Proportions

Proportions are a method of relating one quantity to another

For example: 5 apples cost 30 cents.

 $\frac{apples}{\cos t}=\frac{5}{30}$  

In a directly proportional relationship, the quotient of the two quantities is a constant when all else is held constant.

When one of the quantities increases, the other does also.

Ratios are not always written as fractions. Sometimes they are written with colons as a : c = 5 : 30

 $\frac{pole}{shadow}=\frac{10}{8}$  

 $\frac{pole}{12}=\frac{10}{8}$  

pole = 15

Two quantities are in inverse proportion if they have a constant product when all else is held constant.

When one quantity increases in an inverse proportion, the other quantity decreases.

For example: if the area of a rectangle is 40, the length and width are inversely proportional: lw=40. If the length of the rectangle is doubled and the area is to remain the same, we must devide the width by 2.

The constant product in inverse proportions and the constant quotient in direct proportions are often called the constant of proportionality.

We are told x*y=6*10, x=60/y. so when y=4, we have x=15

x is directly proportional to y and to z. 

also x is inversely proportional to w.

With more than three variations, we can combine them together as  $x=k\frac{yz}{w}$  

 $\frac{xw}{yz}=k$  , $\frac{\left(4\right)\left(6\right)}{\left(8\right)\left(5\right)}=\frac{3}{5}$  

 $x=\frac{3}{5}\cdot\frac{yz}{w}\ =\ \frac{3}{5}\cdot\frac{\left(10\cdot9\right)}{4}=\frac{27}{2}$  

Three variations 'day', 'people' and 'house'

people is inversely proportional to days, (more people less days)

houses is directly proportional to days. (more houses more days)

people is directly proportional to houses. (more people, more houses)

 $\frac{people\cdot day}{house}=\frac{4\cdot3}{5}=\frac{2\cdot x}{6}$  , $x=\frac{36}{5}$  

Why is it related to proportions? (minute and hour hands?)

 $\frac{Minute}{Hour}=\frac{60}{5}$  , for an hour, hour hand move 5 minute distance as minute hand move 60 (whole circle)

At 4 o'clock, hour hand is at 20 minutes point(4/12*60). x = 20 + 5*x/60