Transcript Year 9: Ratio & Proportion
Year 9:
Ratio & Proportion
Dr J Frost ([email protected])
Last modified: 7 th September 2014
Starter Puzzles
(You can work in pairs)
1
My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Frostitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. Whatβs the new ratio of black to yellow fish?
Answer:
π: π ?
2
The ratio of males to females at a party is 3:5. There are twelve more females than males. How many people are at the party?
Answer
= ππ ?
3
In a cardboard box, the ratio of cats to dogs is 3:5 and the ratio of dogs to sharks is 4:7. What is the ratio of cats to sharks?
Make βparts dogβ same in each.
So cats to dogs
?
and cats to sharks
ππ: ππ
. So ratio of cats to sharks
ππ: ππ
.
N There are two regular polygons, π΄ and π΅ . The ratio of their exterior angles is 1: 3 . The ratio of their interior angles is 7: 6 . Prove that polygon π΄ has 30 sides.
(Hint: it may be helpful to start with the exterior angle of
π΄
being
π₯
and working from there)
πππ β π πππ β ππ
Num sides =
πππ ππ = ?
π π = ππ β΄ π = ππ
Direct Proportion
Iain wants to make a Baked Alaska.
He requires 24 eggs to make 700g of the pudding. How many eggs does he require to make 1250g?
We say that the number of eggs and the mass of the pudding are βdirectly proportionalβ.
There is some underlying βscalingβ between the two things.
Method 1: Unitary Method
(Find number of eggs for 1g)
700g require 24 eggs.
24 So 1g requires 700 700
So 43 eggs.
eggs.
So 1250g requires: 24 Γ 1250 = 42.857 β¦ ?
Method 2: Ratio Method !
If
π: π = π: π
then
π π = π π 1250: 700 = π₯: 24 1250 = π₯ 700 24 π₯ = 24 Γ 1250 700 = 42.857 β¦
Test Your Understanding
Solve the following using both the βunitary methodβ (i.e. find quantity for one unit) and the βratio methodβ.
Q
The mass of 16cm 3 20cm 3 of Neoginium is 24g. What is the mass of of the same element?
Unitary Method
16cm 3 is 24g.
1cm 3 20cm 3 is
ππ ππ
is:
ππ ?
ππ Γ ππ = πππ Ratio Method ππ: ππ = ππ: π ππ ππ = π ππ π = ππ Γ ?
ππ ππ = ππ If you finish:
Q
There are 7 Aardvarks and 12 Buffalo in a classroom. The ratio of Aardvarks to Buffalo is π₯ βΆ π₯ + 3 . What is π₯ ?
π + π = π ππ π ?
β π = π. π
Direct Proportion
!
Two things are directly proportional if they in the same ratio.
β’ β’ β’ Can you suggest variables that might be directly proportional?
Speed and distance travelled (if you double your speed, you double the distance travelled).
Total cost and quantity purchased.
?
Length of steel rod and weight.
Why is βhours revised for maths examβ and βmaths exam markβ not likely to be directly proportional?
If we 5 hours revision resulted in 60%, then clearly 10 hours revision
?
are not directly proportional.
Q
Electricity cost is directly proportional to the hours of TV watched. If 2 hours are watched, a cost of 14p is incurred. If 7 hours is watched, what cost is incurred?
14 Γ 7 = 49π 2
Exercise 1
2 4 1 3
The rates of currency exchange published in the newspapers on a certain day showed that 14 kroner could be exchanged for 210 pesos. How many pesos could be obtained for 32 kroner?
480 pesos
?
At a steady speed, a car uses 15 litres of petrol to travel 164km. At the same speed, what distance could be travelled on six litres?
65.6km
?
If a 2kg bag of sugar contains 9 Γ 10 6 crystals, how many crystals are there in: a) 5kg? π. ππ Γ ππ π b) 8kg? π. π Γ ππ π c) 0.03kg? π. π Γ ππ π ?
?
?
The amount of energy carried by an electric current is proportional to the number of coulombs. If five coulombs carry 10 joules of energy, how many joules are carried by 6.5 coulombs?
13 joules
?
5 6 7
N If it takes 3 men 5 hours to dig 12 holes, how many hours does it take for 3 men to dig 84 holes?
35 hours
?
If it takes π men π hours to dig π holes can be dug in π hours?
ππ π ?
holes, how many The number of coins required to cover a circular table is proportional to its area. When the radius of the table is 1m the number of coins is 100. How many coins can cover it when the radius is 2m?
400 coins
?
On Utopia Farm, Farmer Giles has a field in which the amount of grass always increases by the same amount each day. Six cows would take three whole days to eat all the grass in the field; three cows would take 7 whole days to eat all the grass in the field. Assuming that each of Farmer Gilesβ cows eats the same amount of grass per day, how long would one cow take to eat all the grass in the field? (Hint: use variables to represent the initial
amount of grass, the increase in grass each day, and the amount of grass one cow eats per day)
63 days
?
Inverse Proportion Suppose Mo runs at a speed of 8m/s for second, and takes 120 seconds to finish. If he ran double the speed at 16m/s, what happens to his time?
It halves!
?
And what do you notice about the speed multiplied by the time in the two cases?
!
When two quantities are inversely (or βindirectlyβ) proportional, their product remains constant.
Example
Q
Four bricklayers can build a certain wall in ten days. How long would it take five bricklayers to build it?
Method 1: Constant Product Suppose it takes five bricklayers π₯ days to build the wall: ?
π₯ = 8 Method 2: Unitary Method Find how many days one bricklayer would take!
?
1 bricklayer takes 40 days 5 bricklayers take 8 days
Test Your Understanding
Q
Eleven taps fill a tank in three hours. How long would it take to fill the tank if only six taps are working?
If using constant product method:
11 Γ 3 = 6 Γ π₯ 33 π₯ = = 5.5 βππ’ππ 6 ?
If using unitary method:
11 taps fill tank in 3 hours 1 tap fills tank in 33 hours 6 taps fills tank in 33 6 = 5.5
hours If you finish that quicklyβ¦
Q
If it takes 3 men 4 hours to dig 6 holes, how many hours does it take for 6 men to build 8 holes?
3 men 4 hours for 6 holes 6 men 2 hours for 6 holes 6 men 1/3 hours for 1 holes Notice that βmenβ and βhoursβ are inversely proportional.
6 men 8/3 hours for 8 holes (which is 2 hours 40 minutes)
Exercise 2
1 2 3 4 5
The length of an essay is 174 lines with an average of 14 words per line. If it is rewritten with an average of 12 words per line, how many lines will be needed?
?
Nine children share out equally the chocolates in a large tin and get eight each. If there were only six children, how many would each get?
ππ ?
A batch of bottles were packed in 25 boxes taking 12 bottles each. If the same batch had been packed in boxes taking 15 each, how many boxes would be filled?
?
In a school, 33 classrooms are required if each class has 32 pupils. How many classrooms would be required if the class size has reduced to 22?
?
If it takes 6 men 25 hours to dig 5 holes, how many hours does it take to dig 5 holes with 10 men?
?
6 7 8
A marathon runner increases their speed by 25%. What percentage does their time decrease by?
20%
?
If it takes π men π hours to dig π how many holes can be dug by π holes, men in π hours?
π
men
π
1 man
ππ π π
men
ππ
hours for
π ?
ππ
holes holes men 1 hour for
ππ /ππ
holes
π
men
π
hours for
π
holes hours for
ππ π/ππ
holes
A fixed amount of water is filled to the top of a cylindrical container. If the height of the cylinder doubles, but the water still fills to the top, what happens to the radius? (Note that π£πππ’ππ = ππππ ππ ππππππ Γ βπππβπ‘
)
It becomes
π ?