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© T Madas
© T Madas
What do we mean when we say two quantities are in proportion or directly proportional ?It means that if:
one of them doubles,the other one also doubles.one of them trebles,the other one also trebles.one of them x4,the other one also x4.one of them halves,the other one also halves.one of them ÷4,the other one also ÷4.
Can you give examples of directly proportional quantities from every day life?
© T Madas
Directly proportional quantities:They increase or decrease at the same rate
More formally:Two quantities are directly proportional if the ratio between them remains constant.
© T Madas
© T Madas
Laps in Racing Cars
A racing car does 2 laps every 5 minutesHow long does it take to cover 17 laps?How many laps does it cover in 1 hour?
L T 2 : 5
17 laps in 42½ minutes In 1 hour it does 24 laps
1 : 2.5
÷2 ÷2
17 : 42.5
x17 x17
L T 2 : 5
0.4 : 1
÷5 ÷5
24 : 60
x60 x60
© T Madas
© T Madas
Calculating Earnings
A plumber earns £51 in 6 hoursHow much does he earn in a 41 hour week?In many hours does he earn £1000?
H £ 6 : 51
£348.50 in 41 hours approximately 118 hours
1 : 8.5
÷6 ÷6
41 : 348.5
x41 x41
H £ 6 : 51
0.1176: 1
÷51 ÷51
117.6 : 1000
x1000 x1000
© T Madas
© T Madas
Converting Currency
If £5 are worth €7 How many € are £8 worth?How many £ are €3 worth?
£ €5 : 7
£8 = € 11.20
1 : 1.4
÷5 ÷5
8 : 11.2
x8 x8
£ € 5 : 7
0.7143 : 1
÷7 ÷7
2.14 : 3
x3 x3
€ 3 ≈ £ 2.14
© T Madas
© T Madas
In a chemical reaction solution A must be combined with solution B in the ratio of 5 : 8, to form a certain compound.
How many ml of solution B must be used, if 5.5 ml of solution A is to be used?
Chemistry Experiments
A B5 : 8 8.8 ml of solution B must be
mixed with 5.5 ml of solution A
1 : 1.6
÷5 ÷5
5.5: 8.8
X5.5 X5.5
© T Madas
© T Madas
© T Madas
Laps in Racing Cars
A racing car does 2 laps every 5 minutesHow long does it take to cover 17 laps?How many laps does it cover in 1 hour?
L M
:2 5
:1 52
:17 ´ 1752 .= 42 5
17 laps in 42½ minutes
M L
:5 2
:1 25
:60 ´ 6025
=24
In 1 hour it does 24 laps
© T Madas
© T Madas
Calculating Earnings
A plumber earns £51 in 6 hoursHow much does he earn in a 41 hour week?In many hours does he earn £1000?
h £
:6 51
:1 516
:41 ´ 41516 .= 348 5
He earns £348.50
£ h
:51 6
:1 651
:1000 ´ 1000651 » 118
He earns it in 118 hours
© T Madas
© T Madas
Converting Currency
If €7 are worth £5How many £ are €3 worth?How many € are £8 worth?
€ £
:7 5
:1 57
:3 ´ 357 .» 2 14
€3 ≈ £2.14
£ €
:5 7
:1 75
:8 ´ 875 .=11 2
£8 = €11.20
© T Madas
© T Madas
In a chemical reaction solution A must be combined with solution B in the ratio of 5 : 8, to form a certain compound.
How many ml of solution B must be used, if 5.5 ml of solution A is to be used?
Chemistry Experiments
A B
:5 8
:1 85
. :5 5 .´ 5 585
.= 8 88.8 ml of solution B must be used
© T Madas
© T Madas
12 kg of flour cost £15.How much do 28 kg cost?
15´ 2812
5
4
7
= 35
12 kg of flour cost £15.How many Kg do £120 buy?
12´ 12015
4
5
24
= 96
© T Madas
600 Mb require a 72 minute CD.How many minutes do 700 Mb need?
72´700600
12
6
7
= 84
600 Mb require a 72 minute CD.How many Mb correspond to 60 minutes?
600´6072
100
6
5
= 500
© T Madas
In a pie chart 7200 voters are represented by 144°.How many voters do 120° represent?
7200´120144
1200
6
5
= 6000
In a pie chart 7200 voters are represented by 144°.By how many degrees are 1000 voters represented?
144´10007200
2= 20
© T Madas
© T Madas
18 ´ 2045
40y
8024x
The variables in the table are directly proportional.Find the missing value.
80 ´ 4024
10
3
= 4003 . ...» 133 333
8
2045y
18x 2
5
= 84
4003
21.654y
3690x 36´ 5490
2
5
= 1085
.= 21 6
© T Madas
12080y
9060x
The variables in the table are directly proportional.Find the missing value.
90´ 8060
=120
4060y
1218x 12´ 6018
2
3
= 4020
24
5075y
36x50´ 36
752
3
= 2412
15
© T Madas
© T Madas
20 litres of petrol cost £16.
1. Work out the cost of 1 litre of petrol.
2. Work out the cost of 7 litres of petrol.
litres
20
10
1
7
cos
t
16
8
0.8
5.6
:
:
:
:
:
£0.80
£5.60
© T Madas
© T Madas
Toca-Cola costs £1.24 for a 2 litre bottle.Sepsi-Cola costs £0.90 for a 1.5 litre bottle.Work out which brand is the cheapest.
2 lt£1.24
1.5 lt£0.90
lt £:2 1.24:
÷2 ÷2
1 0.62:
Toca-Cola
1 lt costs 62 pence,
lt £:1.5 0.90:
÷1.5 ÷1.5
1 0.60:
Sepsi-Cola
1 lt costs 60 pence,
Sepsi-Cola is the cheapest
© T Madas
© T Madas
18 pens cost £2.16.
Calculate the cost of 24 pens.
pens £:18 2.16:
÷18 ÷18
1 unimportant:
x 24 x24
24 2.88:
pens £:18 2.16:
÷3 ÷3
6 0.72:
x 4 x 4
24 2.88:
£2.88
© T Madas
© T Madas
Look at the nutritional information shown on the label opposite.
How many grams of carbohydrate do 100 grams of these beans provide?
Bono Baked BeansEach 150 g provides
108 kcal6.9 g19.5 g0.3 g0.5 g
EnergyProteinCarbohydrateFatSalt
beans carbs:150 19.5:
÷3 ÷3
50 6.5:x2 x2
100 13:
100 grams of beans provide 13 grams of carbohydrate
© T Madas
© T Madas
Look at the nutritional information shown on the label opposite.
Calculate how many grams of carbohydrate do 100 grams of these beans provide, giving your answer correct to 3 significant figures.
Chilli Baked BeansEach 215 g provides
161 kcal10 g30.5 g0.2 g0.6 g
EnergyProteinCarbohydrateFatSalt
beans carbs:215 30.5:
÷215 ÷215
1 0.1419:x100 x100
100 14.19:
100 grams of beans provide 14.2 grams of carbohydrate
© T Madas
© T Madas
Hojo Apple PieEach 175 g provides
434 kcal6.9 g79.5 g9.9 g0.5 g
EnergyProteinCarbohydrateFatSalt
Mojo Apple PieEach 150 g provides
375 kcal6.7 g70.5 g5.3 g0.5 g
EnergyProteinCarbohydrateFatSalt
Look at the nutritional information shown on the two labels.
Rohan eats the same amount from each apple pie.
Which apple pie provides Rohan with more energy.
grams energy:175 434:
÷175 ÷175
1 2.48:
grams energy:150 375:
÷150 ÷150
1 2.5:
each gram provides 2.48 kcaleach gram provides 2.5 kcal
© T Madas
© T Madas
Mr Green sells a 2.4 kg bag of apples for £1.80 while Mr Red sells a 2.5 kg bag of apples for £2.
Who sells the cheaper apples?
kg £:2.4 1.80:
÷2.4 ÷2.4
1 0.75:
at Mr Green’s
1 kg costs 75 pence,
kg £:2.5 2:
÷2.5 ÷2.5
1 0.8:
at Mr Red’s
1 kg costs 80 pence,
Mr Green sells the cheaper apples
© T Madas
© T Madas
The exchange rate between Sterling and Euros is:
£1 = €1.60
1. How many Euros do we get if we change £150?
2. How many Pounds do we get if we change €32.64?
£ €:
1 1.6:
x150 x150
150 240:
£150 = €240
£ €:
1 1.6:
÷1.6 ÷1.6
Whocares?
1:
€32.64= £20.40
x32.64 x32.64
20.4 32.64:
© T Madas
© T Madas
The exchange rate between Sterling and Euros is:
£1 = €1.60
How many Pounds do we get if we change €32.64?
£
1
2
10
20
40
0.4
20.4
€
1.6
3.2
16
32
64
0.64
32.64
:
:
:
:
:
:
:
:
© T Madas