Maths Quest 9 - Chapter 1 o5v14

MQ9 Vic ch 01 Page 1 Monday, September 17, 2001 10:17 AM

Number skills

Year

World population

1950

2 555 078 074

1960

3 039 332 401

1970

3 707 610 112

1980

4 456 705 217

1990

5 283 755 345

2000

6 080 141 683

2010

6 823 634 553

2020

7 518 010 600

2030

8 140 344 240

2040

8 668 391 454

2050

9 104 205 830

1 There are many factors that affect the environment of our planet. One of these is world population. The data supplied in the table give the estimated or projected world population for the middle of the year. In which ten-year period did (or will) the world population increase the most? In which ten-year period is the percentage increase the largest? This chapter refreshes your skills in working with numbers expressed as fractions, decimals, percentages or in index form and applying those skills to real-life situations.


2

Maths Quest 9 for Victoria

Order of operations Anton has calculated the answer to 5 + 6 × 4 as 44, while Marco insists that the answer is 29. Who is correct? In mathematics, it is important to ensure that everybody obtains the same result from a calculation; so the order in which mathematical operations are worked is important. The order of operations requires that: 1 all brackets are evaluated first, beginning with the innermost brackets 2 then, all multiplication and division are evaluated, working from left to right 3 and finally, any addition and subtraction, working from left to right. To obtain the correct answer to the calculation 5 + 6 × 4, we must complete the operations of + and × in the correct order. That is, × first then +. 5+6×4 = 5 + 24 = 29

WORKED Example 1 Evaluate each of the following without using a calculator. a 4 + 12 − 5 + 6 b 4 + 12 − (5 + 6) c 6 + 21 ÷ 7

d [4 × (5 + 8)] ÷ 2

THINK

WRITE

a

a 4 + 12 − 5 + 6 = 16 − 5 + 6 = 11 + 6 = 17 b 4 + 12 − (5 + 6) = 4 + 12 − 11 =5

1 2 3

b

1 2 3

c

1 2 3

d

1 2 3

Write the calculation. Perform the addition and subtraction from left to right. Write the answer. Write the calculation. Evaluate the bracket first. Perform the addition and subtraction from left to right and write the answer. Write the calculation. Perform the division. Perform the addition. Write the calculation. Remove the brackets by working the innermost bracket first. Divide and write the answer.

c 6 + 21 ÷ 7 =6+3 =9 d [4 × (5 + 8)] ÷ 2 = [4 × 13] ÷ 2 = 52 ÷ 2 = 26

In other examples you will need to read the question carefully to interpret the correct order of operations and the correct way to write the calculation.


Chapter 1 Number skills

3

WORKED Example 2 Mum bought 2 packets of Easter eggs to hide in the garden for her 4 children to find. Each packet contained 20 eggs. While she was hiding them, the dog ate 4 eggs, Dad ate 3, and 1 was squashed. If all the other eggs were found, and each child found the same number of eggs, how many eggs did each child have?

THINK 1

2

3

WRITE

Write a mathematical sentence showing what happened. Find the total number of eggs and subtract the number that were eaten or squashed. Then divide by the number of children looking for eggs. Use order of operations to solve the problem.

Write the answer in a sentence.

[2 × 20 − (4 + 3 + 1)] ÷ 4 = (2 × 20 − 8) ÷ 4 = [40 − 8] ÷ 4 = 32 ÷ 4 =8 Each child found 8 eggs.

Evaluate in the following order. 1. Brackets first, beginning with the innermost pair, then working through to the outermost pair. 2. Multiplication and division in order from left to right. 3. Addition and subtraction in order from left to right.

1A

Order of operations

1 Evaluate each of the following without using a calculator. a 3 + 12 − 5 + 6 b 7 + 5 − 11 + 2 − 3 1 d 18 − 11 + 4 + 12 − 14 e 25 + 5 − 10 + 2 − 10 g 10 × 6 × 4 × 2 h 18 × 4 × 3 × 0 j 25 ÷ 5 × 6 k 8×2÷4×3 m 16 + 2 × 5 n 80 ÷ 2 + 28 p (4 + 6) × 8 q (35 − 11) ÷ 6 s 12 ÷ (9 − 3) t 75 ÷ (12 + 13)

Math

Example

10 − 2 − 3 + 4 32 − 8 + 6 − 7 − 5 80 ÷ 4 ÷ 5 72 ÷ 2 ÷ 6 × 3 12 − 14 × 0 (7 + 2 − 3) × 8

Order of operations Math

cad

c f i l o r

cad

WORKED

Ascending and descending order


4


Mat

d hca

Adding whole numbers DIY Mat

d hca

Subtracting whole numbers DIY Mat

d hca

Multiplying whole numbers DIY Mat

d hca

Dividing whole WORKED numbers Example 2 DIY

2 multiple choice a What is 12 × (4 + 2) ÷ 8 equal to? A 10 B 12 C 9 b What is 36 ÷ 3 ÷ 4 + 2 equal to? A 2 B 72 C 50 c What is 8 × 5 + 3 × (8 − 5) is equal to? A 49 B 192 C 59

D 8

E 1

D 5

E 10

D 129

E 339

3 Evaluate each of the following. a 8 × 9 − 10 × 6 b 14 × (3 + 2) ÷ 7 c 72 ÷ (2 + 7 × 1) d 80 ÷ 5 − 60 ÷ 6 e 35 × (8 + 4 − 6 × 2) f (13 − 3) × 2 + 4 × 6 g (17 − 12) ÷ 5 × 2 h (14 + 7 − 8) × 6 i [14 + (2 × 6 − 3)] × 4 j [(2 + 1) × 7 − 3 × 5] − 6 ÷ 3 k {[(3 + 9) ÷ 12] + 4 × 4} − 17 l {40 − [(8 + 2) × 3 − 5]} ÷ 5 m 16 ÷ 4 + 24 ÷ 6 + 5 × 5 − 19 n 108 ÷ 4 × (4 − 4) × 4 o {11 + (4 + 3) × 2 + 5 × 6 + (8 – 2) × 5} × 4 p [16 × 3 ÷ 2 + 40 ÷ 4 × 2 − 3 × 11 + 14] ÷ 5 + (6 × 2 + 4) × 2 − (7 × 5 + 2) 4 Takiko has brought 3 packs of nut biscuits to share with the 20 of her class. If each pack contains 12 nut biscuits and 3 girls and 5 boys are allergic to nuts or don’t eat biscuits, so don’t have any; how many nut buscuits will each of the other class receive?

5 The Wimbletons wanted to buy a tennis racquet for each of their 3 children. The normal price of a racquet is $100 but the shop is offering a special deal. If two racquets are bought at the same time, the price is reduced by $25 for each one. If the Wimbletons buy one at the normal price and two on the special deal, how much do they pay altogether? Write an equation to show how you could have found the answer.



5

Integers Integers include positive whole numbers, negative whole numbers and zero. They can be represented on the number line. –5 –4 –3 –2

–1

0

1

2

3

4

5

The rules for using integers are: Rule 1 When adding integers with the same sign, keep the sign and add; −3 + −2 = −5. Rule 2 When adding integers with different signs, find the difference and use the sign of the number further from zero; –3 + 4 = 1. Rule 3 When subtracting integers, add the opposite; 5 − −7 = 12. Rule 4 When multiplying integers, the following rules are obeyed. (a) Positive × Positive = Positive

5 × 8 = 40

(b) Positive × Negative = Negative

5 × −8 = −40

(c) Negative × Positive = Negative

−5 × 8 = −40

(d) Negative × Negative = Positive

−5 × −8 = 40

Rule 5 When dividing integers, use the same rules as for multiplication. (a) 16 ÷ 2

=8

(b) 16 ÷ −2

= −8

(c) −16 ÷ 2

= −8

(d) −16 ÷ −8 = 2

WORKED Example 3 Calculate each of the following without the use of a calculator and using the correct order of operations. a −15 × −5 ÷ 3 b 7 + −5 − −8 c 4 − 60 ÷ (−4 − 6) THINK

WRITE

a

a −15 × −5 ÷ 3 = 75 ÷ 3 = 25 b 7 + −5 − −8 = 2 − −8 =2+8 = 10 c 4 − 60 ÷ (−4 − 6) = 4 − 60 ÷ −10 = 4 − −6 =4+6 = 10

1 2

b

1 2

c

1 2 3 4

Write the calculation. Multiplication and division are the only operations; so work from left to right. Write the calculation. Addition and subtraction are the only operations; so work from left to right. Write the calculation. Work the brackets. Perform the division. Perform the subtraction.


6


WORKED Example 4 Insert operation signs to make this equation true. 5 K 3 K 4 K 1 = −2 (Trial and error is a suitable method.) THINK WRITE 5 − 3 − 4 − 1 = −3 ≠ −2 1 The answer (−2) is less than the first number in the question; so try subtraction. 5 − 3 − 4 + 1 = −1 ≠ −2 2 The result of the first try (−3) is a little too small; so change the last sign to +. 5−3−4×1 3 The result of the second try (−1) is too big; so try multiplying the last digit, which is 1, =5−3−4 ing to use the order of operations. = −2

1. When adding integers with the same sign, keep the sign and add. 2. When adding integers with different signs, find the difference and use the sign of the number further from zero. 3. When subtracting integers, add the opposite; for example 5 – –7 = 12. 4. When multiplying and dividing integers, like signs give positive answers, unlike signs give negative answers. 5. When using order of operations, evaluate brackets before multiplication and division, then evaluate addition and subtraction.

1B 1.1

SkillS

HEET

1 Calculate each of the following without the use of a calculator and using the correct order of operations. 3a a −7 + 12 b −14 + 7 c −18 − 8 d 25 − 24 − 2 e −2 − 3 − 6 f −7 − 11 + 5 g 14 − 15 + 11 h 13 − 19 − 6 + 9 i 10 × 2 ÷ 5 j 6 × −3 × −2 k −4 × −3 × −5 l 64 ÷ −16 ÷ 4 m −12 × 4 ÷ 16 n −120 ÷ −10 × 2 o 36 ÷ −6 × −5 p −6 × −1 × −10 ÷ 4

WORKED

Example

1.2

Example

Mat

d hca

Order of operations with integers et

reads L Sp he

EXCE

2 Calculate each of the following without the use of a calculator and using the correct order of operations. 3b a 8 + −7 + −3 b 15 − 18 + −8 c 6 + −7 + −10 d 6 − −7 e −5 − −2 f 7 − −2 − 7 g 4 + −8 − −5 h −7 − −13 i −9 + −9 − −9 j 4 + −6 + −2 − −1 k −3 − 6 − −10 + −5

WORKED

SkillS

HEET

Integers

Arithmetic timer

3 multiple choice a −7 − 8 + 2 − 3 is equal to: A −2 B −16 C −14 b −12 × −8 ÷ −4 × 2 is equal to: A 12 B −12 C 48 c 9 + −5 − −4 + 2 − −1 is equal to: A 9 B 1 C 23

D −20

E 0

D −48

E 316

D 11

E −1



7

4 Calculate each of the following without the use of a calculator and using the correct order of operations. 3c a −3 + 3 × 3 b −9 − 2 × 6 c 15 ÷ 5 − 5 d 7×0−5 e 6 × (0 − 6) f −14 × 2 − 2 × 10 g 2 − 6 × 3 h 8 + 2 × −5 i 3×8−5×7 j 12 × −3 − 4 k 0 × 3 × −6 + 6 l −90 ÷ −5 − 26 m 5 × (−3 + 5) + 7 n 128 ÷ −16 + 3 × −5 o (3 + 7) ÷ −2 + −4 p −60 ÷ −4 × 3 − 43 q 28 ÷ −2 × (2 − 5) r 56 ÷ 7 + 70 ÷ −10 s 94 ÷ 2 + 3 × −3 t 14 − 4 × (5 + −6)

WORKED

Example

5 multiple choice a What does 5 × −4 − 10 × −6 equal? A −40 B 40 C 80 D −80 E 180 b What does 5 × (−4 − 10) × −6 equal? A 420 B −420 C 180 D −180 E −40 c (–2 – –4) × (8 × 5 − 4) is equal to: A 244 B −216 C 16 D −48 E 72 d –64 ÷ 8 – –8 is equal to: A 4 B −4 C 0 D −16 E 16 e The correct operation signs to make 2 K −5 K −2 K −5 = −3 a true statement are: A ×, −, − B ×, +, + C ×, ÷, + D −, +, × E −, ×, + 6 Insert operation signs to make these equations true: a 5 K 6 = 11 b 7 K −4 = −28 4 d −7 K −3 = −4 e 3K4K5=2 g −5 K −4 K 10 = 2 h 6 K 3K 3 = 0 j 2 K 3 K 5 K 4 = 26 k 16 K 8 K 8 = −6 m 12 K 18 K −2 = 21 n −8 K 4 K −2 = 0 p 5 K 2 K −3 K −3 = 2

WORKED

Example

c f i l o

−18 K −2 = 9 7 K 2 K 3 = 17 8 K 5 K 2 = −2 12 K 18 K 2 = 21 10 K 3 K 4 K 2 = 0 GAME time

7 Thanh stands on a cliff top 68 m above sea level and drops a stone into the water. It stops on the bottom 27 m below sea level. How far has the stone fallen?

Number skills — 001

8 The temperature range in Melbourne on 29 April was 7°C. If the maximum temperature was 15°C, what was the minimum temperature?


8


Golf scores In golf, par is the number of strokes considered necessary to complete a hole in expert play. A birdie is a score of one stroke under par and a bogey is one stroke over par. An eagle is a score of 2 strokes under par while 3 strokes under par is called an albatross. A double bogey is 2 strokes over par and a triple bogey is 3 strokes over par. 1 Use integers to represent: a par b a birdie c a bogey d an eagle e an albatross f a double bogey g a triple bogey. 2 Which score for a hole would be the most difficult to achieve? 3 Leon and Dion have finished a round of 18 holes with the following information shown on their scorecards.

Leon pars birdies bogeys eagles double bogeys albatrosses triple bogeys Final score

Dion 4 3 6 1 2 0 2

pars birdies bogeys eagles double bogeys albatrosses triple bogeys Final score

6 2 4 0 2 1 3

What integer represents each person’s final score as a number of strokes over, under or at par? 4 Who wins this round of golf? 5 Two professional golfers achieve overall final scores for 18 holes of −8 and −6. a What does this mean? b Who achieved a better score for this round of golf? c How many strokes did each player make for the 18 holes if the course is considered to be a par 71 course?



9

Estimation and rounding Rounding to a given number of decimal places Ms Shopper’s bill at the supermarket comes to $94.68 and she pays $94.70. Mr Shopper’s bill is $83.72 and he pays $83.70. The bills have been rounded to the nearest 5 cents because the 5-cent is the smallest coin used. Ms Shopper’s bill has been rounded up because 68 cents is closer to 70 cents than to 65 cents. Mr Shopper’s bill has been rounded down because 72 cents is closer to 70 cents than to 75 cents. Measuring distances is another one of the many practical situations where it is necessary to round an answer to a given number of decimal places. For example, the distance between two towns is given to the nearest kilometre. It is not practical or useful to the average motorist that the distance between Melbourne and Sydney by a certain route is 1024.352 km. We give the distance simply as 1024 km. The accuracy of measurement is limited by what is practical and by the accuracy of the instrument being used to take the measurement. For example, with your ruler it would not be possible to measure anything more accurately than to the nearest millimetre. The measurement 5.6713 cm ≈ 5.7 cm because 5.6713 is closer to 5.7 than it is to 5.6. The rounded answer, 5.7, is the closest approximation to the exact answer. To round an answer to a given number of decimal places, consider only the first digit after the required number of decimal places. If that digit is 0, 1, 2, 3 or 4, then leave it and all following digits out of the answer. If that digit is 5, 6, 7, 8 or 9, then the last digit to be written is increased by 1 and all else is left out. Many calculators are able to round off using the FIX function.

On some scientific calculators, you need to press MODE first.

On a TI graphics calculator, press MODE , arrow down to the second row, then arrow across to highlight the number that corresponds to the required number of decimal places. Press ENTER to set this rounding condition. To undo this operation, press MODE , arrow down to highlight FLOAT and press ENTER . Any rounded answer is not an exact answer but a close approximation.

WORKED Example 5

Round 15.439 657 to: a 1 decimal place b 3 decimal places. THINK WRITE a 1 Write the number. a 15.439 657 Look at the second decimal place to determine whether to ≈ 15.4 2 leave it or to round it up. The digit is 3; so rewrite the number without all digits after the first decimal place. b 1 Write the number. b 15.439 657 ≈ 15.440 2 Look at the fourth decimal place to determine whether to leave it or to round it up. The digit is 6; so increase the third decimal place by 1. Note: Adding 1 to 9 gives 10, thus 439 becomes 440 and the zero must be included. Note: The more decimal places, the closer the approximation is to the exact answer.


10


There were 70 000 people at the Melbourne Cricket Ground for Australia’s one-day match against the West Indies.

Rounding to a given number of significant figures Although the caption describes a crowd of 70 000, in reality there may have been 70 246 people. The number has been rounded to 1 significant figure because the rest of the number, 246, has no impact on our image of the size of the crowd. When using very large or very small numbers, rounding to a given number of significant figures is often used. To round to 1 significant figure means having only 1 non-zero digit beginning from the left with the other digits being zeros. The number 367 rounded to 1 significant figure is 400 because 367 is closer to 400 than to 300. To write 452 correct to 2 significant figures, we need to consider whether 452 is closer to 450 or 460. It is closer to 450, and 4 and 5 are the 2 significant figures. The method of deciding whether to leave or round up is the same as rounding to a number of decimal places.

WORKED Example 6

Round 347 629 to: a 1 significant figure b 3 significant figures. THINK WRITE a 1 Write the number. a 347 629 ≈ 300 000 2 Look at the second significant figure to determine whether to leave it or to round it up. The digit is 4, so rewrite the number, replacing all digits after the first significant figure with zeros. b 1 Write the number. b 347 629 Look at the fourth significant figure to determine whether to it ≈ 348 000 2 leave or to round it up. The digit is 6 so write the answer by adding 1 to the third digit and replace all other digits with zeros.



11

Note: The more significant figures taken, the closer the approximation is to the exact answer. When the first non-zero significant figure appears after the decimal point, any zeros before that figure are not significant.

WORKED Example 7 Round 0.004 502 6 to 3 significant figures. THINK 1 2

WRITE

Write the number. The first significant figure is the 4. Round to 3 significant figures beginning with the 4. The last zero must be included in the answer because it is one of the significant figures.

0.004 502 6 ≈ 0.004 50

Estimation Rounding is also used when making an estimation or mental approximation of an answer. Estimation is a method of checking the reasonableness of an answer or a calculator computation. We can estimate an answer by rounding the numbers in the question to simple numbers that can be calculated mentally.

WORKED Example 8 Estimate answers to the following without calculating the exact answer. a 31 × 58 b 46 679 + 2351 × 65 THINK

WRITE

a

a 31 × 58 ≈ 30 × 60 = 1800 b 46 679 + 2351 × 65 ≈ 50 000 + 2000 × 70 = 50 000 + 140 000 = 190 000

1 2 3

b

1 2 3

Write the calculation. Round each number to 1 significant figure. Perform the mental calculation. Write the calculation. Round each number to 1 significant figure. Perform the mental calculation.

1. When rounding to a given number of decimal places, count only those places after the decimal point. 2. When rounding to a given number of significant figures, begin counting from the first non-zero digit. 3. A quick mental estimation can be used to check the accuracy of calculations. 4. Rounding is often used to convey a concept of size rather than an exact number.


12


1C

Estimation and rounding

1 Round the following to: i 1 decimal place ii 2 decimal places iii 3 decimal places. a 5.893 27 b 67.805 629 c 712.137 84 d 81.053 72 e 504.896 352 5 2 Round the following to 0 decimal places. (To 0 decimal places means to the nearest whole number.) a 25.68 b 317.19 c 1027.8 d 19.53

WORKED

Example

3 Round the following to 1 decimal place. a 3047.2735 b 24.7392 c 8.2615

EXCE

et

reads L Sp he

Rounding WORKED Example and 6 significant figures DIY d hca

4 Round the following to: i 1 ii 2 iii 3 iv 4 significant figures. a 574 248 b 430 968 c 28 615 d 1 067 328 e 458 610 5 Round the numbers in question 2 to 2 significant figures.

6 Round the following correct to 3 significant figures. a 0.085 246 b 0.000 580 4 7 d 0.006 765 73 e 0.000 026 973

WORKED

Mat

Example

Rounding

c f

0.000 008 067 3 0.000 352 1

7 Estimate answers to the following without calculating the exact answer. a 183 ÷ 58 b 78 × 11 c 632 + 169 d 1010 ÷ 98 8 e 17 × 19 f 476 ÷ 8 + 52 g (51 + 68) × 12 h 68 + 19 × 9 i 5 × (78 − 59) j 42 × 8 + 18 × 5 k 176 ÷ 18 + 689 ÷ 7 397 l m 473 × 248 n 657 − 239 ÷ 49 o 12 345 + 549 × 146

WORKED

Example

Mat

Estimation

1.1

9 Each of the 178 students who attend the Year 9 Social has to pay $55. If the cost of hiring the band is $1000, estimate how much money would be available to pay for the supper and the security people.

S

QUEST

EN

M AT H

ET SHE

8 multiple choice a The number 49.954 correct to 1 decimal place is: A 49.9 B 49.0 C 50 D 50.0 E 50.1 b The number 3 056 084 correct to 3 significant figures is: A 3 050 000 B 3 056 000 C 3 057 000 D 306 E 3 060 000 c The number 0.008 065 3 correct to 3 significant figures is: A 0.008 B 0.008 065 C 0.008 06 D 0.008 07 E 0.806 d A number rounded to 2 decimal places is 6.83. The original number could have been: A 6.835 B 6.831 C 6.8372 D 6.85 E 6.8

GE

d hca

Work

d 19.9804

CH

AL

L

1 In 1832, a young runner named Mensen Ehrnot reportedly ran nearly 8950 km over a 59-day period. On each of those days he ran 16 hours and rested for 8 hours. Estimate how many kilometres he ran, on average, per hour. 2 In the hundred consecutive whole numbers from 1 to 100, how many times does each of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 occur?

MQ9 Vic ch 01 Page 13 Wednesday, September 19, 2001 8:34 AM


13

1 1 Evaluate 9 − 13 − 14. 2 Evaluate 8 − 8 ÷ 4. 3 Evaluate (13 + 5 × 7) ÷ 12. 4 Evaluate −25 + −10 − −50. 5 Evaluate −84 ÷ 12 × 3. 6 Evaluate −18 + (−9 + 11) × 14. 7 Insert signs to make the following equation true. 5 K 21 K 7 K 5 = 20 8 Round 1.746 582 to 4 decimal places. 9 Round 0.006 059 9 to 4 significant figures. 10 Give an estimate for 78 + 43 + 55 − 86.

Decimal numbers Decimal numbers are so much a part of everyday life that we need to be able to use them, put them in order and convert them to simple fractions and percentages. When using either your graphics calculator or a scientific calculator, enter the calculation as written and the calculator will perform the calculation using the correct order of operations. There are, however, many things that we need to be able to do ourselves with decimals without the aid of a calculator.

Ordering decimal numbers Ascending order means from lowest to highest and descending order means from highest to lowest. This is done by first writing each number with the same number of decimal places, adding zeros where necessary. We then look at the left-most digit. The greater this digit, the greater the decimal number. If the left-most digits are the same, we move to the next digit, and so on.

WORKED Example 9 Write the following decimal numbers in ascending order: 0.66, 0.606, 0.6. THINK WRITE 1 2

3

Write the numbers. Write all numbers with the largest number of decimal places, in this case 3, then compare. Write the original numbers in ascending order after looking at the second and third decimal places.

0.66, 0.606, 0.6 0.660, 0.606, 0.600

0.6, 0.606, 0.66


14


Finite or terminating decimal numbers Finite decimal numbers have a fixed or finite number of decimal places and can be written as a fraction with a denominator that is a multiple of 10. If the decimal number has 1 decimal place, the denominator of the fraction is 10; if there are 2 decimal places, the denominator is 100; if there are 3 decimal places, the denominator is 1000 and so on. In each case the numerator is the decimal number without the decimal point. These fractions are simplified where possible.

WORKED Example 10 Convert each of the following to fractions in simplest form: a 0.65 b 1.2 c 0.6275. THINK WRITE a 1 Write the decimal number. a 0.65

b

c

13

2

There are 2 decimal places, so write as a fraction with a denominator of 100 and simplify by cancelling. (You may use a calculator to simplify.)

3

Write the answer.

1

Write the decimal number.

2

There is 1 decimal place, so write as a fraction with a denominator of 10 and simplify by cancelling. (You may use a calculator to simplify.)

3

Write the answer as a mixed number.

1

Write the decimal number.

2

There are 4 decimal places, so write the fraction with a denominator of 10 000 and simplify. (You may use a calculator to simplify.)

6275 = --------------------400 10 000

3

Write the answer.

251 = --------400

65 = -----------20 100 13 = -----20 b 1.2 6

12 = -------510 1 = 1 --5 c 0.6275 251

Converting decimal numbers to percentages To convert a decimal number to a percentage, we multiply the decimal number by 100 and include the % sign.

WORKED Example 11 Convert 0.357 to a percentage. THINK 1 2

Write the decimal number. Multiply the decimal number by 100 by moving the decimal point 2 places to the right. to include the percentage sign.

WRITE 0.357 = (0.357 × 100)% = 35.7%


15


1. To order decimal numbers, write each with the same number of decimal places and compare. 2. To write finite decimal numbers as fractions, make the denominator an appropriate multiple of 10 and simplify where possible. The number of zeros in the denominator should be the same as the number of digits after the decimal point. 3. To convert a decimal number to a percentage, multiply by 100 and include the percentage sign.

Decimal numbers c f i l

Math

5.6 × 7.04 (8.6 − 4.4) ÷ 7 Operations with 7.2 ÷ 0.12 × 6 decimal 6.2 + 3.5 × 2 numbers

cad

1 Calculate each of the following. a 6.56 + 3.214 b 4.87 − 2.493 d 5.75 ÷ 0.25 e (4.5 + 2.1) × 3.5 g 4.8 − 2.16 ÷ 0.18 h 3.2 × (6.4 + 0.78) j 7.2 ÷ (0.12 × 6) k 5.8 × (3.1 ÷ 0.4)

HEET

9

1.4

3 Write each of the following sets of decimal numbers in ascending order. a 0.66, 0.4, 0.71 b 2.3, 0.23, 23 c 0.7, 1.32, 1.04 d 1.02, 1.1, 1.22 e 0.5, 0.56, 0.06 f 0.323, 0.4, 0.35

HEET

1.5

SkillS

Example

1.3

SkillS

2 Calculate each of the following, rounding your answers to 2 decimal places. a 6.46 × 2.356 b 8.12 × 5.4 ÷ 9.6 c 8 ÷ 0.35 + 2.1 d (6.509 + 4.804) ÷ 0.341 e 3.2 × 4.057 − 13.91 ÷ 2.43 WORKED

HEET

SkillS

1D

4 Write each of the following sets of decimal numbers in descending order. a 0.24, 0.204, 0.2004 b 0.062, 0.081, 0.11 c 0.7, 0.77, 0.707 d 0.082, 0.09, 0.0802 e 1.2304, 1.23, 1.204 f 0.359, 0.39, 0.3592 5 multiple choice a The expression 6.43 × 2.356 ÷ (2.1 − 0.365) correct to 2 decimal places is equal to: A 0.36 B 6.85 C 87.31 D 8.73 E 6.84 b The false statement is: A 0.67 < 0.7 B 0.506 < 0.51 C 0.735 > 0.73 D 0.203 < 1.3 E 0.085 > 0.85 c The expression −0.9 + 6.5 × 0.004 − 1.2 ÷ 0.6 is equal to: A −1.074 B −2.874 C −2.64 D −20.874 E −0.84 d A good estimate for 5.2 × 0.2 + 1.18 ÷ 0.012 is: A 101 B 11 C 110 D 99.373 E 1010

WORKED

11

7 Convert each of the following to percentages. a 0.72 b 0.31 c 0.89 d 0.57 f 0.06 g 0.782 h 0.6175 i 0.0094 k 1.602 l 11 m 2.3 n 5.75

e 0.9 j 1.35 o 2.485

L Spread XCE

Converting decimals to percentages

HEET

1.6

SkillS

E

Converting decimals to fractions sheet

Example

L Spread XCE

HEET

SkillS

10

6 Convert each of the following to fractions in simplest form. a 0.9 b 0.6 c 0.16 d 0.27 e 0.78 f 0.15 g 0.08 h 1.5 i 2.84 j 0.125 k 0.484 l 0.963 m 0.775 n 0.0625 o 0.8875

sheet

Example

E

WORKED

1.7


16


8 multiple choice a In simplest form and as a fraction 0.3125 is equal to: 3125 5 A --------------B 31.25 C ----D 31 1--410 000 16 b As a percentage 0.0875 is equal to: 7 -% A 0.875% B 8.75% C 87.5% D ----80 c The number 0.656 25 is equal to: --------------------A 13 B 53 C 11 D 21 20 80 16 32

E

13 -----40

E 875% E

5 --8

9 Francis is paid $11.50 an hour for babysitting. If he works for 7 hours over the weekend, how much does he earn altogether?

QUEST

GE

S

EN

M AT H

10 Yvette babysits for 5 hours after school each Friday. She is paid $10 an hour. a How much does she earn each week? b If she banks $3.25 of the money each week, how much does she have left to spend?

CH

AL

L

1 Allison, Bhiba, Chris and Dinesh ordered one box of apples to share equally between them. However, no one was present when the box was 1 delivered. Allison arrived and took --4- of the apples. Later, Bhiba came 1 and took --3- of the apples left in the box. Then Chris came and did the same. Finally Dinesh arrived and took his rightful share of the remaining apples. If 9 apples remained in the box, how many apples were in the box originally? 2 Mitchell has mown 0.6 of the lawn. He still has 50 m2 of lawn to mow. What is the total area of the lawn? 3 A train 0.5 km long is travelling at a speed of 80 km/h. How long will it take the train to go completely through a tunnel which is 1.5 km long?


17 What type of creatur creature e is a KATYDID KATYDID and where where are are its ears? Chapter 1 Number skills

Answer the decimal questions to find the puzzle’s code. 3 – 4

=

= 8.6 – 4.9

as a decimal

=

= 2.8 + 3.6 = = 0.5 × 8.4 =

= 10% as a decimal =

= 0.3 + 0.4 =

= 93% as a decimal = = 5 × 0.3 =

=

5 – 2

=

23 –– 50

as a decimal

=

=

= 12.7 – 9.87

= 8.34 – 6.54

7 –– 20

= 6.3 ÷ 0.63 =

= 1.2 – 0.8 =

=


= 0.67 + 0.53


= 12 ÷ 0.5 =

= 1.1 × 0.8 =

= 0.2 × 20 = = 0.87 + 1.33

=

= 1.64 ÷ 0.4 =

= 5.26 + 1.87 =


=

1.6 – 0.95

= 6 × 0.8 =

=

17 –– 4

=

=

4 – 5

=

as a decimal

= 4.5 × 1.2 =

as a decimal

=

2.374 + 3.926

as a decimal

=

7.63 – 3.23

3 – 8

as a decimal

=

5.4

2.5

4.0

3.7

1.8 0.46 2.2 0.375 4.8

4.1 0.93 0.75 7.13 0.88 0.6

0.1 0.51 4.2

0.22

24

1.5 0.65

0.4 10

0.8 0.7

6.3

6.4

4.4

1.2 4.25 2.83 0.35


18


Fractions There are many essential skills that you will need with fractions. You can review them in the exercise below and by the matching SkillSHEET. You should be able to simplify fractions and convert between improper fractions and mixed numbers. You should also be able to use your calculator efficiently.

an answer Graphics Calculator tip! Obtaining expressed as a fraction

▼

▼

As with any calculation involving fractions, if you wish to have an answer expressed as a fraction then each calculation needs to end by pressing MATH , selecting 1: Frac and pressing ENTER . ------ on your graphics calcuFor example, to simplify 28 44 lator, enter 28 ÷ 44 then press MATH choose option 1: Frac, then press ENTER . This can be seen in the screen at right. Note: The graphics calculator gives all answers as improper fractions and will not give answers as mixed numbers. It is important that we know how to perform calculations using fractions both with and without a calculator. Without a calculator, we would simplify

28 -----44

by dividing both the numerator and the

denominator by the highest common factor (HCF) of both. The HCF of 28 and 44 is 4. 7

28 28 ------ = --------11 44 44 7 = -----11

WORKED Example 12 Evaluate the following. a 3--4- + 5--6b

3 --4

×

5 --6

c 2 1--4- ÷

3 --5

THINK

WRITE

a

a

b

+

5 --6

Write both fractions with the same denominator by using equivalent fractions.

=

9 -----12

Add the fractions and simplify the answer by writing it as a mixed number.

=

19 -----12

1

Write the fraction calculation.

2 3

1

Write the fraction calculation and cancel where applicable.

2

Multiply numerators and multiply denominators.

3 --4

+

7 = 1 ----12

b

31 ----4

× =

5 ----62 5 --8

10 -----12



THINK

WRITE

c

c 2 1--4- ÷

19

3 --5

1

Write the fraction calculation.

2

Change the mixed number to an improper fraction.

=

9 --4

3

Times and tip, (change the division sign to a multiplication sign and tip the second fraction) and cancel.

=

93 ----4

4

Multiply numerators and multiply denominators; then simplify the answer by writing the fraction as a mixed number.

=

15 -----4

÷

3 --5

×

5 ----31

= 3 3--4-

Graphics Calculator tip! Fraction calculations

▼

▼

▼

To perform the calculations in worked example 12 on a graphics calculator, the following steps need to be followed: (a) Enter 3 ÷ 4 + 5 ÷ 6, press MATH , choose 1: Frac then press ENTER . The result is given ------ . The graphics calculator gives all answers as as 19 12 improper fractions. (b) Enter 3 ÷ 4 × 5 ÷ 6, press MATH , choose 1: Frac then press ENTER . (c) Enter (2 + 1 ÷ 4) ÷ (3 ÷ 5), press MATH , choose 1: Frac then press ENTER .

WORKED Example 13

Find

3 --7

of 98.

THINK 1 Write the calculation. 2 3

WRITE 3 --- of 98 7

Change the ‘of’ to ×, write the whole number over 1 and cancel if applicable. Multiply numerators and multiply denominators.

=

3 ----71

×

98 14 ----------1

= 42

Writing fractions with the same denominator allows us to compare the size of fractions.

WORKED Example 14

Write the fractions 2--3- , 8--9- ,

5 --6

in ascending order.

THINK 1 Write the fractions. 2 3

Write all fractions as equivalent fractions by finding the lowest common denominator, in this case 18. Rewrite the original fractions in the correct order.

WRITE 2 8 5 --- , --- , --3 9 6 =

12 16 15 ------ , ------ , -----18 18 18

= 2--3- , 5--6- ,

8 --9


20


Another way of writing fractions in order is to convert each fraction to a decimal number before comparing them.

Converting fractions to decimal numbers To convert a fraction to a decimal number, divide the numerator by the denominator.

WORKED Example 15

Convert

7 --8

to a decimal number.

THINK 1 Write the fraction.

WRITE 7 --8

2

Divide the numerator by the denominator.

0.875 8 ) 7.000

3

Write the fraction and the equivalent decimal number.

7 --8

= 0.875

Converting fractions to percentages To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign.

WORKED Example 16

Convert

23 -----40

to a percentage.

THINK 1 Write the fraction.

WRITE 23 -----40 23 - × = ( ------2

2

Multiply by 100, include the percentage sign and cancel if applicable.

3

Multiply the numerators and multiply the denominators.

=

4

Simplify by writing as a mixed number.

= 57 1--2- %

40

100 5 ----------1

)%

115 --------- % 2

1. To write fractions in simplest form, divide the numerator and the denominator by the highest common factor (HCF) of both. 2. To change improper fractions to mixed numbers, divide the numerator by the denominator and express the remainder as a fraction in simplest form. 3. To change a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator and write the result over the denominator. 4. To add or subtract fractions, form equivalent fractions with the same denominator, then add or subtract the numerators. 5. To multiply fractions, cancel if possible, then multiply the numerators, multiply the denominators and simplify if appropriate. 6. To divide fractions, times and tip, then simplify if possible. 7. To add, subtract, multiply or divide mixed numbers, change the mixed numbers to improper fractions first. (When subtracting, an alternative method is to make the second fraction into a whole number after writing the fractions with the same denominator.) 8. To write fractions in order, express them as equivalent fractions and compare. 9. To find a fraction of an amount, multiply the fraction by the amount. 10. To convert a fraction to a decimal number, divide the numerator by the denominator. 11. To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign.


21


1E

Fractions

1 Write the following fractions in simplest form.

Simplifying fractions

8 -----12

b

24 -----30

c

14 -----28

d

72 -----81

e

45 -----50

f

35 -----49

g

24 -----64

h

14 -----22

i

21 -----36

j

36 --------108

k

108 --------144

l

75 --------500

m

16 -----20

n

25 --------100

o

33 -----99

HEET

2 Convert the following to mixed numbers in simplest form. 22 -----5

b

31 -----7

c

49 -----4

d

37 -----6

e

21 -----9

f

68 -----16

g

55 -----20

h

80 -----15

i

98 -----10

j

94 -----12

HEET

3 Convert the following mixed numbers to improper fractions.

Example

h 1 5--8-

a

1 --4

+

3 --5

b

2 --3

e

5 --7

+

2 --5

f

9 -----10

i

7 --8

−

5 --9

j

2 1--4- + 3 1--3-

+

1 -----10

Example

13

WORKED

Example

14

WORKED

−

5 -----12

c

2 --7

g

7 -----10

×

3 --4

×

11 -----14

9 k 5 2--5- − 4 ----10

n (4 1--4- + 3 3--5- ) ×

o − 1--6- −

4 --5

2 --3

d

8 -----15

h

5 --8

÷ ÷

GC pro

5 --6

Fractions

5 --6

p

1 --2

1.12

2 --5

l 3 7--8- × 1 1--6-

+

HEET

HEET

× − 1--3- ÷ − 1--4- ÷

1.13

1 --5

5 Find the following. a

5 --8

of 72

b

3 --4

f

4 --9

of 117

g

7 -----10

of 28 of 150

c

5 --6

of 36

d

2 --3

h

1 --7

of 98

i

11 -----12

of 81

e

1 --5

of 192

j

3 -----16

of 65 of 480

6 Write each of the following sets of fractions in ascending order. a

1 1 3 --- , --- , --4 2 8

b

3 1 7 ------ , --- , -----10 3 20

c

1 3 1 --- , ------ , --6 20 5

d

7 2 13 1 ------ , --- , ------ , --10 3 20 2

e

2 7 11 19 --- , ------ , ------ , -----5 20 25 50

f

7 -, 1 1--4- , 1 5--6- , 1 ----12

g

7 3 1 2 ------ , ------ , --- , --30 15 3 5

h

5 5 5 ------ , ------ , -----18 19 17

i

− 1--8- , − 1--5- ,

j

31 79 ------ , − ------ , − --------− 19 25 40 100

k

1 ------ , 10

l

− 2--3- ,

− 1--9- , 1--8- , − 1--7-

d

g

5 -----16

h

3 -----80

k

5 -----32

l

1 1--2-

c

f

7 --------100

j

141 --------200

b

e

31 -----40

i

3 -----10

a

− 3--4- ,

4 --5

GC pro 14 -----25

9 -----20

4 --5

3 --4

1 --4

7 2 ------ , --- , 10 3

7 Convert each of the following fractions to a decimal number.

11 -----16

Converting fractions to decimals

L Spread XCE

Converting fractions to decimals

sheet

15

5 -----12

gram

Example

−

1.11

-----3 14 17

j

4 Evaluate the following.

m 6 3--4- ÷ 3 1--2WORKED

-----5 13 20

i

HEET

9 e 3 ----10

SkillS

9 g 2 ----11

d 5 5--6-

gram

12

9 1--3-

E

WORKED

5 6--7-

c

SkillS

f

b 4 4--5-

1.10

SkillS

a 3 3--4-

1.9

SkillS

a

1.8

SkillS

a

HEET

SkillS

cad

Math


22 EXCE

et

reads L Sp he

Converting fractions to percentages

WORKED

Example

16

8 Convert each of the following fractions to a percentage. a

1 --4

b

1 --5

c

3 --8

d

7 -----16

e

79 --------100

f

18 -----25

g

59 -----80

h

11 -----20

i

5 --6

j

7 --9

k

5 --7

l

4 -----11

Mat

d hca


Converting fractions to decimals or percentages

9 multiple choice a The fraction A

112 --------192

9 -----16

is equal to: B

25 -----48

5 - − b The expression (− ----11 575 A − ----------9504

c

C 5 ------ ) 12

25 B − ----------9504

5 --8

A 0.58 e The fraction A 0.6%

5 --8

E

7 -----12

317 C −1 ----------3168

C

5 --8

<

4 --7

857 D −1 ----------3168

D

4 --5

>

3 --4

E 0 E

5 --6

>

5 --7

as a decimal number is: B 1.6

2 --3

D

× ( 5--8- − 5--9- ) is equal to:

The false statement below is: 3 A 1--5- > ----B 2--3- < 3--420

d The fraction

13 -----24

C 0.625

D 62 1--2-

E 5.8

D 66 2--3- %

E 67%

as a percentage is: B 66.67%

5 -% C 66 ----11

10 Easisell High had 100 boxes of lollies to sell to raise money for the Children’s Hospital. If Year 9 sold 1--2- of the boxes and Year 10 sold 1--5- of what was left: a how many did each year level sell? b how many boxes were left for the other year levels to sell? c what fraction of the total was left for the other year levels to sell? 11 Alexa made a cake for the family to share. As soon as it was iced, Mum and Alexa each ate 1--6- of it, Dad ate 1--4- , and Freddi and Elliot 1 - each. What ate ----12 fraction of the cake was left for the next day?



23

Percentages In this section, we will be converting percentages to decimal numbers and to simple fractions. We will be finding the percentage of an amount, expressing one amount as a percentage of another, finding the full amount given some other amount as a percentage of it, and increasing and decreasing amounts by a given percentage.

Converting percentages to decimal numbers To convert a percentage to a decimal number, divide the percentage by 100. % means ‘out of 100’.

WORKED Example 17 Convert 56.25% to a decimal number. THINK 1 2

WRITE

Write the percentage. Divide the percentage by 100. The number will be smaller so change the position of the decimal point 2 places to the left.

56.25% = 0.5625

Converting percentages to fractions in simplest form To convert a percentage to a fraction in simplest form, place the percentage over 100 and simplify where appropriate. If the percentage includes a fraction, change to an improper fraction then divide the percentage by 100 and simplify where appropriate.

WORKED Example 18 2

Convert 22 --9- % to a fraction in simplest form. THINK 1 2

3

WRITE

Write the percentage. Change the percentage to an improper fraction and divide by 100. Simplify.

22 2--9- % =

200 --------9

÷ 100

=

200 --------9

×

=

2 --9

1 --------100

Finding a percentage of an amount To find a percentage of an amount, change the percentage to a fraction, the ‘of’ to × and perform the operation.


24


WORKED Example 19 Find 34% of 950. THINK 1

Write the calculation.

2

Change the percentage to a fraction, the ‘of’ to × and perform the operation.

WRITE 34% of 950 =

34 ----------100 2

×

950 19 -------------1

= 17 × 19 = 323

Expressing one amount as a percentage of another To express one amount as a percentage of another is the same as to convert a fraction to a percentage. Write a fraction with the first amount as the numerator and the second amount as the denominator, then multiply the fraction by 100.

WORKED Example 20 Write 2.4 as a percentage of 12.8. THINK 1

2

Write a fraction with the first amount as the numerator and the second amount as the denominator. Change the fraction to a percentage by multiplying by 100 and including the %.

Simplify. Note: The answer could also be left as a fraction. 3

WRITE 2.4 ---------12.8

2.4 - × = ( --------12.8

100 --------1

)%

= 18.75%

Finding the full amount, given a percentage of it If Fred knows that his cheque for $50 000 was 25% of his uncle’s estate, can he work out the value of the estate? To do this, first calculate 1% of the total amount, then multiply by 100 to find 100% of the amount. This is known as the unitary method of percentages.

WORKED Example 21 Find the number, if 62% of the number is 186. THINK 1 Write the given information. 2 Find 1% by dividing both the percentage and the number by the percentage. (That is, ÷ 62.) 3 Find 100%, or the whole amount, by multiplying the percentage and the number by 100. 4 Write the answer in a sentence to show what is meant.

WRITE 62% is 186. 1% is 3. 100% is 300. 62% of 300 is 186.



25

Increasing and decreasing by a given percentage To increase an amount by a given percentage, add the increase to 100% to find a new percentage and then find the new percentage of the original amount. Similarly, to decrease an amount by a percentage, subtract the decrease from 100% and find the new percentage of the original amount.

WORKED Example 22 Increase 300 by 17%. THINK 1 2

3

4 5

WRITE

Add the increase to 100% to find the new percentage of 300. Write the calculation using the new percentage which is greater than 100% because it is an increase. Write the percentage as a fraction out of 100, multiply by the amount and cancel if appropriate. Simplify. Write a sentence.

(17 + 100)% = 117% 117% of 300 117 300 3 = -----------1 × ----------1 100 = 351 If 300 is increased by 17%, it becomes 351.

Alternatively, the percentage increase could be found and added to the original amount. To decrease an amount by a given percentage, subtract the decrease from 100% to find a new percentage and then find the new percentage of the original amount. For example, to decrease 300 by 17% is to find 83% of 300. The answer must be less than the original amount because it has been decreased.

1. To convert a percentage to a decimal number, divide by 100. 2. To convert a percentage to a fraction in simplest form, divide by 100 or write the percentage as a fraction out of 100, then simplify. 3. To find a percentage of an amount, divide the percent by 100 and multiply by the amount. 4. To express one amount as a percentage of another, divide the first amount by the second and multiply by 100. 5. If we need to find an amount, given the percentage of that amount, find 1% and multiply the result by 100. 6. To increase an amount by a given percentage, add the percentage to 100% and find the resulting percentage of the amount. 7. To decrease an amount by a given percentage, subtract the percentage from 100% and find the resulting percentage of the amount.


26


1F 1.14

SkillS

HEET

WORKED

Example

17

d hca

Mat

WORKED

Example

18

Percentages

1 Convert each of the following percentages to a decimal number. a 62% b 41% c 38% d 93% e 10% f 2% g 36.7% h 21.25% i 250% j 315.7% k 800% l 0.6% 2 Convert each of the following percentages to a fraction in simplest form. a 97% b 42% c 40% d 70% e 55% f 50% g 25% h 30% i

EXCE

et

reads L Sp he

Converting percent- WORKED ages to Example 19 fractions and decimals EXCE

Finding the percentage WORKED Example of an 20 amount et

reads L Sp he

EXCE

62 1--2- %

9 -% m 81 ----11

One amount as a percentage of another

8 1--3- %

33 1--3- %

k 47 1--2- %

l

n 28 4--7- %

o 44 4--9- %

p 16 2--3- %

j

3 Find the following. a 71% of 8 d 13% of 54

b 65% of 320 e 83% of 27

c f

52% of 1700 24% of 175

g 12.5% of 104.48

h 42.5% of 55

i

58 1--3- % of 15.6

88 8--9- % of 3.69

k 23 1--2- % of 150

l

33 1--3- % of 300

et

reads L Sp he

Percentages

j

4 Write each of the following number where appropriate. a 45 out of 60 d 32 out of 50 g 21 out of 48 j 0.63 out of 1.25

as a percentage, giving your answer as an exact decimal b e h k

c f i l

27 out of 100 37.5 out of 60 9.6 out of 15 15.5 out of 60

6 out of 20 0.3 out of 12 18 out of 25 62.8 out of 80

5 multiple choice a The percentage 123.5% as a decimal number is: A 1.235 B 12.35 C 0.1235 D 12 350 b As a fraction in simplest form 67 1--4- % is: A c

27 -----40

B

11 -----16

E 123.5 E

2 --3

D 61%

E

49 -----80

C 10.44

D 10.5

E 10.4

C 58.3%

D 21%

E 25%

C

53 -----80

D

29 -----50

269 --------400

Which of the following is largest? A 57%

B 0.6

C

d What is 23 1--5- % of 45? A 10.4625 B 10.575 e What percentage of 60 is 35? A 58 1--3- % WORKED

Example

21

B 171 3--7- %

6 Find the number in each of the following examples. a 12% of the number is 156 b 23% of the number is 368 c 15% of the number is 690 d 82% of the number is 328 e 16% of the number is 1.44 f 120% of the number is 5.4 g 13% of the number is 32.5 h 68% of the number is 138.72 i 2.5% of the number is 22.5 j 31 1--4- % of the number is 37.5


22

7 Increase each of the following numbers by the given percentage. a 45 by 15% b 5800 by 42% c 65 by 20% d 72 by 70% e 106 by 53% f 670 by 3% g 880 by 62 1--2- % h 2.5 by 27% i 84 by 41 2--3- % 8 Decrease each of the following numbers by the given percentage. a 45 by 15% b 76 by 35% c 120 by 40% d 2722 by 53% e 6530 by 30% f 104 by 7% g 1.2 by 11% h 640 by 42 1--2- % i 96 by 16 2--3- %

L Spread XCE

sheet

Example

Increasing or decreasing by a percentage

time

9 multiple choice GAME a If 20% of a number is 80, what is the number? A 400 B 16 C 25 D 96 E 64 Number skills — 002 b If 480 is decreased by 27 1--2- %, the result is: A 612 B 132 C 348 D 72.5 E 1745 c If 60 is increased by 15%, by what percentage does the result have to be decreased to obtain 54? 6 ------ % -% A 15% B 21 17 C 25%% D 78 ----E 20% 23 23 10 If Fred knows that his cheque for $50 000 was 25% of his uncle’s estate, what was the value of his uncle’s estate? 11 The Sunflower Clothing Store was having a 15% off sale. If Sam wanted to buy a new pair of jeans, how much would they cost if the original price was $75?

ET SHE

Work

WORKED

27 E


1.2


28


Career profile GREG McINTYRE — Production Manager

Name: Greg McIntyre Profession: Production Manager Qualifications: Certificate of Business Studies (Advertising) Employer: Idea Communications I decided to undertake the Business Studies (Advertising) course because I was interested in advertising and commercial art. It was also recommended to me by my high school art teacher. At Idea Communications I oversee all work produced in the art studio and by outside suppliers. I check that production specifications are complied with, that budgets are adhered to and deadlines are met. All new jobs require ‘opening’ which entails filling in production brief forms, estimating costs and preparing schedules. Quotations are obtained for all outside supplier costs. A choice is made, but not always because of cost, and orders are issued. A lot of the work I do involves calculations with fractions, decimals and percentages. For example: 1. Estimates involve calculating the number of hours each task or job will involve, multiplying by the hourly rate for staff required, adding outside costs (printer,

photographers and so on) and adding GST. 2. Schedules are produced by working backwards from the delivery deadline. I determine the sequence of tasks, how long each task will take and the studio required. All staff time sheets must correlate with the task and the time allocated. Work is often ‘juggled’, depending on priority, to ensure that deadlines are met. 3. Invoice approvals are checked against purchase orders and take into unquoted items such as freight, tax and corrections. It seems to me that mathematical skills are vital to a productive enterprise of any kind. Computers and calculators are a useful aid, but without an understanding of the principles, not much can be achieved. Questions 1. What does ‘opening’ a new job entail at Idea Communications? 2. Why is it more beneficial for Greg to ‘work backwards from the delivery deadline?’ 3. What other courses offer an advertising certificate or degree?



29

Interesting times Candice has the choice of investing $1000 at 6 2--3- % p.a. or at 6.65% p.a. 1 Write 6 2--3- % as a decimal number. 2 Write 6.65% as a decimal number. 3 Which of the two interest rates is greater? 4 Using the greater interest rate, how much interest would Candice earn in 1 year? 5 If she kept the money (including the interest) in the bank for 3 years, how much would she have at the end of the 3 years? (Hint: It is not $1200.00.)

2 1 Evaluate −3 − 4 ÷ −2 × (6 − 9). ------ . 2 Evaluate 4--7- + 1--4- − 11 14

15 5 ------ × ------ ÷ --- . Evaluate 11 20 22 6 Round 0.003 950 01 to 4 significant figures. Convert 67% to a decimal. ------ as a percentage. Express 17 20 Write 0.62 as a simple fraction. Evaluate 49.312 − 183.8 + 701.6511. Change 5 6--7- to an improper fraction. Decrease $349 by 83%.

3 4 5 6 7 8 9 10

Index notation, square roots and higher order roots Index notation An index is a power to which a number is raised. It is the number of times that the base number is multiplied together. In the case of 24, 2 is the base and 4 is the index. On a calculator this is calculated by using the ^ function or the xy function. The keys to press on a graphics calculator to calculate 24 are 2^4 ENTER.

WORKED Example 23 3

Express  4--9- as a fraction in simplest form. THINK

WRITE 4 ---  9

3

1


2

Remove the brackets (optional).

=

3

Simplify both numerator and denominator.

=

43 ----93 64 --------729


30


fractions Graphics Calculator tip! Simplifying raised to a power

▼

Fractions need to be entered using the ÷ key. However, to ensure that both the numerator and the denominator are raised to the given power, enter the fraction with brackets. If you wish the answer to be expressed as a fraction, to press MATH and select 1: Frac before pressing ENTER . The screen at right shows the calculation for worked example 23.

Square roots The square root of a number is a value which, when multiplied by itself, gives the original number. For example 64 = 8 because 8 × 8 = 64. Taking the square root of a number is the opposite operation to squaring a number. That is, 82 = 64 and 64 = 8. Generally, on a scientific calculator you would key. On a graphics calculator, enter 64 then the you would press 2nd [ ] then 64 and ENTER .

WORKED Example 24 Evaluate 458 and write the answer, correct to 2 decimal places. THINK 1 2 3

WRITE

Write the given square root. Use a calculator to find the square root. Round the answer as required.

458 ≈ 21.40

Higher order roots The cube root of a given number is a value, which when written 3 times and multiplied, is equal to the given number. For example, 3 27 = 3 because 3 × 3 × 3 = 27. The fourth root of a given number is a value which, when written 4 times and multiplied, is equal to the given number. For example, 4 625 = 5 because 5 × 5 × 5 × 5 = 625. On a scientific calculator this is done by using the

x

1 ---

or x y key.

cube roots Graphics Calculator tip! Finding and higher order roots To find the cube root of a number, press MATH , select 4: 3 (, enter the number under the root sign and press ENTER . The screen to the right shows the calculations for the following cube roots: 3 27 , 3 2197 , 3 12.56 .



31

To find a higher order root, press the number of the root required (4 for fourth root, 5 for fifth root and so on) then MATH , select option 5: x , enter the number under the root sign and press ENTER . (Note: This option can also be used with square roots and cube roots.) The screen at right shows the calculations for the following higher order roots: 4 625 , 5 781 , 6 42 .

WORKED Example 25 Calculate

5

649 correct to 3 decimal places.

THINK 1 2 3

WRITE

Write the given root term. Use a calculator to find the answer. Write the answer correct to 3 decimal places.

5

649 = 3.651

The square root of a fraction can be evaluated by finding the square root of both the numerator and the denominator. This also applies to higher order roots.

WORKED Example 26 Express

36 -----49

as a fraction.

THINK

WRITE 36 -----49

1

Write the given square root.

2

Rewrite as the square root of both the numerator and the denominator.

=

36 ---------49

3

Evaluate, keeping the answer in fraction form.

=

6 --7

On a graphics calculator,

36 -----49

would be entered as ▼

] 36 ÷ 49). Press MATH and select 1: Frac (to express your answer as a fraction) then press 2nd [

ENTER .

1. Use a calculator to evaluate numbers with indices and to find square roots and higher order roots. 2. If the number is a fraction, calculate the numerator and denominator separately.


32


Index notation, square roots and higher order roots

1G 1 Calculate the following. a 27 b 35 2 f 1.7 g 2.54

1.15

SkillS

HEET

WORKED

Example

23

reads L Sp he

EXCE

et

Mat

d 44 i 3.052

WORKED

Example

24 Index notation, square roots and higher order WORKED roots DIY Example 25

WORKED

Example

26

2 Express the following as fractions in simplest form. a

4 ---  5

2

b

2 ---  3

3

c

1 ---  2

6

d

3 4  ---- 10

f

8 ---  9

5

g

1 ---  2

7

h

3 ---  4

3

i

6 ---  7

4

EXCE

et

Work

ET SHE

1.3

7 2  ---- 12

j

3 ---  5

5

441

b

0.09

c

81

d

1.44

e

2116

f

0.0529

g

676

h

132.25

i

0.0576

j

7.29

4 Evaluate the following, correct to 2 decimal places. a

465

b

65.87

c

2354

d

0.986

e

19.9

f

8624

g

1.75

h

56.78

i

21.45

j

5.6

5 Calculate the following. a 3 13.824 b 5 5.378 24 c 4 70.7281 6 Calculate the following, correct to 3 decimal places.

d

6

729

a 8 46 b 3 869 c 5 149.0642 7 Express the following as fractions in simplest form.

d

7

8975

a

1 -----81

b

9 -----16

c

121 --------169

d

4 --------121

e

289 --------729

f

0.36 ---------9.61

g

0.25 ---------1.44

h

0.49 ---------2.25

c

0.84 × 1.23

9 multiple choice a To 3 significant figures, 3 4583 is: A 67.7 B 16.617 C 67.698 D 17.0 b Rounded to 3 decimal places, (1.2)5 is equal to: A 2.488 B 2.49 C 1.037 D 1.04

Scientific notation DIY

e

a

8 Calculate the following: b 6.12 – 2.13 a 2 4 + 53

reads L Sp he

e 53 j 0.83

3 Evaluate the following.

Square roots DIY d hca

c 106 h 3.13

10 A large number can be expressed in the form 3.56 × 104. What is the number? 11 The planet Jupiter is approximately 7.78 × 108 km from the sun. Write the number of kilometres as a whole number.

d 64 ÷ 43

E 16.6 E 6



33

Calculator computations A calculator can be used to compute more difficult examples. Generally we enter the calculation as it is seen; however, we have to to bracket everything which is under a square root sign as well as bracketing the numerator and denominator in a fractional calculation.

WORKED Example 27 Calculate the following, expressing the answers correct to 3 decimal places. 3.5 4 – 9.8 3 a 6.5 2 + 1.7 3 b ----------------------------10.7 2 – 53 THINK

WRITE

a

a

b

1


2

Perform the calculation using either a scientific or graphics calculator. (On a graphics calculator, press 2nd [ ] 6.5 ^ 2 + 1.7 ^ 3), then ENTER .)

3

Write your answer correct to 3 decimal places.

1


2

Perform the calculation using a calculator. (On a graphics calculator, press (3.5 ^ 4 − 9.8 ^ 3) ÷ 2nd [ 10.7 ^ 2 − 53) then ENTER .)

3

6.5 2 + 1.7 3

= 6.868 3.5 4 – 9.8 3 b ----------------------------10.7 2 – 53

]

Write your answer correct to 3 decimal places.

= −100.889

1. Use a calculator to evaluate more difficult examples. 2. Rounding to a given number of significant figures begins at the first non-zero digit. 3. Rounding to a given number of decimal places begins with the first digit after the decimal point.


34


1H Mat

d hca

WORKED

Example

27

Calculator computations DIY

Calculator computations

1 Calculate the following, expressing the answers correct to 3 decimal places. a d

516 – 204 3

65 – 97 46.7 g ---------------------21 – 18.6 j

−6.53 + 2.66

65 + 35 m -----------------72 – 98

b

516 – 204

c

e

9.6 × 4.1 + 6.8 5.9 – 2.4 --------------------3.7 –2.7 – 3.9 --------------------------– 4.6 × – 3.2

f

h k n

75 + 9.2 -----------------------61 – 3.7

i l

516 – 204 3

7.8 2 1 1 ------- + ------9.7 3.4 1 ---------------------59 + 75

416 – 324 o ---------------------------5.8 + 7.2

2 Calculate the following, correct to 4 significant figures. 1 a -----------------47 – 29

b

1.28 2 + 3.15

e

25.8 2 – 4.1 2 g -------------------------------5.9 3 – 6.4 4

h

d

56 + 99 -----------------28 + 11 12 2 + 8 2 – 7 2 ------------------------------2 × 12 × 6 963 150 – 29.3 ------------------------ × ------------2.7 4.1 2

c

5.32 − 4.43

f

4.6 4 – 2.1 5 ------------------------3.4 3 – 1.9 5

i

96.5 + 67.5 -----------------------------( 5.1 – 2.8 ) 2

3 multiple choice a What is 3 49.6 – 5.6 × – 3.1, correct to 4 significant figures, equal to? A 24.40 B −5.026 C 21.03 D −13.69 E 22.65 3 b What is ( 5.1 – 2.78 ) ( 1.6 + 1.05 ) , correct to 2 decimal places, equal to? A 17.66 B −0.68 C 7.84 D 5.15 E 20.08

World population Year 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

World population 2 555 078 074 3 039 332 401 3 707 610 112 4 456 705 217 5 283 755 345 6 080 141 683 6 823 634 553 7 518 010 600 8 140 344 240 8 668 391 454 9 104 205 830

At the start of the chapter we looked at estimates of the world population over a 100-year period. 1 Calculate the increase in population for each 10-year period. 2 Which 10-year period has the largest increase in population? 3 What is the predicted increase in population from 1950 to 2050? 4 Round each of the populations given in the table to 2 significant figures. 5 Use your answers to part 4 to calculate the percentage increase in population for each 10-year period. 6 Which 10-year period has the highest percentage increase in population? 7 What is the predicted percentage increase in population from 1950 to 1960? 8 Write a few sentences describing your results. How does the increase in world population affect the environment? 9 Parts 5, 6 and 7 used rounded results to calculate the percentage increase. Design a spreadsheet to perform these calculations with the original population figures.



35

Which piece of Australian curr currency featur eatures es scenes from from aviation aviation history? history? Use a calculator to answer the questions to find the puzzle’s code. =

=

2.37 5.06 + 8.98

=

6.2 + 3.05 + 1.79 + 8.43 + 0.95 + 3.781 + 2.62

= =

2.68 x 5.45

=

= =

=

3.672 x 4.85

=

84.7 less 57.359

= =

15.73 – 8.218

the mean of 21.8, 15.24, 7.62 and 31.8

38% x 35.4

= the average of 2.7, 3.6, 9.4 and 3.8

=

21.3 2.5

=

=

=

= 16.48 ÷ 4

=

=

18.9 3.2

= =

the product of 2.75 and 3.84

= =

= =

=

21% of 126.4

=

12.4 ÷ 1.6

=

= the mean of 6.78, 13.24, 8.91 and 3.74

72% of 35.8

=

= =

=

= =

=

18.237 – 4.87

=

= the sum of 8.34, 9.27, 5.416, 18.203 and 1.94

=

10.34 multiplied by 1.265

=

=

18.9 ÷ 2.4

=

5.28 x 3.65

=

=

2.3846 5.078 + 4.9424

the sum of 8.73, 9.21, 2.643, 8.4, 2.976 and 1.425

=

=

= the difference between 7.054 and 21.9

34.293 less 17.84

the average of 16.74, 13.29 and 4.89

=

=

the sum of 3.74, 8.94, 0.87, 2.64, 3.17 and 1.43

=

=

= =

103.8 x 0.345

=

81% of 25.8

24% of 158.6

=

=

=

7.512

14.846 25.776

20.898 4.875

16.41

13.367 14.606

8.52 20.79 17.8092

13.452

8.1675 26.544

33.384 38.064

11.64

10.56

27.341 26.821 16.453 5.90625

12.405

35.811 13.0801

43.169 19.115

19.272 7.875

7.75

4.12


36


Applications The skills learnt in this chapter can now be applied to problems relating to real-life situations. In this part the problems will be simplified to basic mathematical expressions so we can use mathematical skills to determine the answers.

WORKED Example 28 The temperature at Mt Buller drops steadily at night by 1.8°C per hour. The temperature at 6 pm is 4.5°C. What is the temperature at 2 am?

THINK 1 Find the number of hours between 6 pm and 2 am. 2 Write a mathematical expression for the total drop in temperature. 3 Write a mathematical expression for the temperature drop from 4.5°C. 4 Use the order of operations to solve the problem. Write the answer in a sentence. 5

WRITE Hours between 6 pm and 2 am: 6 + 2 = 8 Temperature drop: 8 × 1.8 Temperature at 2 am: 4.5 − 8 × 1.8 = 4.5 − 14.4 = −9.9 The temperature at 2 am is −9.9°C.

WORKED Example 29 In a school of 460 students, half of them buy their lunch from the canteen, while 2--5- bring lunch from home. The rest do not eat lunch. How many students do not eat lunch? THINK

WRITE

1

Find the number of students who buy lunch. Students who buy lunch:

2

Find the number of students who bring their lunch. Find the number who eat lunch by adding these amounts. Find the number who do not eat lunch by subtracting these amounts from the total number of students. Write the answer in a sentence.

3 4

5

Students who bring lunch: Students who eat lunch:

1 --2 2 --5

× ×

460 --------1 460 --------1

= 230 = 184

230 + 184 = 414

Students who don’t eat lunch: 460 – 414 = 46

The number of students who do not eat lunch is 46.



37

WORKED Example 30 Tessa’s wage is increased by 3.2%. If her old wage is $675 per week, what is her new wage? THINK

WRITE

2

Find the new percentage by adding the percentage increase to 100%, which is the original wage. Find the new wage.

3

Answer the question in a sentence.

1

100% + 3.2% = 103.2%

New wage: 103.2% of 675 = 696.60 The new wage is $696.60.

1. 2. 3. 4. 5.

Read the question carefully. Highlight or underline important information. Write a mathematical expression or calculation for the given situation. Use mathematical skills to evaluate the expression. Write a sentence to answer the question.

1I

WORKED

Example

28

1 a

Applications

The temperature in Young drops steadily by 1.7°C per hour. The temperature at 5 pm is 8°C. What is the temperature at 2 am? b The temperature in Doblin at 4 am is −5°C. The temperature rises steadily by 2.6°C per hour. What is the temperature at 10 am? c The winter temperature at Dubbo drops from 13°C at 3 pm to −6°C at 4 am. How many degrees does the temperature drop? d If the minimum temperature in Canberra on Monday is −7°C and it rises by 16°C to the maximum temperature, what is the maximum temperature that day? e The temperature range on Tuesday in Milday was 24°C. If the maximum temperature was 10°C, what was the minimum temperature?


2 a Roger wins $150 on a poker machine but then loses $340 the rest of the night. How much worse off is he than when he started? b A bank statement shows a balance of −$53.76. Fran deposits $156.80. What is the new balance?

3 a What is the cost of 36 litres of petrol at 71.9c per litre? b Potatoes cost $1.80 per kilogram. How much will 8 kilograms of potatoes cost? c Donna buys 5 exercise books at $1.35 each and 4 pens at 45c each. What is her change from $10? d Tony is paid $15.60 per hour. How long must Tony work in order to earn $140.40? e Steve earns a salary of $41 000 per annum. What is his weekly salary if 1 year is 52.143 weeks? f Janice pays for the following amounts of petrol on a trip from Sydney to Melbourne: 23 L at 81.9c per L, 36 L at 85.8c per L, 31 L at 90.5c per L. What was Janice’s total petrol bill for the journey? g A certain body is 68% fluid. The body’s volume is 5.8 L. How much of this body is fluid? Give the answer in millilitres. 3 - of the students choose soccer, 4 A school has 570 students. For sport ----10 ball and the rest play tennis. How many play tennis? 29 5 a Yan has a choice of 3--5- of $40 or 2--3- of $39. Which choice would give him more money? b Of the 1881 people who live in Galaxy, 4--9- are women. How many of the inhabitants are men? c At a factory, 1 out of 150 light bulbs are faulty. If the factory makes 1200 light bulbs, how many are faulty? d Teri is paid $480 per week. 9 2 - goes to rent, --- to Of this, ----5 1 20 - to other essentials food, ----10 and the rest is saved. How much per week does Teri save? e In a biscuit mixture, 3--8- is sugar. If there are 120 grams of sugar in the mixture; how many grams of mixture is there?

WORKED

Example

WORKED

Example

30

1 --3

choose foot-

6 Jay earns $12.50 per hour and works a 35-hour week. He obtains a 4.5% pay increase. What is his new weekly wage?



39

7 a In a township, 68% of the people go to church. There are 5500 people in the town. How many go to church? b In an election, 43% vote Liberal, 41% Labor, 8% Democrat and the rest vote for minor parties. i What percentage of voters vote for minor parties? ii If there are 15 500 ed voters, of which 95% vote at the election, how many voters do not vote for either of the 2 major parties (Liberal or Labor)? c Roald achieves 39 out of 60 for a test while Serena achieves 25 out of 40. Who performed better and by what percent? d A car is discounted by 15%. If the customer pays $16 150, what was the price of the car before it was discounted? 8 The distance from Earth to our closest neighbouring galaxy, the Large Magellanic Cloud, is approximately 1.608 × 1018 km. Write this distance as a basic numeral. 9 multiple choice a The temperature at Fresia is 4.8°C at midday. The temperature falls steadily by 1.3°C per hour. What is the temperature in Fresia at 1 am the following morning? A −12.2° B –12.1° C −12° D −9.5° E 3.11° b At a school, 3--5- of Year 9 students study history, 1--4- study geography and the rest study commerce. If there are 180 students in the whole of Year 9, how many students study commerce? A 153 B 80 C 27 D 21 E 45 c The Johnson family pay council rates at 1.008 12 cents in the dollar. If their land is valued at $60 000, how much do they pay in rates? A $60 487.20 B $604.87 C $6048.70 D $604.90 E $60.50 d Jeans are discounted by 20%. The discounted price is $30 less than the usual price. How much are the discounted jeans? A $24 B $45 C $120 D $150 E $50


40


summary 1

2 3 4

5 6 7 8 9 10 11 12

13 14

Copy the sentences below. Fill the gaps by choosing the correct word or expression from the word list that follows. The order of operations is first, followed by multiplication or division left to right, then finally addition or subtraction, left to right. When using a scientific calculator, work the whole question from left to right. Rounding to a given number of decimal places begins with the first digit after the decimal . Rounding to a given number of figures begins with the first non-zero digit of the complete number. To convert a decimal number to a fraction, place the digits over 10, 100, 1000 . . . (depending on the number of decimal places), and simplify if appropriate. To convert a fraction to a decimal number, divide the by the denominator. To convert a decimal number to a percentage, multiply the number by 100. To convert a percentage to a decimal number, the percentage by 100. To convert a fraction to a percentage, multiply the by 100. To convert a percentage to a fraction, put the over 100 and simplify if appropriate (or divide by 100 if the percentage has a fraction). To find a percentage of an amount, divide the percentage by 100 and multiply by the . To express one amount as a percentage of another, divide the first amount by the second amount and multiply by . To find the full amount, given some other quantity is a percentage of the amount, determine 1% by dividing, then by 100 to find the full amount. To increase an amount by a given percentage, the percentage to 100% and find that percentage of the amount. To an amount by a given percentage, subtract the percentage from 100% and find that percentage of the amount.

WORD add decrease divide point

LIST decimal significant numerator brackets

100 multiply percentage finite

amount fraction



41

CHAPTER review 1 Evaluate each of the following. a (16 + 12) ÷ 7 b 7 × 6 − 12 ÷ 3

c [3 × (7 − 5)] ÷ 2

2 Evaluate each of the following. a 5 × 9 + (60 − 6) ÷ 9 c [(8 × −4 + 7) + 7] ÷ −3

d 12 + 8 × 5 ÷ 4

1A 1B

b −6 − 4 × −2 d 36 ÷ −6 − 4 × −5 + 6

4 Round 5689.7143 to: a 2 decimal places

b 2 significant figures.

1B 1C

5 Round 2156.586 to: a the nearest ten

b the nearest tenth.

1C

3 Insert operation signs to make the equation true: 6 K −5 K 3 K 2 = −12

6 multiple choice A rounded number is 13.50. The original number could have been: A 13.5 B 13.52 C 13.507 D 13.495 7 Evaluate these expressions. a 0.375 + 4 × 9.06

1C E 13.4938

1D

b 14.4 ÷ 1.2 − 0.65 × 23

8 Convert each of the following decimal numbers into a fraction in simplest form. a 0.875 b 0.24 c 0.55 d 0.365 e 0.248 f 0.13 g 0.75 h 0.575 i 0.372 j 0.4

1D

9 Evaluate these expressions. 7 1 - − --- ) ÷ a 1--4- + 3--5- × 5--8b 2--3- × ( ----12 4

1E

5 -----18

10 Convert each of the following to a percentage. -----a 0.71 b 2.4 c 15 16

d

1E

1 --6

11 multiple choice Which of the following statements is false? A

3 --4

<

19 -----25

B

5 -----16

>

3 -----10

C

12 Evaluate the following. a 3--7- of 56 b 65% of 590

c

13 -----40

1E <

7 -----20

D

8 -----11

<

18 -----25

6 --7

>

17 -----20

1F

0.89 of 420

13 Convert each of the following to a decimal number. a 12% b 34.6% c 7--8-

E

d

33 -----4

1F

14 Convert each of these percentages to a fraction in simplest form. a 52% b 130% c 28 4--7- %

1F

15 Decrease 240 by 24%.

1F 1F

16 If 45 is 3% of a number, what is the number? 17 Place the following in ascending order. a 57%, 0.6,

29 -----50

b

6 ------ , 25

0.245, 23 1--2- %

1F


42 1F


18 Place the following in descending order. a

4 --- , 5

82%, 0.83

b

5 ------ , 12

41%, 0.416

1G

19 Evaluate each of the following.

1G

20 Evaluate each of the following. b 2.5 × 102 × 3.5 × 103 a 2.52 × 3.53

1H

21 Calculate each of the following correct to: i 3 decimal places ii 3 significant figures.

a

1296

87 + 54 a -----------------87 – 54

4.2025

c

3

1.331

3 789 – 614 b ---------------------------654 – 437

1I

22 multiple choice The temperature in Warialda at 7 pm is 3.5°C and at 3 am is −3.7°C. What is the average drop in temperature per hour between the two readings? A 0.72° B 0.02° C 0.025° D 0.9° E 0.8°

1I

23 multiple choice In the first semester test Dominic scored 37 out of 50, while in the second semester test he scored 59 out of 80. Which of the following statements correctly compares his second semester result with his first semester result? A increased by 1--4- %

B decreased by 1--4- %

D increased by 28.7%

E decreased by 28.7%

C same result

1I

24 Rhonda pays $27.93 for petrol. If the price of petrol is 66.5 cents per litre, how many litres did Rhonda put into her car?

1I

25 A dam is 58% full. It presently holds 302 470 L. How many more litres of water will fill the dam to 100% of capacity?

1I

26 Simon visits his parents in a country town 1575 km ------ of the way away from his home. He drives 16 25 the first day and intends to reach the town the next day. a How far does he need to drive on the second day? b Simon drives for 12 hours on the first day. What is his average speed on that day? c Simon’s car averages 15 km per litre of petrol. What is his petrol cost for the trip if the price of petrol is 69.9 cents per litre?

test yourself CHAPTER

b

1

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