Other Proportion Problems Slideshow 32, Mathematics Mr Richard Sasaki

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

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

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.

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

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

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

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

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 1 5 of the time of 1 man. 1:5=𝑥:1 ⇒𝑥= 1 5 of an hour 60∙ 1 5 =12 minutes

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=916. 6 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