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Representing Ratio and Proportion – Original Image










































An original image has width 8 inches and length 12 inches. These dimensions can be represented by the point (8, 12) on a graph. How can you use the graph to figure out if a change in dimensions is proportional?

Comments for Representing Ratio and Proportion – Original Image

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Sep 19, 2012
Resizing an Original Image
by: Staff


Answer:

Part I

Ratios and proportion can be applied to many routine problems encountered in everyday life.


Direct Proportions:

         A direct proportion between two variables means that one variable is a constant multiple of the other variable.

         The terms “proportional relationship”, “direct variation” and “direct proportion” all mean the same thing. The terms are interchangeable.

         The equation for a “proportional relationship” ALWAYS looks like this:

             y = kx

         k is a constant. It is called the “constant of variation”.


      I’m going to rearrange that equation so it looks like this:

             y/x = k

         In other words, a direct proportion means that DIVIDING the value of ONE VARIABLE BY THE value of the OTHER variable is a CONSTANT value (k, the constant of variation). (I’ll apply this idea to your problem shortly).

         For example, if you have a job and are paid by the hour, your pay is proportional to the number of hours you work.

             The more hours you work, the higher your pay.

             The fewer hours you work, the lower the pay.

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Sep 19, 2012
Resizing an Original Image
by: Staff


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Part II

             However, your total pay divided by the total number of hours you work is a constant value.

             If your rate of pay is $25 per hour and you work 10 hours, you will earn $250.

             $250 ÷ 10 hours = $25 per hour

             If your rate of pay is $25 per hour and you work 1000 hours, you will earn $25,000.

             $25,000 ÷ 1000 hours = $25 per hour

             Pay divided by hours always EQUALS the same number, the CONSTANT of variation. In this example the constant of variation is 25. The constant of variation (25) will not change, regardless of the number of hours you work.

         The equation for resizing the image in your problem statement is the same equation discussed in the preceding paragraphs. It will look like this:

             y = kx

             (image length) = k * (image width)


             You need to calculate k, the constant of variation, before the proportion can be graphed.

             The constant of variation for the image is:

             (image length) / (image width) = k


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Sep 19, 2012
Resizing an Original Image
by: Staff


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Part III

             Since the original image has width 8 inches and length 12 inches:

             (12 inches) / (8 inches) = k

             (12 inches) / (8 inches) = 1.5

             k = 1.5


         The final equation is:

             (image length) = k * (image width)

             (image length) = 1.5 * (image width)

         For the sake of simplicity, let y = “image length” and x = “image width”:

             y = 1.5 * x

             y = 1.5x


Ratio and Proportion






Final Answer

         How can you use the graph to figure out if a change in dimensions is proportional?

             If a plot of the image dimensions (width and length) fall on the graph of the equation y = 1.5x, they are in the correct proportion.

             If the image dimensions do not fall on graph of y = 1.5x, the proportion is incorrect.






Thanks for writing.

Staff
www.solving-math-problems.com



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