1. Year 9: Ratio & Proportion
Dr J Frost
Last modified: 3rd September 2015

2. Starter Puzzles ? ? ? ? (You can work in pairs)
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 1 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 𝟏𝟖𝟎−𝟑𝒙 = 𝟕 𝟔 ∴𝒙=𝟏𝟐
Num sides = 𝟑𝟔𝟎 𝟏𝟐 =𝟑𝟎 ? 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 ? ?

3. 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.
So 1g requires eggs.
So 1250g requires:
×1250=42.857…
So 43 eggs.
Method 2: Ratio Method
! If 𝒙:𝒚=𝒂:𝒃 then 𝒙 𝒚 = 𝒂 𝒃
1250:700=𝑥: = 𝑥 24 𝑥=24× =42.857…
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.

4. Test Your Understanding
Solve the following using both the ‘unitary method’ (i.e. find quantity for one unit) and the ‘ratio method’. Q
The mass of 16cm3 of Neoginium is 24g. What is the mass of 20cm3 of the same element?
Unitary Method
Ratio Method ? ?
16cm3 is 24g.
1cm3 is 𝟐𝟒 𝟏𝟔
20cm3 is:
𝟐𝟎× 𝟐𝟒 𝟏𝟔 =𝟑𝟎𝒈
𝟏𝟔:𝟐𝟎=𝟐𝟒:𝒙 𝟐𝟎 𝟏𝟔 = 𝒙 𝟐𝟒 𝒙=𝟐𝟒× 𝟐𝟎 𝟏𝟔 =𝟑𝟎
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 𝒙 = 𝟏𝟐 𝟕 → 𝒙=𝟒.𝟐

5. 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 2 ×7=49𝑝

6. Exercise 1 ? ? ? ? ? ? ? ? ? (see provided sheet) 5 1 6 2 7 3 4
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× crystals, how many crystals are there in:
5kg? 𝟐.𝟐𝟓× 𝟏𝟎 𝟕
8kg? 𝟑.𝟔× 𝟏𝟎 𝟕
0.03kg? 𝟐.𝟕× 𝟏𝟎 𝟓 =𝟐𝟕𝟎 𝟎𝟎𝟎
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
[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: B 20:31 C 11:20 D 5: E 9:20 Solution: B
[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) 1 ? ? 6 2 ? ? 3 7 ? ? ? 4 ? ?

7. Exercise 1 ? ? ? ? ? (see provided sheet)
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 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 E it is not possible
Solution: E
[IMC 2003 Q25] The width : height ratio of 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: B 3:2 C 3: 2 D 4: E none of these
Solution: A 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
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 N ? N ? 9 ? N ? ?

8. 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.

9. 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
5𝑥=40 𝑥=8
Find how many days one bricklayer would take!
4 bricklayers take 10 days.
1 bricklayer takes 40 days
5 bricklayers take 8 days

10. Test Your UnderstandingQ
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 𝑥= 33 6 =5.5 ℎ𝑜𝑢𝑟𝑠
If using unitary method:
11 taps fill tank in 3 hours
1 tap fills tank in 33 hours
6 taps fills tank in =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? ?
3 men 4 hours for 6 holes
6 men 2 hours for 6 holes
6 men 1/3 hours for 1 holes
6 men 8/3 hours for 8 holes (which is 2 hours 40 minutes)
Notice that “men” and “hours” are inversely proportional but “hours” and “holes” are directly proportional.

11. Exercise 2
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
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
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
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
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 6
A marathon runner increases their speed by 25%. What percentage does their time decrease by?
20%
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
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. 1 ? ? 2 7 ? ? 3 ? 4 8 ? 5 ? ?