Proportions

Proportions

1. Direct and Inverse

2. Manipulating Proportions

3. Conversion Factors

4. Percent

What are proportions?

• Proportions are a method of relating one quantity to another

• For example: 5 apples cost 30 cents.

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

1.1 Direct proportion

• 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

Direct proportion example:

A ten foot pole casts an eight foot shadow. How long is a pole which casts twelve foot shadow?

•  $\frac{pole}{shadow}=\frac{10}{8}$  

•  $\frac{pole}{12}=\frac{10}{8}$  

• pole = 15

Inverse proportion

• 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.

Inverse proportion example:

If x and y are inversely proportional and x=10 when y=6, what is x when y=4?

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

Combined proportion

• 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  $x=k\frac{yz}{w}$  

Combined proportions example

Given that x is directly proportional to y and to z and is inversely proportional to w, and that x = 4 when (w, y, z) = (6, 8, 5), what is x when (w, y, z) = (4, 10, 9)?

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

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

Example:

It takes 3 days for 4 people to paint 5 houses. How long will it take 2 people to paint 6 houses.

• 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)

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

Example:

It is 4 o'clock now. How many minutes will pass before the minute and hour hands of a clock are coincident(at the same exact place)?

• 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