Finding and Working with RatiosDate: 12/26/2001 at 13:16:54 From: Hima Mehta Subject: Ratios and proportions 1. What is the ratio of the circumference of a circle to its radius? 2. A snail can move i inches in m minutes. At this rate, how many feet can it move in h hours? Date: 12/26/2001 at 16:50:47 From: Doctor Achilles Subject: Re: Ratios and proportions Hi Hima, Thanks for writing to Dr. Math. A ratio is just a fraction, where the numerator and denominator can have different units. For example, if you travel 60 miles in 2 hours, we can write that as a ratio: 60 miles -------- 2 hours and we can simplify that ratio, using the same techniques you've learned for simplifying fractions: 60 miles 30 miles -------- = -------- 2 hours 1 hour So to find the ratio of a circumference of a circle to the radius of the same circle, we can start by writing expressions for those quantities, and using them to form a fraction. Suppose we have a circle with radius r. The circumference of the circle is pi times the diameter of the circle, and the diameter is twice the radius, so the circumference is pi times twice the radius: circumference = 2 * pi * radius So now let's form the fraction: circumference 2 * pi * radius ------------- = --------------- radius radius which simplifies to just 2 * pi So the ratio of the circumference of a circle to its radius is 2*pi. Your second question asks you "A snail can move i inches in m minutes. At this rate, how many feet can it move in h hours?" We're told something about inches and minutes and we're supposed to find something about feet and hours. How can we do this? We'll need to use a special kind of ratio, called a 'conversion ratio'. A conversion ratio is a fraction where the numerator and denominator express the same quantity using different units. The ones we'll need for this problem are 1 hour 1 foot ---------- and --------- 60 minutes 12 inches Note that since each of these has the same quantity in both the numerator and the denominator, multiplying by either ratio has the same effect as multiplying by 1 - which is to say, it may change the appearance of a quantity, but it won't change the value. (This is sort of the same idea as when we multiply a fraction like 1/2 by a fraction like 3/3 to get 3/6. Since 3/3 is just another way to write 1, 1/2 and 3/6 have the same value, even though they look different.) Let's go back to the problem. We are told that a snail moves i inches in m minutes. Then we are asked to figure out something about his rate. Here is one of the most important equations in math and physics: distance = rate * time Or, dividing both sides by time: distance / time = rate The snail's distance is i inches, and the time is m minutes, so let's put those into the equation: i inches ---------- = rate m mins Now we have a couple of conversion ratios that we can use. We can multiply the left side of this equation by one of our ratios: i inches 1 foot ---------- * ----------- = rate m mins 12 inches Remember that in multiplying fractions, you take the numerator (top) of the first and multiply it by the numerator of the second, and you take the denominator (bottom) of the first and multiply it by the denominator of the second. i inches * 1 foot --------------------- = rate m mins * 12 inches Now, you can treat units (such as inches, feet, hours, whatever) like constants. That means they can cancel each other out. So since we have inches on top and on the bottom of the fraction, it cancels out: i * 1 foot ------------- = rate m mins * 12 Or, more simply: i feet ----------- = rate 12*m mins Let's use our second conversion ratio: i feet 60 mins ----------- * --------- = rate 12*m mins 1 hour And so we multiply the numerator by the numerator and the denominator by the denominator: i feet * 60 mins -------------------- = rate 12*m mins * 1 hour And since "mins" is in both the numerator and the denominator, we can cancel it out: i feet * 60 --------------- = rate 12*m * 1 hour And simplify a bit: 60*i feet ----------- = rate 12*m hours Now, let's just look at the numbers for a minute. We have 60 on top and 12 on the bottom. They both have 6 as a factor, so let's factor 6 out: 6 * 10*i feet --------------- = rate 6 * 2*m hours And the 6's cancel: 10*i feet ----------- = rate 2*m hours Now, both the top and bottom have 2 as a factor, so let's get that out also: 2 * 5*i feet -------------- = rate 2 * m hours And cancel: 5*i feet ---------- = rate m hours So the question we were trying to answer is: how many feet can the snail move in h hours? Well, we know from this that it can move 5*i feet in m hours. How do we figure out h hours? We need to get an h on the bottom. The only way to do that is to multiply the top and bottom of the fraction by h. (Notice that this is just another ratio, h/h is a fraction that equals one.): 5*i feet h ---------- * --- = rate m hours h Becomes: 5*i feet * h -------------- = rate m hours * h Becomes: 5*i*h feet ------------ = rate h*m hours We're almost done. Now we just have to get rid of that pesky little m on the bottom. There's one more ratio we can multiply by: 5*i*h feet 1/m ------------ * ----- = rate h*m hours 1/m Becomes: 5*i*h feet * 1/m ------------------ = rate h*m hours * 1/m Becomes: 5*i*h/m feet -------------- = rate h*m/m hours And finally, we have: 5*i*h/m feet -------------- = rate h hours So our little snail (who is probably all tired out from doing so many ratios), can move 5*i*h/m feet every h hours. Be sure you understand all the steps for this. If you understand how to use conversion ratios, you will find that they are very useful for any type of math or science. I hope this helps. If you have other questions about this or you're still stuck, please write back and I'll be glad to help you some more. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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