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Ratio and Proportion

A ratio is an ordered pair of real numbers,

written ab, with b ? 0.

Remark You should remember that a fraction is

also an ordered pair of numbers a and b with b ?

0, except that a and b must be integers. In other

words is a ratio, but is not

a fraction.

Ratio and Proportion

- Recall that a fraction is always used for

part-to-whole comparison, but a ratio can be used

for - part-to-part comparison
- part-to-whole comparison
- other comparisons such as length-to-width.

Ratio and Proportion

Equality of ratios Two ratios ab and cd are

equal if and only if a

d b c i.e. a b c d if and only

if insideinside outsideoutside

Proportions

A proportion is a statement that two given ratios

are equal.

- Practical examples
- If a cocktail recipe calls for 1 part of 7-up and

2 parts of orange juice, then you need to use the

same ratio (no matter how much of cocktail want)

in order to keep the taste consistent. - If you are mixing paint to paint your house, you

need to keep the ratio (of color pigments to

white paint) constant to ensure that the color

will remain exactly the same.

Proportions

- Practical examples
- If city tax rate is 7.75 to every 100 of

purchase, then you have to use the same ratio no

matter how much your purchase is (because it is

the law). - If you want to enlarge a 45 picture, then you

should keep the same ratio in dimension to avoid

distortion.

Proportions

Practical examples 5. In music, it is the

ratio of frequencies that determines the scale.

The actual frequencies can vary a small amount.

In C major scale, we have the following ratios

(using the just-intonation scale)

Some situations where proportions are less

understood. 1. Can Giants exist?

Most of us like to watch movies and animations of

giants or gigantic animals. But can we simply

increase the size of a person (or an animal) and

yet expect all other physical properties to

remain the same?

Some situations where proportions less

understood. 1. Can Giants exist?

If we triple the size of a cube using the same

material, the pressure (lb/ft2) on the bottom of

the cube will become 3 times as big even though

the foot print of the big cube is bigger.

Some situations where proportions are less

understood. 1. Can Giants exist?

Consequently, if a giant has the same body shape

as a normal person and yet 3 times as high, his

feet will take 3 times of the normal pressure,

and his blood pressure has to be 3 times as high

to pump blood from the feet back to his

heart. This will put too much stress on the body

and hence giants cannot live long if they ever

exist.

Some situations where proportions are less

understood. 2. Why babies can craw on their

knees for a long time?

On the other end of the spectrum, if a baby is

only 1/3 of our height, then the pressure on its

knees will be only 1/3 of ours, and such a small

pressure will not cause pain in the knees.

Some situations where proportions are less

understood. 3. Are babies just miniature

adults?

No, a babys body parts have proportions very

different from those of adults. For instant, a

baby has much smaller arm length / body

length ratio than that of an adult.

Some situations where proportions are less

understood. 3. Are babies just miniature

adults?

No, a babys body parts have proportions very

different from those of adults. For instant, a

baby has much smaller arm length / body

length ratio than that of an adult.

Some situations where proportions are less

understood. 4. How does the face change

with age?

The skull does not grow proportionally, hence the

face does not simply enlarge when a child grows

up.

Can Giants exist?

Tallest man ever lived. Tallest man in Iceland

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