part of rice; hence the amount of rice is 120 g.

Find the amount of water needed. We need 2 parts, so 2 × 120 g = 240 g of water is needed.

1 2

3 360

3

---4

5

17

Ratios

1 Express each of the following ratios in the simplest form.

2 Express each of the following ratios in simplest form.

3 Write each of the following ratios in the simplest form.

4

The ratio not equivalent to 3:5 is:

5 Find the ratio of each pair of quantities and write the answer in the simplest form.

6

The ratio of 36 seconds to 6 hours is:

7 Divide each of the following quantities into the ratio given in brackets.

a 12:18 b 8:56 c 9:27 d 14:35

e 88:66 f 16:60 g 200:155 h 144:44

i 32:100 j 800:264

a 1.2:0.2 b 3.9:4.5 c 9.6:2.4 d 18:3.6

e 1.8:3.6 f 4.4:0.66 g 0.9:5.4 h 0.35:0.21

i 6:1.2 j 12.1:5.5 k 8.6:4 l 0.07:14

a 1 :2 b 2:1 c 1 :2 d 1 :1 e 3:

f :2 g 5:1 h 2 :1 i 3 :2 j 1 : 6

A 18:30 B 0.6:1 C : D 2.1:3.5 E :

a 5 cm to 20 cm b 12 mm to 1 cm c 2 m to 78 cm

d 4.6 km to 400 m e 250 ml to 3 l f 504 kg to 1 tonne

g 20 kg to 1050 g h $12 to 60 c i 4 month to 5 years

j 18 min to 100 sec

A 36:6 B 6:1 C 1:6 D 1:60 E 1:600

a 270 kg (4:5) b 600 m (1:11) c 215l (2:3)

d 5000 mm (3:5) e 420 g (4:3) f 3.6 tonnes (8:1)

g $4000 (2:3:3) h 250 km (2:3:5) i 700 ml (2:3:9)

j 48 h (11:3:2)

remember

1. A ratio compares two (or more) quantities in the same units. 2. A ratio itself does not contain any units.

3. The ratio a:b is in the simplest form if both of its terms (a and b) are whole numbers and have been reduced to their lowest terms.

4. Order of numbers in the ratio is important.

5. To write the ratio of two quantities, convert them to the same units first, then put them as a ratio in the same order as they were given to you in the question and omit the units.

6. To divide a quantity in a given ratio, find the total number of parts, the size of one part and then the size of each share.

remember

1E

EXCEL Spreadshe_et

Ratios WORKED Example 15a WORKED Example 15b WORKED Example

15c _SkillSHE

ET

1.3

multiple choiceultiple choice

2 5 --- 4 6 --- 1 3 --- 1 5 ---WORKED Example

16 _SkillSHE

ET

1.4

m

8 Michelle is studying for her Maths and Science exams. She decides to divide the 8 hours that she has for study in a ratio 1:3. How many hours will Michelle study for each exam?

9 A brother and a sister are sharing a packet of chewing gum in the ratio 5:7. If a packet contains 60 pieces of chewing gum, how much will each receive?

10 Three friends, Lena, Vicky and Margaret, always order household goods in bulk from the home delivery service to save on costs to suit their families’ needs. They then divide the goods in the ratio 1:2:3. On one occasion, their order contained a carton of tissues (36 packets per carton) valued at $39.60 and a box of 240 garbage bags valued at $18. Find:

a the number of packets of tissues each of the friends received

b the number of garbage bags that each received

c the total amount of money each has to contribute to pay for the order (the delivery is free).

11 Leon and Igor invested $18 000 and $22 000 respectively in International Independent shares. Calculate how they should divide a $3000 dividend if they agreed to share it in the ratio of their investments.

12 The angles of a quadrilateral are in the ratio of 3:4:4:5. Find the size of each angle and hence name this quadrilateral.

13 To make wholegrain bread with a bread-maker, I must use water, bread mix and yeast (in that order) in the ratio 35:50:1. If the total weight of the mixture is 0.86 kg, find the amount of each ingredient.

14 The estimated volume of the earth’s salt water is about 1285 600 000 cubic kilo-metres. The estimated volume of fresh water is about 35 000 000 cubic

kilometres.

a What is the ratio of fresh water to salt water (in simplest form)?

b Find the value of x, to the nearest whole number, when the ratio found in a is expressed in the

form 1:x

WORKED Example 17

Wor

Proportion

A proportion is a statement about the equality of two ratios. Consider the following example. To make a cup of coffee a person combines 1 teaspoon of instant coffee and 2 teaspoons of sugar; hence the ratio of coffee and sugar is 1:2. Clearly, to make 3 cups of coffee, this person will use 3 times as much coffee and sugar, namely 3 teaspoons and 6 teaspoons respectively. The ratio in this case is 3:6. Obviously the two ratios are equivalent and hence can be written as a proportion:

1:2 = 3:6. This should be read as ‘1 is to 2 as 3 is to 6’.

When three terms of a proportion are known, the fourth term can be found.

The Maths Quest Excel file ‘Ratio’ may be used to solve problems like the one in the previous worked example.

Find the value of x in each of the following:

a x:4 = 9:18 b 3:10 = 7.2:x

THINK WRITE

a Write the equation. a x:4 = 9:18

Rewrite each of the two ratios as fractions. = Solve for x:

(a) Cross-multiply.

(b) Divide both sides by 18.

18x = 36 x = x = 2

b Write the equation. b 3:10 = 7.2:x

Rewrite each of the two ratios as fractions. =

Solve for x: (a) Cross-multiply.

(b) Divide both sides by 3.

3x = 72 x = x = 24 1

2 x

4 --- 9

18

---3

36 18

---1

2 3

10 --- 7.2

x

---3

72 3

---18

WORKED

E

xample

EXCEL Spreadshe_e

t

Find the value of x in the proportion (x− 3):(x+ 5) = 3:4.

THINK WRITE

Write the question. (x − 3):(x + 5) = 3:4

Rewrite the ratios as fractions.

Solve for x: cross-multiply. 4(x − 3) = 3(x + 5)

Expand. 4x − 12 = 3x + 15

Subtract 3x from both sides and simplify.

4x − 3x − 12 = 3x − 3x + 15 x − 12 = 15

Add 12 to both sides and simplify. x = 15 + 12

x = 27 1

2 (x–3)

x+5 ( ) --- 3

4 ---=

3 4 5

6

19

WORKED

E

xample

If a set of 5 identical books costs $31.50, how much will 3 of these books cost?

THINK WRITE

Since we are talking about the same books, the ratio of the number of books to the total cost will be equivalent; therefore a proportion can be formed. Remember to keep the number of books and the total cost in the same order for both ratios.

5:31.50 = 3:x

Rewrite the ratios as fractions.

=

5x = 31.5 × 3 Solve for x. Cross multiply then divide

both sides by 5.

x = x = 18.9

Interpret the result. The total cost of three books is $18.90. 1

2 5

31.50 --- 3

x

---3 31.5×3

5

---4

20

WORKED

E

xample

remember

1. Proportion is a statement about the equality of two ratios. 2. When solving a proportion:

(a) Write the two ratios as fractions. (b) Solve for x.

3. When forming a proportion, remember to keep the same order on both sides.

Proportion

1 Find the value of x for each of the following proportions.

2

If 5:70 = 9:x, the expression to use to find x is:

3

If 4:x= 5:8, the expression which is not true is:

4 Form a proportion from the following numbers: 12, 3, 32 and 8. How many solutions are possible?

5 Find the value of x in each of the following proportions.

6 If 12 identical pens cost $11.64, how much would 5 such pens cost?

7 Two 100-g cans of tuna in spring water cost $2.70. How many such cans can be bought for $10.80?

8 A large box contains chocolate biscuits and shortbread biscuits in the ratio of 4:9. If there are 28 chocolate biscuits, find:

a the number of shortbread biscuits

b the total number of biscuits in the box.

9 The diagram at right shows a pair of similar triangles. If the lengths of the corresponding sides of similar triangles are in the same ratio, find the values of x and y.

10 Challenge

For each of the following proportions find x, if x is a positive number.

a x:5 = 45:x

b (x+ 2):2 = 9:(x− 1)

c 3:(x− 5) = (x+ 7):15

a x:2 = 9:4 b x:7 = 24:56 c x:3 = 21:9 d 6:x= 4:3

e 4:x= 6:8 f 3:x= 9:8 g 5:9 =x:15 h 10:3 =x:12

i 6:5 = 18:x j 4:11 = 22:x

A B C D E

A B C D E x× 5 = 4 × 8

a (x+ 6):(x− 2) = 12:5 b (x− 8):x= 3:8

c x:(x+ 4) = 7:9 d (x− 5):(x + 7) = 3:10

e (x− 2):(x− 5) = 6:4 f (x+ 3):5 =x:3

g (x− 6):4 = (x+ 5):3 h (2x − 9):2 = (x− 3):12

i (7x− 3):(2x+ 9) = 3:2 j (5 + 3x):9 = (4 − 2x):5

1F

WORKED Example

18 EX

CEL Spre_adsh e_e

t

Proportion

m

multiple choiceultiple choice

5×70 9

--- 5×9 70

--- 9×70 5

--- 70 9×5

--- 5

70×9

---m

multiple choiceultiple choice

4 x --- 5

8

---= 8

5 --- x

4

---= x 4×8

5

---= x

5 --- 4

8 ---=

WORKED Example 19

W WORKEDORKED E Examplexample

20

y

12

9

x

5 2.5

Gears and drive belts

Proportions are also used in solving problems which involve gears and drive belts (or chains).

The proportion used in the case of a gear drive (or a chain drive) is:

Note the inverse relationship between the number of teeth and the speed.

The speeds of the driver and follower are measured in revolutions per minute (rpm). The proportion used in the case of the belt dive is:

1 The driving wheel of a chain drive has 42 teeth and rotates at a speed of 600 rpm. Find the number of teeth of the follower if its speed is 400 rpm.

2 The driver of a belt drive has a diameter of 16 cm and rotates at a speed of 120 rpm. Find the speed of the follower if its diameter is 4 cm.

3 The follower of a belt drive has a diameter of 18 cm and rotates at a speed of 360 rpm. Find the speed of the driver if its diameter is 2.4 times larger than that of the follower.

4 The speed of the driver and the follower in a water pump are 800 rpm and 1200 rpm respectively. If the diameter of the driver is 24 cm, find the diameter of the follower.

Number of teeth on the driver Number of teeth on follower

--- Speed of the follower Speed of the driver ---=

Gear drive

Chain drive

Diameter of the driver Diameter of the follower

--- Speed of the follower Speed of the driver ---=

Percentages

The term per cent means per hundred. For instance, 29% means 29 parts out of 100. Any percentage can be changed into a fraction by dividing it by 100.

Note: As was shown in the preceding example, a percentage can be a whole number (for example 76%), a common fraction (for example %) or a decimal (for example 2.25%). When the percentage contains a fraction (for example %) be careful not to confuse it with the common fraction (in this case, , which is 25%).

Write the following percentages as: i common fractions ii decimals.

a 12% b 10.3% c 76 %

THINK WRITE

a i To express 12% as a fraction, put it over a hundred and remove the % sign.

a i 12%=

Simplify. =

ii To write 12% as a decimal, divide it by 100 (that is, move the decimal point 2 places to the left) and remove the % sign.

ii 12% = 0.12

b i Put 10.3 over 100 and remove the % sign. b i 10.3%= Multiply the numerator and the denominator

by 10 to get rid of the decimal point.

= ii Divide 10.3 by 100 by moving the decimal point 2

places to the left and removing the % sign.

ii 10.3%= 0.103

c i Change the fractional part of the percentage into a decimal by dividing the numerator by the denominator (3 ÷ 8).

c i 76 %= 76.375%

Put this over 100 and remove the % sign. =

Multiply the numerator and the denominator by 1000 to get rid of the decimal point.

=

Simplify. =

ii Change the fractional part of the percentage into a decimal.

ii 76 %= 76.375% Divide by 100 by moving the decimal point 2

places to the left and dropping the % sign.

= 0.763 75

3 8

---1 12

100

---2 ₂₅---3

1 10.3

100

---2 103₁₀₀

---1   38

---3 8

---2 76.375---₁₀₀

3 _{100 000}---76 375

4 611₈₀₀

---1 3

8

---2

21

WORKED

E

xample

1 4

---1 4 ---1

---The table below contains percentages and corresponding fractions in everyday use. Knowing them by heart helps in solving problems as well as in many practical situations.

In order to change a percentage into a fraction we had to divide by 100 and remove the % sign. To change a fraction into a percentage, we need to do the opposite, that is, multiply the fraction by a 100 and add a % sign.

To find a percentage of a quantity, express the percentage as a fraction (either common, or a decimal) and multiply by that quantity. Remember that a percentage represents a certain part of a quantity, so the answer should be in the same units as the quantity in question. (That is, when finding a percentage of a certain amount in dollars, the result is in dollars; when the given quantity is in kilograms, the percentage of that quantity is also in kilograms.)

To express one quantity as a percentage of another, form a fraction from the two quan-tities and multiply by 100%. Remember to keep both quanquan-tities in the same units.

% 1% 10% 12.5% 20% 25% 33 % 50% 66 % 100%

Fraction 1

Decimal 0.01 0.1 0.125 0.2 0.25 0.5 1.0

1 3

--- 2 3 ---1

100 --- 1

10

--- 1 8

--- 1 5

--- 1 4

--- 1 3

--- 1 2

--- 2 3

---0.3˙. 0.6˙.

Express each of the following as a percentage.

a b 1.02

THINK WRITE

a Multiply the fraction by 100 and add a % sign.

a = × 100% = % Change the improper fraction into a

mixed number. = 42 %

b Multiply by 100 by moving the decimal point 2 places to the right and add a % sign.

b1.02 = 1.02 × 100% = 102%

3 7

---1 3₇--- 3

7 ---300

7

---2 6

7

---22

WORKED

E

xample

Find 23% of $72.

THINK WRITE

Change 23% into a decimal fraction. 23% of $72 = 0.23 × 72 Multiply by $72 and add a $ sign to the

answer. = $16.56

1 2

23

Express 3 months as a percentage of 2 years.

THINK WRITE

Change years into months. 2 years= 2 × 12 months = 24 months Form a fraction from the two quantities

and multiply it by 100%.

× 100%

Evaluate. = %

= 12.5% 1

2 ₂₄---3

3 300---₂₄

24

WORKED

E

xample

Career

profile

D A R R E N B L O O D — A c c o u n t E x e c u t i v e

Qualifications: Completing Advanced Diploma in Business Marketing

Employer: De Bortoli Wines Pty Ltd

Company website: http://www.debortoli.com.au

s I have a passion for wines, I wanted to work in the wine industry and wrote letters to companies outlining my experience. A position was available at De Bortoli Wines and I have been there for about 5 years. I didn’t want an office job, and sales seemed to provide variety, challenge and opportunity.

A typical day includes travelling to liquor stores to sell products, check stock and provide merchandise (T-shirts, caps, prints, aprons and so on). I conduct wine tastings for store customers and set up window displays. At the end of the day, sales are added up and faxed through to the warehouse. Calculations I need to perform which relate to using percentages include: 1. percentage discounts of wholesale prices of wine sold to liquor outlets — discounts apply with orders above a certain amount

2. percentage increases or decreases in monthly and yearly sales

3. setting yearly budgets based on the preceding year’s sales

4. calculating rebates — once a store has sold a certain amount, they receive a 2% rebate of monthly sales

5. using Excel spreadsheets to calculate which stores stock particular products

6. calculating sales tax (41%) and adding it to the wholesale price.

Other calculations involve:

1. rounding off customers’ change to the nearest 5c 2. measurement — weights of bottles, size

(dimen-sions) of pallets (64 cartons in a pallet and 12 bottles in a carton), different sized bottles — a mag-num is 1.5 L and a jeraboam is 3L — and conver-sion of volume from millilitres (mL) to litres (L). I once took an order from a customer for 400 cartons when the customer meant to order 400 bottles. He was a little surprised when a semi-trailer delivered the stock the next day.

Mathematical calculations are also very useful in analysing my total sales. I can check that I have been paid the right commission.

Questions

1. List three ways Darren uses percentages in his work.

2. Calculate the sales tax on a particular carton of wine if the wholesale cost of each bottle is $8.50.

3. Find the rebate calculated by Darren for a store that has sold over its monthly quota with sales of $11 224.00.

4. Where could you complete an Advanced Diploma in Business Marketing?

Percentages

1 Write the following percentages as:

i common fractions

ii decimals.

2 Change each of the following into percentages.

3 Change each of the following into percentages.

4 Copy and complete the following table.

a 15% b 27% c 58% d 112%

e 7.3% f 19.8% g 42.06% h 100.75%

i 10 % j 44 % k 67 % l 95 %

a b c d

e f g h

i 1 j 2

a 0.63 b 0.29 c 0.51 d 0.723

e 0.05 f 0.003 g 0.6 h 0.1998

i 1.2 j 4.03

Decimal Fraction Percentage

a 0.18

b

c 46%

d 0.9

e 1

f 7.4%

g 1.003

h

i 15 %

remember

1. To change a percentage into a fraction, divide it by 100 and omit the % sign. 2. To change a fraction into a percentage multiply it by 100%.

3. To find a percentage of a quantity, express the percentage as a fraction and multiply it by the given quantity.

4. To express one quantity as a percentage of another, form a fraction and multiply it by 100%.

remember

1G

Mathc

ad

WORKED Example 21

3 5

--- 1 2

--- 1 2

--- 3 4

---WORKED Example

22a 4₉--- 7

10

--- 5 8

--- 2 3 ---5

12

--- 29 50

--- 7 8

--- 19 20 ---4

5

--- 3 4

---WORKED Example 22b

7 8

---2 3

---32 40

---3 20

5

Which of the following is the same as 23 %?

6

Which of the following is the same as 1.06?

7 Find the following, correct to 2 decimal places.

8 Express the first quantity as a percentage of the second.

9 Rachael scored 87 points on her French test out of a possible 90 points. What percentage is that?

10 In a kindergarten group of 24 children, 16 are boys. What percentage of the group are girls?

11 A secondhand-furniture shopkeeper pur-chased a coffee table at a street market for $40 and later sold it for $95.

a Find the profit that she made.

b Express the profit as a percentage of the purchase price.

c Express the profit as a p