Adding and subtracting fractions :

02 Section2 3R pp014-031.qxd 2.1 23/11/04 8:22 am Page 14 Key words Adding and subtracting fractions fraction equivalent fraction denominator c...
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Key words

Adding and subtracting fractions

fraction equivalent fraction denominator common denominator lowest common multiple mixed number improper fraction

Add and subtract fractions by converting them to equivalent fractions with a common denominator Add and subtract algebraic fractions

Adding and subtracting fractions : Type A: Same denominators: Simply add or subtract the numerators, e.g. 1139  1179  3109 or 11119. Type B: Different denominators: Convert them to equivalent fractions which all have the same (common) denominator . Choose the lowest common multiple (LCM) of the denominators. 1 3 5 1   e.g. 12  8  6 2 2 2 0 9  24  24  24  1214 LCM of the denominators is 24.

e.g.

3x 2y    yz xz 3x2 2y2     LCM of the denominators is xyz. xyz xyz

Adding and subtracting mixed numbers (part whole numbers and part fractions): Method A: Calculate the whole number part and the fraction parts separately, and then combine them. Method B: Convert each mixed number to an improper fraction , then add or subtract them as fractions.

Example 1 Work out 213  335  456. 2341 1  3

Method A: When adding or subtracting mixed numbers, add or subtract the whole number parts first.

 35  65  3100  3180  3205  330 

So

231

Convert the fraction parts to equivalent fractions with the same denominator. The LCM of 3, 5 and 6 is 30.

1  10

 335  465  1101

Write the fractions as equivalent fractions with the same denominator: x3 x2

Example 2 Simplify the following expressions: a b 2ab2 3ab b)    a)    2 3 a2 b a b 3a 2b a)        2 3 6 6 3a  2b   6

14

Maths Connect 3R

2b 3a 2b2 3a3 b) 2    2  2 a b ab ab 2b2  3a3  2 ab

a 2

3a 6

b 3

2b 6

x3

x2

The LCM of a2 and b is a2b: xa2

xb 2b2 a2b

2b a2 xb

3a2 a2b

3a b xa2

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Exercise 2.1  Work out: a) 114  235

b) 213  157

c) 323  216  178

d) 114  2130  245 3 5

 Find the amount of wire needed to make this shape:

cm 3 5

1 2

cm

cm

 Jack has bought 3 pizzas to share with his family. His mum eats 56 of a pizza, his sister eats 5  7

of a pizza and his dad eats 114 pizzas. Jack eats the rest. How much pizza does Jack eat?

 Simplify the following expressions: a b a)    3 3

5 3 b)    a a

5a 2a c)    b b

4 b d) 2  2 a a

2a 2b e) 2  2 ab ab

c) 3a and 2b

d) 3y and y2

e) xy and xyz

d a c)    3 t

2a 5a d) 2  3 de d

 Find the LCM of the following: a) 7 and 4

b) p and q

 Simplify the following expressions: a b a)    4 4 3x2y yz

4 3 b)    w y

You can simplify an algebraic fraction by cancelling. Look for a common factor in the numerator and denominator.

2xy2 yz

 Simplify    as much as possible. 2

 Which of the following are true for all values of a and b: 2 1 1 a)      a b ab

1 1 ba b)      a b ab

ab  1 c)   b  1 a

a2b  2ab d)   a  2 ab

Investigation

a) Copy and complete this table showing the first 4 terms of a sequence: Term number

1

Working

1  2

Term

1  2

2 1  2



3 1  4

1  2

1  4

 

4 1  8

3  4

b) Find the 5th and 6th terms of the sequence. c) What do you notice about the terms of this sequence? 2n means 2  2  2  …  2. n times

          

d) You can write the general term of this sequence as: 1 1 1     …  n 2 2 4 Write down the general term as a single fraction. e) Prove that no term in the sequence is larger than 1.

Adding and subtracting fractions 15

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Key words

Multiplying fractions

fraction cancelling numerator denominator mixed number improper fraction

Multiply together two or more fractions Begin to multiply algebraic fractions

To multiply fractions : Cancel by dividing any numerator and any denominator by the same number. When no further cancelling is possible, multiply the numerators together, and the denominators together: 1 3

1

e.g.

5 18 11 5 18 11 11                6 35 24 16 357 248 56

e.g.

xxx5y8z 8z x3 5y 2x2z     2     43x5yy 4 3x 5y 3y

1

1

1

1

1

1

2

1

To multiply mixed numbers : First change the mixed numbers to improper fractions , then multiply the fractions 5

5

3

1

8 1 35 15 35 15 25 1 3  2            8 9 7 9 7 9 7 3 3

e.g.

Example 1 Find the volume of a cuboid with height 216 m, width 178 m and depth 135 m. 261  187  135  163  185  85 13   6 2

1 3

1

1

1

1 5   8

Write mixed numbers as improper fractions before multiplying. Simplify the multiplication by cancelling. It sometimes helps to write each number as a product of prime factors: 13 35 2  2 2       2 3 222 5

  85  123

The volume of the cuboid is 123 m3 or 621 m3.

Example 2 a2 3b Simplify   . ab 2a2

aa a2 3b 3b   2     ab 2aa ab 2a 3   2a

Simplify the multiplication by cancelling, then multiply together the numerators and the denominators.

Exercise 2.2  Simplify these fractions by cancelling: a) 16

2 0 6 390

Maths Connect 3R

b)

1 7 51

2ab c)  4b

x2 d) 3 x

x2y3 e) 32 xy

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 Complete the following multiplications: a)

3  5

 47

5  8

b)

d) 445  318

 23  190

c)

e) 335  1152  1314

1 0 13

 1425  58

f) 317  1191  245

 A concrete block is 634 m long, 513 m wide and 213 m high. What is its volume?  This kitchen tile is coloured black and white: A whole number of tiles are used to cover a floor with area 957 m2. Find the area of the floor which is a) black

b) white

 a) What is the area of the shaded triangle A in this

C B

diagram? 1m

b) What are the areas of triangles B and C?

A

c) The pattern of shaded triangles continues infinitely. What fraction of the large triangle is shaded in total? Explain how you arrived at your answer.

2m

 Simplify the following expressions: a 3 a)    6 a

t2p a2b 6bt d)       4a2b t4p a2p

a2b ab2 c)   3 ac bc

ab 3c b) 2   a b

 Write expressions for the areas of these shapes. Simplify your answers as much as possible. 7x y

a)

b 6a

b)

4y x

1 a

3a2 b a b2

 Calculate the value of: a)

3

(23)

b)

2

(14)

c)

2

(125)

d)

2

(57)

Simplify the following expressions: a)

a  2



2

b)

4  2c

 

3

c)

2b  4ab

 

2

Investigation

a) Make 6 cards showing the following expressions: x2

x2y

y2

y2x

2xy

4xy2

Shuffle your cards and make 2 fractions using four of your cards. Now multiply your fractions. b) Use your cards to make fractions with the following products: 1 4 x i)  ii) 2 iii)  iv)  2x y y

For example: x2

2xy

y2

4xy2

Multiplying fractions 17

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Key words

Reciprocals

2.3

reciprocal product

Recognise and use reciprocals Use the reciprocal key on a calculator The reciprocal of 5 is 15 : The reciprocal of 67 is 76 : y 2x The reciprocal of  is  : 2x y To find the reciprocal of 8 on a calculator, use the reciprocal key: press:

8



1

/x

1

/x :

0.125

The product of a number and its reciprocal is always 1 x2 2y   2  1. 2y x

For example 2  12  1

4  14  1 so 4  54  5 and 45  54  1 5

5

Example 1 Find the reciprocal of: a)

3  5

a)

c) 134

b) 7 5  3

1  7

b)

c)

7  71, so its reciprocal is 17.

4  7

2x d) 2 y y2 d)  2x

134  74, so its reciprocal is 47.

Example 2 Find the reciprocal of 0.23. Method 1: convert into a fraction 23 8 100    0.23   100, so its reciprocal is 23  423

Method 2: 0.23    1

The product of a number and its reciprocal is always 1.

So the reciprocal is 1  0.23  4.34782…  4.35 to 2 d.p. Method 3: using the reciprocal key on a calculator

0

.

1

2

3

1

/x

 4.34782 …  4.35 to 2 d.p.

18

Maths Connect 3R

The reciprocal key on a calculator looks like this: 1

/x

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Exercise 2.3  Find the reciprocal of: a)

6 11

c) 327

b) 12

d)

1 10

 Convert each decimal to a fraction or mixed number and find its reciprocal: a) 0.5

b) 4.25

c) 1.6

d) 0.88

 Use the reciprocal key on your calculator to find the reciprocals of each of the following. Give your answers correct to 3 d.p. a) 27

b) 0.035

c) 12.1

d) 0.9

 For each part of Q3, multiply your answer by the original number. Why are the products not equal to 1?

 Copy and complete the following: a) 215 

p2t b) 2  ab

1

1

c) 0.4 

1

d) 2.2 

1

 a) Find a number which is equal to its reciprocal. b) Are there any other numbers which are equal to their reciprocals?

 Find the reciprocal of 0.12. Give your answer correct to 2 d.p.  Chloe has written the reciprocal of a whole number less

0.0337

than 100 in her exercise book, correct to 4 d.p., but one of the digits is covered with ink: Find a whole number and the missing digit which fits.

Who am I? a) My reciprocal is one quarter of my value.

b) My reciprocal is twice my value. 1 x

a) Complete this table of values for the equation y  : x

0.1

1 y   x

10

0.2

0.4

0.5

1

2

4

5

10

b) Draw axes from 0 to 10 in both directions and plot the graph of this equation.

Use a pencil to join your points with a smooth curve.

Investigation

Do all numbers have a reciprocal? Using the reciprocal key on your calculator, investigate for positive and negative numbers. Try whole numbers, decimals, and fractions. Use numbers that have truncating decimals, recurring decimals and decimals that go on for ever.

Reciprocals 19

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Key words

2.4 Dividing fractions

fraction inverse numerator denominator mixed number improper fractions cancelling

Divide one fraction by another Begin to divide one algebraic fractions by another

To divide one fraction by another: Multiply by its inverse For example:

1 0 7

 58 is equivalent to 170  85

9 2x2 2x2 10x For example:    is equivalent to    10x 3 3 9 Dividing by a fraction is equivalent to multiplying by the same fraction with the numerator and denominator exchanged. This works because: 1 0 7

 58  170  85 2

1

1

1

Multiply the top and bottom by the reciprocal, 85.

 88  88 

10  7

 85

The denominator can be ignored because division by 1 has no effect.

To divide one mixed number by another: First change the mixed numbers to improper fractions , then divide the fractions For example: 216  158  163  183  163  183, and then cancel .

Example 1 Work out:

258  134

285  134  281  47

Convert mixed numbers to improper fractions first.

21  4      7 8

 32  121

Example 2 Simplify

a2b bc     6 4

2 1 7 2 1    8  4 is equivalent to 8

4 a2b bc a2b        bc 6 4 6

2

Exchange the numerator and denominator and multiply.

a  a  b   4    6  b  c 3

2a2   3c

Exercise 2.4  Work out: a) 20

2 3

 76

Maths Connect 3R

b)

5 9

 56

c)

3 8

 190

 47.

d)

7 15

 2215

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 Complete the following divisions: a) 179  49

b) 11115  235

c) 715  225

d) 367  11134

 How many lengths 78 m long can I cut from a piece of wood 514 m long?  A circle has a circumference of 1727 cm. Taking  as 272, find the diameter of the circle.  Simplify the following expressions: a 3 a)    b b

2a 6b b) 2   5b 10

9pn 3mn2 d) 2   2mnp 4pm

4t3p3 6tp2 e)    5p2t2 15p3

3xy2 2xy

ab bc c) 2  2 ab bc

9x2y 4x y

 a) Simplify   2. 2 b) Substitute x  2 and y  3 into the original expression, and into your simplified expression. Are the answers the same? c) Simplify the following expression and find its value when a  5 and b  2: 3ab2 2ab2    ab 2ab

 Which of the following are true for all values of x and y? x a)   y  x y

2x 1 1 2x2  y c) 2       xy y y x

1 1 x b)      y x y

 Simplify the following expressions: 1 pq a)    2 2r p

a 2 a2 b) 2     3b b b

y2 y3 2y c)   2   2x xy x

Draw axes from 10 to 10 in both directions.

Simplify the equations as much as possible before drawing the graphs.

Draw and label the graphs of the following equations: 4y2 1 y 3xy 4 b) 2  2    1 a)     2x  8 2xy 2x x y 2x y

Investigation

a) Make six cards showing the following expressions: 2x2

x2y2

xy

2y2

2x2y

Use four of your cards to complete this expression and then simplify.

4y2x



For example: 2x2 xy 2 2  2 xy 2y

b) Use your cards to make expressions which simplify to each of the following: x2 2 8 i)  ii) 1 iii)  iv)  8y y y

Dividing fractions 21

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Key words

Mental methods

calculation multiplier power of 10 divisor

Learn strategies for multiplying and dividing with decimals

There are nearly always several methods of doing calculations . To multiply 0.26  0.3, for example: Method 1 Change the decimals to whole numbers 0.26  0.3  26  3  1000  78  1000  0.078 Method 2

Change the multiplier to include a power of 10 , i.e. 0.26  0.3  0.26  0.1  3  (0.26  10)  3  0.026  3  0.078

Method 3

Change to fractions and multiply by cancelling, i.e. 26 3 26  3 78 0.26  0.3          0.078 100 10 100  10 1000  10

Make the divisor a To divide 0.84  0.4, for example: whole 0.84 8.4 Method 1 Multiply both numbers by the same amount:     2.1 number by 0.4 4 multiplying by a power  10 of 10. Method 2 Change the divisor to include a power of 10, i.e. 0.84  0.4  (0.84  0.1)  4  (0.84  10)  4  8.4  4  2.1 Method 3

Change to fractions and divide by converting it to a multiplication: 84 4 84 10 84  10 21 0.84  0.4           which cancels to   2.1 100 10 100 4 100  4 10

Example 1 Calculate mentally:

a) 3.4  0.7

a) 3.4  0.7  34  7  100 b)

b) 

0.23  0.3

c) 5.2  0.4



0.23  0.3  0.23  0.1  3

 238  100

 0.23  10  3

 2.38

 0.023  3

c) 5.2  0.4  52  4  13



Multiply both numbers by the same number:

 0.069

 10

Change the decimals to whole numbers.

5.2 52    0.4 4

Change the multiplier to include a power of 10.

 10

Example 2 Calculate mentally the volume of this cuboid: 2.5  0.75  24    24 5  2

3  4

533  45 m3

0.75 m

24 m 2.5 m

Change the decimals into fractions then multiply using 3 cancelling. 12  5   3   24  2 4   1

22

Maths Connect 3R

1

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Exercise 2.5  Calculate mentally: a) 0.38  0.6

b) 0.23  0.4

c) 0.64  0.8

d) 0.35  0.3

e) 0.42  0.7

f) 0.24  0.6

 How many 1.6 ml spoonfuls can I get from a 56 ml medicine bottle?  Find the area of a rectangular path 2.3 m long and 0.45 m wide.  How many 0.7 m lengths can be cut from 7.35 m of ribbon?  Calculate mentally the volumes of these cuboids: a)

b)

3.5 cm

7.1 m 13 m

32 cm 0.25 cm

5.8 m

 Calculate mentally: a) (42)2

b)

0.21  0.11

 2 

c) (1.6)3

d) 1.44 

0.35) (0.29   e)  0.2

 a) Find the product of 0.021 and 0.007. b) If 28  32  896, what is 2.8  0.32? c) What number, when divided by 0.5, gives the answer 3.2? d) If 327  3  109, what is 327  0.3?

 Make mental estimations of the following calculations by rounding the numbers to 1 decimal place: a) 0.338  0.592

b) 0.771  5.2091

c) 2.382  0.5925

d) 14.08  0.34009

Use a calculator to work out the calculations in Q8. Compare the answers with your estimates.

Solve these equations mentally: a) 0.2x  6.8

b) 0.05y  3.5

c) 0.15a  0.8  2.75

d) (x  9)2  36

There are two answers to part d).

Mental methods 23

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Key words

Percentages

percentage increase percentage decrease compound interest

Solve problems involving percentage change Solve problems involving compound interest

If an amount has a percentage increase of 812%, then it is 108.5% of the original amount. If it has a percentage decrease of 712%, then it is 92.5% of the original amount. When you invest money, you earn compound compound interest if you leave the interest in the account. Compound interest for any year is the interest paid on the combined a) total amount invested, plus b) any interest earned in previous years. e.g. an investment of £300 at a compound annual interest rate of 812%, is worth: after 1 year: £300  1.085  £325.50 after 2 years: £325.50  1.085  £353.17 after 3 years: £353.17  1.085  £383.19, and so on. An alternative method of calculation is: after 1 year: £300  1.085  £325.50 after 2 years: £300  (1.085)2  £353.17 after 3 years: £300  (1.085)3  £383.19.

Example 1 A record shop is offering 20% off all CDs in a sale. Find the original cost of a CD costing £14 in the sale. Method 1 £14 represents 80%

Find the value of 1% first.

Method 2 If the original cost in pounds is x, then:

£14  80 represents 1%

x  0.8  14

£14  80  100 represents 100%

So x  14  0.8  17.5

So the original cost was £14  80  100  £17.50

So the original cost was £17.50

Write an equation using the original cost as the unknown. Solve your equation using inverse operations.

Example 2 If you invest £250 in a bank with an annual compound interest rate of 4.5%, how much will it be worth after 3 years? Give your answer correct to 2 decimal places. After 1 year:

24

£250  1.045  £261.25

After 2 years:

£261.25  1.045  £273.01

After 3 years:

£273.01  1.045  £285.30

Maths Connect 3R

You can work this out using a single calculation: £250  (1.045)3  £285.30

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Exercise 2.6  Karim’s wage was increased from £210 to £225 per week. What was the percentage increase?  Copy and complete this table showing

Original price

Sale price

Dishwasher

270

220

a) Which item has the largest percentage decrease in price in the sale?

Toaster

18

15

DVD player

190

165

b) Which item has the smallest percentage decrease in price in the sale?

Food processor

99

78

the original price and the sale price of four items in an electrical shop.

Item

 An airline adds a 5% fuel surcharge to all its fares. What percentage of the total cost is the fuel surcharge? Give your answer correct to 2 decimal places.

 A standard 500 g pack of breakfast cereal costs £1.28. A large pack contains 35% more, and costs £1.70. Which pack is better value?

 An elastic band is stretched so that its length increases by 25%. If the stretched elastic band is 16 cm long, find its unstretched length.

 A car is reduced by 17% and now costs £2300. Find the original price of the car to the nearest pound.

 An antique chair increases in value by 7.5% each year. a) If it is now worth £187.50, how much was it worth 1 year ago? b) How much was it worth 3 years ago?

 Carla saved £12 when she bought a pair of jeans in a sale. The sale offered a 30% reduction. Find the original selling price of the jeans.

If the length and width of this cuboid increases by 15% and the depth increases by 12.5%, what is the percentage increase in volume?

width  10 cm length  20 cm

depth  5 cm

Ali invests £500 in a bank with an annual compound interest rate of 3.2%. a) How much will he have after 3 years? Give your answer to the nearest penny. b) Write a single calculation to work out how much he will have after 10 years.

A strain of bacteria increases at a rate of 12.5% per day. a) If there were 1000 bacteria originally, calculate the number there will be at the end of each day for 7 days. Round each calculation to the nearest whole number. b) Use a single calculation to find the number of bacteria at the end of 7 days. Round your answer to the nearest whole number. c) What do you notice about your answers to parts b) and c)? Explain your findings.

Percentages 25

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Key words

Direct proportion

direct proportion directly proportional to scale

Solve problems involving direct proportion Recognise notation y  x for direct proportionality

If y and x are in direct proportion , then as one increases, the other increases proportionally. $ is the symbol If £1 is worth US$1.8, then £10 is worth US$18 and £20 is worth US$36. for US dollars. The value in UK pounds is directly proportional to the value in US dollars. If y is directly proportional to x, we write: y  x, or y  kx, where k is a constant value. The graph of y against x is always a straight line, passing through the origin. Distances on a scale drawing are directly proportional to the true distances. If a distance of 4 cm on the drawing represents 12 m in real life, then a distance of 8 cm represents 24 m in real life.

Example 1 At a hardware shop, the cost of wood is in direct proportion to the length bought. 4 m of wood cost £2.24. a) Find the cost of 8 m of wood. b) How much wood can you buy for £10? Give your answer to the nearest cm. a) Cost of wood  length bought.

 means ‘is proportional to’. Cost of wood is proportional to length bought, so if the length bought doubles, the cost doubles.

b) 1 m of wood costs £2.24 ÷ 4 = £0.56

4 m costs £2.24

10 ÷ 0.56 = 17.86

So 8 m costs £4.48

17.86 m  100  1786 cm

Find the cost of 1 m of wood first.

You can buy 1786 cm of wood for £10.

Example 2 These two prisms have the same cross-sectional area: Prism A has a volume of 172 cm3. Find the volume of prism B. Because the prisms have the same cross-sectional area, volume will be in

8 cm

5 cm

direct proportion to height. Prism B is 62.5% of the height of prism A, so it will have 62.5% of the volume. 172  0.625  107.5 Prism B has a volume of 107.5 cm3.

Height  volume, so the percentage change in height will be the same as the percentage change in volume.

Exercise 2.7  a) A prism has the same cross-sectional area as the prisms in Example 2. It has a height of 9.6 m. Find its volume. b) You can write V  kh where V represents the volume of the prism in cm3, h represents the height of the prism in cm and k is some number. Find the value of k. c) Check this formula with the prisms in part a) and Example 2. 26

Maths Connect 3R

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 Ned is converting pounds (£) into euros (€) for his holiday in France. Pounds are in direct proportion to euros, and £1  €1.39.

You can write Pounds  Euros.

Convert each of the following, giving your answer correct to 2 d.p. a) £7 into euros

b) £3.50 into euros

c) €8 into pounds

d) £27.32 into euros

e) €23.20 into pounds

f) €127 into pounds

 A map uses a scale of 1 cm  3 km.

1 cm  10 cm would be written as 1 : 10.

a) Write this scale as a ratio. b) What distance would 3.2 cm on the map represent? c) What distance on the map would represent 11 km. Give your answer to the nearest mm.

 This is a recipe for fruit cocktail. a) How much of each ingredient do you need for 5 people? b) If you had 250 ml of orange juice, how much of the other ingredients would you need? c) If I want to make a litre of fruit cocktail, how much of each ingredient do I need? Give your answers to the nearest ml.

Serves 3 300 ml Orange Juice 250 ml Cranberry Juice 120 ml Soda water

 A man runs 1 mile in 4 minutes. Tom says ‘This is direct proportion. I can write this as t  4m and work out his time for any number of miles’. Ali says he is wrong. Who is correct? Why?

 This is a formula for converting pounds (m) into US dollars (d): d  1.72m a) Are pounds in direct proportion to U.S. dollars? Explain your answer. b) How many pounds would I get for $5.30? c) How many dollars would I get for £26? d) On a different day the exchange rate is £1  $1.68. Write this in the form d  __m.

 True or False? a) If g  b then b  g. b) If a  6b then b  6a. c) If s  r, then s  kr, where k is some number. d) If one can of cola is 20p and 5 cans cost 90p, this is direct proportion. e) Direct proportion always produces a straight line graph.

 Explain in words what the following represent: a) t  p, where t  total mass of the pins and p  mass of a pin b) d  s, where d  distance and s  length of a stride c) c  7g, where c  cost and g  weight in grams d) d  35t, where d  distance and t  time in hours.

A snowplough can clear 5 km of road in 1 hour. The snowplough uses 1 litre of fuel every 22 minutes. Find the amount of fuel the snowplough uses per kilometre of road cleared. Direct proportion 27

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Key words

Inverse proportion

inverse proportion constant

Solve problems involving inverse proportion Recognise the graphs of variables which are inversely proportional to each other

Two variables, x and y, are inversely proportional to each other if their product xy is always the same, or constant . We can write xy  k, where k is a constant. If y is inversely proportional to x then as y doubles, x halves. If we consider rectangles which all have an area of 12 cm2, then the length, x, of the rectangle is inversely proportional to the width, y, i.e. as the length increases, the width decreases. Similarly, as the length decreases, the width increases. For example:

x  12 cm

y  1 cm

area = 12 cm2

x  6 cm

y  2 cm

area = 12 cm2

x  4 cm

y  3 cm

area = 12 cm2, and so on.

We can see that as the width is doubled, the length is halved. 12 We can find the width of this rectangle if we know the length because y  . x 12 Similarly, we can find the lengthby using x  . 12 is the constant here. y

Example 1

Example 2

It would take 7 people 10 days to paint a cinema. How many people would be needed to complete the job in 2 days if they worked at the same rate?

If a farmer has enough hay to feed 5 horses for 6 days, how long would the hay last for 3 horses?

The number of days is inversely

The number of days is inversely

proportional to the number of people.

proportional to the number of horses.

10 days 1 day 2 days

7 people

5 horses

6 days

10  7  70 people

1 horse

5  6 days

3 horses

56   10 days 3

70  2

 35 people

10 times as many

You can write this as: 1 Number of horses   Number of days

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Exercise 2.8  It takes 1 man 120 days to build a house. Copy and complete the table to show how long it would take to build the same house with different numbers of men. Men (m)

1

Days (d)

120

2

3

4

5

6

8

10

12

15

12

 4 professors can mark a set of exam papers in 5 hours. a) How long would it take 1 professor working at the same rate? b) How long would it take 10 professors working at the same rate?

 It takes 6 people 8 hours to clean the Royal Albert Hall. a) How many people would be needed to clean it in 3 hours? b) If only 4 people turned up for work how long would it take to clean?

 It takes a jet 8 hours at a constant speed of 400 mph to cross the Atlantic. a) How long would it take at a constant speed of 500 mph? b) If it took 11 hours, what would have been the constant speed?

 Five shipwrecked people have enough food for 6 days. a) How long would the food last if only 3 people had been shipwrecked. b) How many people can the food support for 15 days?

 A triangle has an area of 24 cm2. Draw a graph plotting base against height. This graph represents inverse proportion.

 500 phones can be manufactured by 4 machines working for 3 hours. Draw a graph 1 of m against  where m  number of machines and h h  number of hours.

Make a table of base and height measurements that give an area of 24 cm2 first. Find the constant using mh  k first, then make a table of values for m and h.

 Sketch a graph that represents direct proportion. Label the axes with suitable variables and explain what the graph might show. Now do the same for inverse proportion.

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Key words

Ratio

ratio cancelling enlargement area volume

Compare ratios by changing them to the form 1 : n or n : 1 Recognise the ratio of lengths, areas and volumes when comparing similar shapes

Ratios can be simplified, like fractions, by cancelling , i.e. dividing all numbers by the same number, e.g. 8 : 12 simplifies to 2 : 3. To compare one ratio with another, reduce them to the form 1 : n. If one 2-D shape is an enlargement of another by a scale factor of k, then ● the lengths of corresponding sides are in a ratio of 1 : k. ● the areas of the shapes are in a ratio of 1 : k2. 6 cm 5 cm 3 cm Ratio of corresponding sides is 1 : 2 4 cm Ratio of areas is 1 : 4

10 cm 8 cm

If one 3-D shape is an enlargement of another by a scale factor of k, then ● the lengths of corresponding edges are in a ratio of 1 : k. ● the areas of the corresponding faces are in a ratio of 1 : k2. ● the volumes of the shapes are in a ratio 1 : k3. Ratio of corresponding sides is 1 : 3 6 cm 2 cm Ratio of corresponding areas is 1 : 9 6 cm 6 cm 2 cm Ratio of volumes is 1 : 27

18 cm

Example 1 In Johnsons’ Jam there is 55 g of fruit to 45 g of sugar. In Sunny Jam there is 240 g of fruit to 205 g of sugar. Which Jam has the highest fruit content? Fruit : Sugar Johnsons’

Write the ratios in the form 1 : n

Fruit : Sugar

55 : 45

Sunny

240 : 205

1 : 0.82

55

1 : 0.85

55 : 45 1 : 0.82

There is less sugar to one unit of fruit in Johnsons’s Jam, therefore it has the highest fruit content

Example 2 The area of two squares are 9 cm2 and 64 cm2. a) Find the ratio of their sides. b) Find the scale factor of the enlargement from the smaller square to the larger. a) Areas Side lengths

9 : 64 9  : 64  3:8

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b) Sides

3:8 1 : 2.6

Scale factor  2.6 to 1 d.p.

55

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Exercise 2.9  The ratio of boys to girls in Ali‘s school is 3 : 5. The ratio of boys to girls in Jane’s school is 5 : 9. Which school has the greater proportion of boys?

 a) Find the ratio of the sides of the squares A and B. b) Find the ratio of the sides of the areas of A and B. c) What do you notice about your answers to parts a) and b)?

A

B

5 cm

6 cm

 Two rectangles, p and q, are similar. p is 6 cm long and 2 cm wide and q is 9 cm long and x cm wide. a) What is the ratio of the lengths of the sides p and q? b) Write this ratio in the form 1 : n. c) What is the scale factor of the enlargement from the smaller rectangle to the larger? d) Use the scale factor to find the length of side x. e) Calculate the area of each rectangle. f) What is the ratio of the areas? g) Compare your answer to part a) to your answer to part f). What do you notice?

 The side lengths of these cubes are in the ratio 3 : 5. a) b) c) d)

Work out the volume of cube S. T S Work out the volume of cube T. Find the ratio of the volumes of the cubes. 3 cm 5 cm Compare your answer to the ratio of the side lengths. What do you notice?

 In a bank I changed £18 and received €24. In the travel agents I changed £20 and received €27.50. Assuming that neither the bank nor the travel agents charge a fee for changing money, which one gives better value?

 These two similar triangles have bases of 4 cm and 10 cm. The smaller triangle has an area of 13 cm2. a) Write the ratio of their bases in the lowest terms. b) What is the ratio of their areas? c) Use your answer to part b) to find the height of the larger triangle.

 The volumes of two similar cuboids are in the ratio 512 : 1331. Find the ratio of their areas.  A regular tetrahedron with volume 343 cm3 is enlarged. The enlarged tetrahedron has a volume of 1728 cm3. Find the scale factor of the enlargement.

True or False? a) If two shapes have a pair of sides in the ratio 3 : 2, then all pairs of sides are in the ratio 3 : 2. b) If the areas of two similar shapes are in the ratio 16 : 64, then the sides are in the ratio 8 : 32. c) If two similar shapes have areas in the ratio of 4 : 9, then the volumes are in the ratio 43 : 93. d) If three similar shapes have areas in the ratio of 1 : 2 : 3, then their areas are in the ratio 12 : 22 : 32 and their volumes are in the ratio 13 : 23 : 33. Ratio 31

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