Year 9: Ratio & Proportion
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
www.drfrostmaths.com
Last modified: 3rd September 2015
Starter Puzzles
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.
What’s the new ratio of black to
yellow fish?
Answer: 𝟓: 𝟏
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 = 𝟒𝟖 ?
(You can work in pairs)
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 ‘parts dog’ same in each.
So cats to dogs 𝟏𝟐: 𝟐𝟎 and cats to sharks
𝟐𝟎: 𝟑𝟓. So ratio of cats to sharks 𝟏𝟐: 𝟑𝟓.
?
N
There are two regular polygons, 𝐴 and 𝐵. The
ratio of their exterior angles is 1: 3. The ratio of
their interior angles is 7: 6. Prove that polygon 𝐴
has 30 sides.
(Hint: it may be helpful to start with the exterior angle
of 𝐴 being 𝑥 and working from there)
𝟏𝟖𝟎 − 𝒙
𝟕
=
𝟏𝟖𝟎 − 𝟑𝒙 𝟔
Num sides =
𝟑𝟔𝟎
𝟏𝟐
?
= 𝟑𝟎
∴ 𝒙 = 𝟏𝟐
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?
Method 1: Unitary Method
(Find number of eggs for 1g)
700g require 24 eggs.
24
So 1g requires 700 eggs.
So 1250g requires:
24
× 1250 = 42.857 …
700
So 43 eggs.
?
Method 2: Ratio Method
𝒙
𝒂
! If 𝒙: 𝒚 = 𝒂: 𝒃 then 𝒚 = 𝒃
We say that the number of eggs and the mass
of the pudding are ‘directly proportional’.
There is some underlying ‘scaling’ between
the two things.
1250: 700 = 𝑥: 24
1250
𝑥
=
700
24
1250
𝑥 = 24 ×
= 42.857 …
700
Test Your Understanding
Solve the following using both the ‘unitary method’ (i.e. find quantity for one
unit) and the ‘ratio method’.
The mass of 16cm3 of Neoginium is 24g. What is the mass of
Q
20cm3 of the same element?
Unitary Method
Ratio Method
16cm3 is 24g.
𝟐𝟒
1cm3 is 𝟏𝟔
20cm3 is:
?
𝟐𝟒
𝟐𝟎 ×
= 𝟑𝟎𝒈
𝟏𝟔
𝟏𝟔: 𝟐𝟎 = 𝟐𝟒: 𝒙
𝟐𝟎
𝒙
=
𝟏𝟔 𝟐𝟒?
𝟐𝟎
𝒙 = 𝟐𝟒 ×
= 𝟑𝟎
𝟏𝟔
If you finish:
Q
There are 7 Aardvarks and 12 Buffalo in a classroom. The ratio of Aardvarks to
Buffalo is 𝑥 ∶ 𝑥 + 3. What is 𝑥?
𝒙 + 𝟑 𝟏𝟐
=
𝒙
𝟕
?→
𝒙 = 𝟒. 𝟐
Direct Proportion
! Two things are directly proportional if they in the same ratio.
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 “hours revised for maths exam” and “maths exam mark” not likely
to be directly proportional?
If we 5 hours revision resulted in 60%, then clearly 10 hours revision
wouldn’t result in 120%! So while the two
? things are ‘correlated’, they
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
× 7 = 49𝑝
2
Exercise 1
(see provided sheet)
The rates of currency exchange published in
1 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
?
2 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
?
6
3 If a 2kg bag of sugar contains 9 × 10
crystals, how many crystals are there in:
a) 5kg?
𝟐. 𝟐𝟓 × 𝟏𝟎𝟕
b) 8kg?
𝟑. 𝟔 × 𝟏𝟎𝟕
c) 0.03kg? 𝟐. 𝟕 × 𝟏𝟎𝟓 = 𝟐𝟕𝟎 𝟎𝟎𝟎
?
?
?
4
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 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
?
6 [IMC 2009 Q9] Joseph’s flock has 55% more
sheep than goats. What is the ratio of goats to
sheep in the flock?
A 9:11
B 20:31
C 11:20
D 5:9
E 9:20
Solution: B
?
7 [IMC 2005 Q6] A shop advertise ‘Buy one, get
one at half price’. For this offer, the average
cost per item is the same as:
A Two for the price of one
B Three for the price of one
C Three for the price of 2.
D Four for the price of three
E Five for the price of four
Solution: D (make the original cost of each
item £1 for simplicity, and go from there)
?
Exercise 1
8 [IMC 1997 Q18] Seven rolls weigh the
same as four crumpets. Five scones
weigh the same as six crumpets. If
each crumpet weighs 𝑐 grams, and
each roll weighs 𝑟 grams, and each
scone weighs 𝑠 grams, which of the
following must be true?
A 𝑟<𝑠<𝑐
B 𝑠<𝑟<𝑐
C 𝑠<𝑐<𝑟
D 𝑐<𝑠<𝑟
E 𝑟<𝑐<𝑠
Solution: E
?
9 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
?
(see provided sheet)
N If it takes 𝑎 men 𝑏 hours to dig 𝑐 holes, how
many holes can be dug in 𝑑 hours?
𝒄𝒅
𝒃
[IMC 2010 Q22] Curly and Larry like to have their
N
orange squash made to the same strength.
Unfortunately, Moe has put 25ml of squash with
175ml of water in Curly’s glass and 15ml of
squash with 185ml of water in Larry’s glass. How
many millilitres of the mixture in Curly’s glass
must be put into Larry’s glass so that they end up
with drinks of the same strength?
A 5
B 7
C 10
D 12
E it is not possible
Solution: E
[IMC 2003 Q25] The width : height ratio of
N
television screens is changing from the
traditional 4 : 3 to the widescreen 16 : 9. If a
traditional screen and a widescreen have the
same area, then what is the ratio widescreen
width : traditional width?
A 2: 3
B 3: 2
C 3: 2
D 4: 3
E none of these
Solution: A
?
?
?
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?
It remains the
? same.
! When two quantities are inversely (or
‘indirectly’) 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
Method 2: Unitary Method
Suppose it takes five bricklayers
𝑥 days to build the wall:
5 × 𝑥 = 4 × 10
? 40
5𝑥 =
𝑥=8
Find how many days one bricklayer
would take!
? days.
4 bricklayers take 10
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 × 3 = 6 × 𝑥
33
𝑥=
= 5.5 ℎ𝑜𝑢𝑟𝑠
6
If using unitary method:
11 taps fill tank in 3 hours
1 tap fills tank in 33 hours
33
6 taps fills tank in 6 = 5.5 hours
?
If you finish that quickly…
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?
Notice that “men” and “hours” are inversely
3 men 4 hours for 6 holes
proportional but “hours” and “holes” are directly
6 men 2 hours for 6 holes
?
proportional.
6 men 1/3 hours for 1 holes
6 men 8/3 hours for 8 holes (which is 2 hours 40 minutes)
Exercise 2
1
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?
𝟐𝟎𝟑
6
A marathon runner increases their speed
by 25%. What percentage does their time
decrease by?
20% ?
7
If it takes 𝑎 men 𝑏 hours to dig 𝑐 holes,
how many holes can be dug by 𝑑 men in 𝑒
hours?
𝒂 men 𝒃 hours for 𝒄 holes
1 man 𝒃𝒂 hours for 𝒄 holes
𝒅 men 𝒃𝒂 hours for
? 𝒄𝒅 holes
𝒅 men 1 hour for 𝒄𝒅/𝒃𝒂 holes
𝒅 men 𝒆 hours for 𝒄𝒅𝒆/𝒃𝒂 holes
8
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 𝑣𝑜𝑙𝑢𝑚𝑒 =
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡)
It becomes 𝟐 times smaller.
?
2
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?
𝟏𝟐
?
3
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?
𝟐𝟎
?
4
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?
𝟒𝟖
?
5
If it takes 6 men 25 hours to dig 5 holes,
how many hours does it take to dig 5 holes
with 10 men?
𝟏𝟓 𝒉𝒐𝒖𝒓𝒔
?
?