Tuesday, 8 September 2015

Ratio and Proportion

Ratio and Proportion

Real life applications of ratio and proportion are numerous! The concept occurs in many places in mathematics

When you prepare recipes, paint your house, or repair gears in a large machine or in a car transmission, you use ratios and proportions.

Say a recipe to make brownie requires 4 cups of flour for 6 persons

You may want to know how much flower to put for 24 persons

As a former math teacher, I used to tell my students to set it up as you see below:

You just cannot go wrong when you take this first step:

The setup you see above can be translated into the following proportion:

4 cupsx

The lessons below are geared toward setting up problems like these and solving them

Say you need 4 gallons of paint to paint 1000 square feet of area in your house.

You may want to know how much paint, you will need for 1500 square feet

Try setting up that one yourself as shown above!

Probably one of the best applications we can find is that of gears in a car transmission.

A gear look like a circle and it has teeth all around it.

Gear theory in a vehicle transmission is a complex study. We won't discuss this here in details, or I am going to have to read again my old mechanics books.

Having said that, I will only show you the forest. You will need to do your own research to see all the trees.

My goal is to introduce you to an interesting application of the topic.

In sum, transmissions contain several combinations of large and small gears.

Say for instance a small gear (20 teeth) drives a large gear(40 teeth). 

The large gear will turn at half the speed of the small gear.


However, this situation increases the turning force (or torque) of the large gear.

In general, the larger the gear the bigger the torque.

Therefore, you can see that knowing the ratio of gears, can help us determine the speed and how much torque each gear will deploy.

This concept is important when putting a transmission together. The situation we just describe may for instance set the proper gear ratio for moving a load.

However, at cruising speed, gears may have the same ratio so that the amount of torque that enter the transmission equal the amount of torque that goes out.

Automotive engineers do lots of gear ratios calculation before they can put a transmission together.

This concludes our introduction. Dig deeper by reading the following lessons:

The following lessons about ratio and proportion follow a logical order, so try your best to learn them in order

Some formal definitions of ratio are given to include continued ratio. Learn how to write ratios given some real life situations 

Learn what a proportion is and how to set it up. Learn also about fourth proportional and how to find equivalent proportions

Solving proportions
Shows you how to set up a proportion and solve for x. Some special techniques to quickly simplify a proportion are also introduced

Proportion word problems
Solve some carefully chosen real life proportion word problems.

Working together word problems
Find out how long it takes to do a task with someone else.

Scale drawings
Learn how you can apply proportions to scale drawing

What is the golden ratio
Lastly, challenge yourself. Learn what the golden ratio is and how to derive a formula