  Direct variation is a mathematical term to quantify a relationship between a set of variables. Direct Variations often use Algebra to substitute the numbers in an equation for clarity. Also, this allows you to input unknowns in to your equations to help find out more about a given equation. You can find a quick tune up course in Beginner’s Algebra and a more advanced course Algebra One.

Algebra isn’t a difficult subject to learn, you just have to think logically and clearly. This is especially true when dealing with complex equations like Direct Variation Equations. Making sure you have a good, solid background in mathematics is a great helping hand in being able to use these equations and many more.

Usually, direct variation is applied to a set of two variables where as one increases, so does the other. The variables must increase or decrease in the same amounts as their ratio predetermines. This means if your ratio is 2:1, your first variable must always be twice as large as your second value.

X ∝ Y x 2

If variables are not related as their ratio describes, they are no longer examples of direct variation.

Direct variation can be shown with any number of values.

X ∝ Y x 2 ∝ Z x 4 or

X = Zx4

Y = Z x 2

Subtracting the algebraic values for a real life situation, imagine your local superstore has a sale on. The details of the sale are a little unorthodox, but here they are anyway.

You can buy a single box of chocolates for \$2. The chocolates are usually \$4.

So money saved is \$2 per box of chocolates bought. The ratio of Original Cost:

Discounted Cost is 2 : 1.

Number of Chocolate Boxes                                   Money Saved

1                                                                                       2

2                                                                                       4

4                                                                                       8

6                                                                                       12

20                                                                                     40

With any simple observation technique and analysis, you can easily find a pattern between the two numbers. Most problems in Mathematics will have patterns and training your eyes to see them will make you a better mathematician in no time at all. Pattern recognition is a very important part of mathematics and will allow you to be able to do math in your head with little struggle. Improving your mental math skills will also help to solidify your mathematical understanding and using this Secrets Of Mental Maths course, you can do things in your head that you didn’t think was possible.

In the example, it is shown that the amount of Chocolate Boxes bought give a linear increase in the money saved from buying the chocolate boxes, no matter how many boxes are being purchased at that one time. If we wanted to work out how much of a discount that you might think you would get after buying 512 boxes of chocolate, we can simply use the ratio that we can find out from the table. In this case, the ratio is 2 : 1, because for every box of chocolate bought, two pounds will be saved of the original purchase price of four pounds.

Let x be number of chocolate boxes

Let y be money saved

Our formula would be

(x) x 2 = y

If our ratio were 16 : 1  “x” would be multiplied by 16, meaning you saved 16 times the price.

So following our formula to answer the 512 boxes of chocolate problem gives us

(x) x 2 = y.

512 x 2 = 1 024.

512 boxes x 2 = £1 024 saved.

Obviously, due to the example shown, these numbers are very easily calculated and the equation is simple to write.

This pattern shows that we can just divide the number of boxes of chocolate by the amount of the money off, which will always give us the same number. In our case, it is 2.

We can further demonstrate the linearity of the relationship between the two variables by drawing a graph. If you plot all of the points on a graph with Cost over Money saved, you will have a perfectly straight 45 degree line to show you the linearities in a visual way. This graph line should have the line starting at the coordinates 0,0 and extending in to infinity. We always find that graphs are often poorly drawn by hand and often don’t show us exactly what we need or are prone to errors. To remedy this, we often build our graphs in Microsoft Excel and then export them as PDF or JPEG files. If you don’t like to draw graphs freehand, it might be worth looking at this course for Microsoft Excel to learn the basics and get started – Microsoft Excel 2010 course.

Linear Direct Variations can be tested and proven by trying different ratios and comparing the results. Try graphing the results of all the different ratios you try and you will see that each graph, assuming your ratios are all proportional should make very similar shaped lines that fit the data. This proves that direct linear equations work and you can extrapolate any number of pieces of data from these graphs. Obviously because these graphs are linear, the amount of data that can be found is infinite.

Often, people find that applying formulas like this in real life difficult and only realize that they are using Direct Variation after the fact and have deduced the patterns with their own brain. Our brain is wired to look for many patterns on a day-to-day basis. Unfortunately, it isn’t wired to naturally be able to understand formulae and put them to use in the real world. This means that when you learn a few new formulas, you must take the time to memorize and practice the formulas and write the down. Every so often, checking over the old formulas to refresh yourself will help you apply your knowledge to the real world. These Direct Variation Equations will come in to play throughout your life.

Page Last Updated: February 2020

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