1. Other Proportion Problems
Slideshow 32, Mathematics
Mr Richard Sasaki

2. Objectives Being able to calculate mixed worded proportional values
Understand how lengths and areas relate to one another

3. Notation
Recently, some new notation is being used so I want to make sure that you all understand what it means.
(𝑒𝑔:4∙5)
Multiplied by ∙
Implies that ⇒ ∴
Therefore

4. Mixed proportion questions
We need to be able to identify what kind of question is being asked in a particular context.
We need to understand whether the question refers to a directly proportional relationship or an inversely proportional relationship.
Note: 𝑥 1 and 𝑦 1 refers to 𝑥 and 𝑦 at a specific point. These are unknowns, not variables.

5. Example
A man bakes 3000g of bread in 6 hours. He then bakes 𝑦 1 g of bread in 8 hours.
What kind of proportion is this?
Direct Proportion
What do the two variables represent?
𝑦= grams of bread, 𝑥= number of hours
Calculate 𝑦 1 .
As 𝑦∝𝑥, 𝑦=𝑘𝑥.
At a point, 𝑥=6, 𝑦=3000.
So 3000=6𝑘∴𝑘=
500 ⇒
𝑦=500𝑥
⇒ 𝑦 1 =500∙8⇒
𝑦 1 =4000 g

6. Direct Proportion
The total cost of 𝑥 cars
1200=8𝑘⇒𝑘=150
𝑦 1 =150∙3=450 Yen
She is always drinking at the same rate.
4=16𝑘⇒𝑘= 1 4
𝑦 1 = 1 4 ∙12=3
15=60𝑘⇒𝑘= 1 4
8= 1 4 ∙ 𝑥 1 ⇒32
Inverse Proportion
𝑦 refers to the spice, 𝑘 refers to the cayenne pepper
𝑘=5, 𝑦= 5 𝑥

7. Proportion in shape
Relationships in shape are clear as long as you know the formulae!
If the area 𝐴 remains constant, what is the relationship between 𝑥 and 𝑦?
𝐴 𝑐𝑚2
𝑦 𝑐𝑚
Inverse, 𝑦= 𝐴 𝑥
𝑥 𝑐𝑚

8. Direct Proportion
Inverse Proportion
Inverse Proportion
Direct Proportion
𝑘=1 (they are the same size)
𝑘=4
Directly proportional
21 𝑐𝑚
Directly proportional

9. Additional Help saves time
If it takes one man an hour to unload bricks from a van, how long would it take five men? (The men all work at the same rate as each other.)
5 men would take of the time of 1 man.
1:5=𝑥:1
⇒𝑥= 1 5 of an hour
60∙ 1 5 =12 minutes

10. 6:3=8:𝑥⇒3÷6×8=4
6:3=𝑥:8⇒6÷3×8=16
8:3=6:𝑥⇒3×6÷8=2.25
(2 hours, 15 min )
500:6=𝑥:11⇒500÷6×11= g
2 processors, 30 minutes
1 processor, 60 minutes
5 processors, 12 minutes
5 processors, double the length, 24 minutes
3 cats, 4 minutes, 160𝑔 of food
3 cats, 1 minute, 40𝑔 of food
3 cats, 6 minutes, 240𝑔 of food
1 cat, 6 minutes, 80𝑔 of food
5 cats, 6 minutes, 400𝑔 of food