Proportion to Powers of a Variable

Slideshow 24, MathematicsMr Richard Sasaki, Room 307

Objectives

Review Direct Proportion (Grade 7) Understand how to calculate a constant for where Be able to solve problems where variable

Direct Proportion

As you know, there are two main types of proportion.

and .Direct Proportion Inverse ProportionDirect proportional is represented by the symbol . ‘’

An equation representing direct proportion is in the form

where .𝑦=𝑘𝑥

If is unknown, but we have a value for and at a point, we can calculate (which is constant for all points).

Direct Proportion

Example

Two variables exist where . At a point, when , . Show there exists a specific equation for in terms of and calculate when .

As, there exists some relation for all .

For when , , we get .As holds for all and , for all .When .

𝑦=𝑥When

𝑦=9 𝑥When

𝑦=𝑥9When

𝑦=−𝑥7When

A pencil costs the same, no matter how many are

bought.

𝑦=18 𝑥

𝑦=18 ∙40=720𝑌𝑒𝑛

As, there exists some relation for all .For when , , we get .

As holds for all and , for all .

When .

Direct Proportion with Other Powers

We know that if two variables exist where , this implies for all .

If is directly proportional to the square of , this can be represented as .𝑦 ∝𝑥2

For some unknown value where , this relationship implies that .𝑦=𝑘𝑥2

In fact, the same implication applies for any power. If , where , we can say that .𝑦=𝑘𝑥𝑎

Direct Proportion with Other Powers

Example

Two variables exist where . At a point, when , . Find a specific equation for in terms of and calculate when .

As when and , . 7, we get . When , . 56

Try the next worksheet!

わんわんわん。

Meow.

𝑦=3 𝑥2When 𝑦=6 𝑥2When

𝑦=7 𝑥2When 𝑦= 𝑥2

3888When

𝑦=13 𝑥2

2548𝑚

𝑦=25 𝑥3

6

18𝑐𝑚

Real World Regular Proportion Problems

Try to solve some of the problems like the one below applying your general understanding of proportion.Example

Tojiro walks at a constant speed for 15 minutes and covers three-quarters of a kilometre. How fast was he walking? Please give your answer in kilometres per hour.

Hi! I’m Tojiro!3415×60¿34×4¿3𝑘h

− 1

My name is Ryoma.

Hi! I’m Tojiro!

My name is Ryoma.

24 𝑘𝑚

0.012𝐵𝑇𝐶

270𝑔

185 𝑠𝑒𝑐𝑜𝑛𝑑𝑠𝑖𝑛𝑡𝑜𝑡𝑎𝑙⇒ 65𝑠130𝑤𝑜𝑟𝑑𝑠

7 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 h𝑠 𝑜𝑡𝑠

40𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑠𝑒𝑟𝑣𝑒𝑠