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Math examples and help needed

I need examples for the topic tags and homework help with those topics.

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Please break these up into separate questions.  That will allow tutors to handle topics they feel more comfortable with.  Also, the answer will become very unwieldy if all of them are handle in one answer.

 

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A ratio is any two numbers of the same "kind" that can be expressed as a relationship. 2:3 is a ratio that can also be expressed as the fraction 2/3. Proportion has to do with the scale of the ratio being multiplied by a common factor; i.e. 2/3 is proportional to 4/6 and 8/12 and 100/150.

Scale drawings have to do with maintaining a ratio of size proportional to the objects in the drawing. You can see this on maps when they tell you that 1 inch = 1 mile. The true scale would be found by converting miles to inches: 12 in/ft*5280ft/mi = 63360 in/mi. So the scale in this case could be expressed as the ratio 1:63360.

Estimating solutions could be described as using a simple model to make an approximate solution that is not necessarily exact. Using the last example, if I wanted to make a quick calculation of map scale I could've said 12in/ft*5300ft/mi yielding an estimate of (12*53)*102 = (530+106)*102 = 63600. Clearly not an exact value but maybe something workable for the problem at hand that is easier to calculate in your head with a % error of [(63600-63360)/63360]*100 = 0.378%. To tie in to the previous concept, the error expressed as a ratio here would be approximately 38/10000 or 38:10000.

I have no idea what symbolize problem solutions is referring to, I'm assuming it would be a pictorial representation of the problem designed to ease the process of solving it. i.e. Make a picture defining the values involved and how they relate to each other.

Similarity, using triangles as an example, is when the ratios of the sides of two triangles are proportional, making their internal angles equivalent. e.g. We know that a 45/45 triangle has sides whose lengths are of a ratio 1:1:sqrt(2). A triangle that has sides of length 2:2:2sqrt(2) is similar even though the "size" of the triangle is larger. Congruent triangles are exactly the same size. Hope that helps you out!