direct proportionality
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Direct ProportionalityMonday 4 January 2010
Understand the idea
Know how to checkthat two quantities aredirectly proportional
Find a missing number
Level 7
Why am I
doing this?
It is an important idea thathas many real life
applications
A favourite SAT and
GCSE question
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Here is an example of direct proportionality
Some DIY stores sell electrical wire by the
metre.Suppose that one store sells it at 25pper metre. The cost for different lengths isshown on this table.
Length (m) 0 1 2 3
Cost (p) 0 25 50 75
If I want to make an electricallead that is 4 metres long, whatwill be the cost of the wire?
Answer
100p
£1
The relationshipbetween the
length and the
cost is directly
proportional.
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Checking if a relationshipis directly proportional
You need to be able todecide if a relationship isdirectly proportional
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When are quantities are DIRECTLY PRPORTINAL?
Hours 4 7 9
Wages 25 43.75 56.25
Here is a tableshowing the wagesearned for thenumber of hoursworked.
You would expect this to be a directlyproportional relationship. You can checkwhether or not it is directly proportional bydividing like this:
25 ÷ 4 = 6.25
43.75 ÷ 7 = 6.25
56.25 ÷ 9 = 6.25
If you always get the sameanswer when you divide, therelationship is directlyproportional.
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When are quantities are DIRECTLY PROPORTINAL?
Hours 4 6 9
Wages 28 42 72
Here is another tableshowing the wagesearned for thenumber of hoursworked.
Are the wages directly proportional to thehours worked?
28 ÷ 4 = 7
42 ÷ 6 = 772 ÷ 9 = 8
If even one result does not give
the same answer, the relationshipis NOT directly proportional.
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Which of the relationships shown beloware directly proportional?
Length 3 4 10
Weight 15 21 50
Cost 2 5 9
Length 7.4 18.5 33.3
Base 2 5 9
Height 12.4 31 55.8
15 ÷ 3 = 5
21 ÷ 3 = 5.25
15 ÷ 3 = 5
7.4 ÷ 2 = 3.7
18.5 ÷ 5 = 3.7
33.3 ÷ 9 = 3.7
12.4 ÷ 2 = 6.2
31 ÷ 5 = 6.2
55.8 ÷ 9 = 6.2
NO
YES
YES
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Solving Direct
ProportionalityProblems
Monday 4 January 2010
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Finding Missing Numbers
A common question is to give you a table showing two quantities.Unfortunately, the table has some missing numbers and you are
asked to find these values. Here is an example:
The table shows a directly proportional relationship between
LENGTH and COST. Calculate the two missing values in thistable.
Length 12 15 21 40
Cost 42 52.5 73.5 60
The next two slides will help you
understand how to do this
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Length (m) 0 1 2 3 4 5
Cost (p) 0 25 50 75 100
How to find amissing number
First choose any of thenumbers on the top row.
I’ve chosen 2
Find the number that youhave to multiply 2 by to get 5.
(5 ÷ 2 = 2.5) 2 x 2.5 = 5
Multiply 50 by 2.5 to
get the missing number
x 2.5
x 2.5
125
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Length (m) 0 1 2 3 4
Cost (p)
0 25 50 75 100 525
Missing numbers on thetop row can be found
by a similar method
First choose any of thenumbers on the bottom
row. I’ve chosen 50
Find the number that you have tomultiply 50 by to get 525.
(525 ÷ 50 =10..5) 50 x 10.5 = 525 Multiply 2 by 10.5 to
get the missing number
x 10.5
x 10.5
21
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Direct Proportionality – Test
1) The relationship between A and L is directly proportional.
Copy and complete this table:
A 1.4 1.6 1.72 3.2
L 12 18 1.60 114
2) The relationship between K and V is directly proportional.
Copy and complete this table:
K 7.2 1.18 1.6 1.9
V 0.2 0.5 0.48 0.25
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Direct Proportionality
Getting the Idea
Model Plane Real Plane
When there is direct proportionality betweentwo things, it means that the relationship is the
same as the model’s lengths and the real plane’slengths.
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A scale for model planes is 1/72. This means thatthe real plane’s lengths are 72 time bigger than
the model’s.Complete the numbers in this chart,
Model Plane 1 3 4 10 12 10
Real Plane 72 144 576 504 936
Model Plane 1 3 2 4 10 8 7 12 10 13
Real Plane 72 216 144 288 720 576 504 864 720 936
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