One of the powers of algebra is that it teaches you how to look at information in a way that causes you to see how to get at the answer. This may sound a bit funny, but it’s true. When answer isn’t quite as evident as 2+2 it might not be as easy to see the = 4. However, thanks to algebraic techniques when you do have all the information that you need, you can find ‘the missing piece’. We’ll go through this by example.
An airline passenger has taken an overseas flight across the North Atlantic. She has noticed that the time it took was not always the same. On one flight she went up to the cockpit, where she asked the pilot about this. The men in the small, cramped chamber smiled and glanced at each other.
“Well,” one of them laughed, “I won’t tell you directly, but I will tell you this: It takes 4 hours to fly 1710 km when we have a tail wind. If we’re going against the wind, however, it takes us 5 hours just to go 1370 km.”
The pilot chuckled and asked, “How fast is the wind up here at 25,000 km blowing?”
The lady passenger looked at them all and replied, “Well, that it is a bit funny, but since its such a long flight I might as well go ahead and figure it out.” She knew enough to know that her answer would be inexact; but you have to had enough physics lessons to know just and how and why.
She went back to her seat where she took out a pen and a notebook. She determined the following steps. First, she would find out how fast the plane was flying when the wind was behind them. 1710 km in 4 hours. Well, 1710 ÷ 4 = 427.5 km/hr.. Okay, she thought, this tells me something, but it doesn’t tell me how quickly the wind is blowing at all. Next, she tried dividing 1370km by 5 hours. As it happens: 1370 ÷ 5 = 274.4km/hour. Now she knew more, but this still wasn’t it. She gazed briefly out the window at the clouds and wondered, how fast is the wind going? She leaned back in her chair…it occurred to her: “what’s the difference?” Well, obviously the difference in speed was caused by the wind direction and speed. Then she remembered something that was covered in both physics and linear equations and scales…things balance out…forces balance each other out. Then suddenly it dawned on her.
Although strange, she blurted aloud, “it’s just the other half of the same equation”, and began to calculate the difference between the speed of the aircraft under the two conditions. A difference, remember, is an arithmetic relationship. So: 427.5 – 274 = 153.5 km / hour. That’s the difference in the speed of aircraft with the wind blowing along with the plane, compared to when the aircraft flies against the wind. As long as you can safely assume that the amount of energy the plane is using is the same, and is the very aircraft, then what is causing the difference is the wind….but since forces always balance out, you can tell that that difference is the ‘same difference’ as the one that is causing the plane to go at the given pace. Therefore, you really can conclude, that the wind is blowing at that very 153.5 km.hr..
Once you see that the air speed is the exact same value as the difference in aircraft speed, then the hidden mystery of the wind’s force is suddenly revealed and is in fact quite obvious. The thing is that you do ‘need to see it’….In this sense algebra can ‘make you feel funny’ in that this kind of transformation happens all the time when you apply this kind of math. That is what makes it so very powerful and useful. Every time that happens it reinforces the significance of interrelationships within nature and in human systems.
