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Question 483096: I am struggling with solving the following word problem and need to know how to set up the equation:
Suppose 1 person tells a story to 4 people in 20 minutes. Then each of those 4 people tells the story to 4 other people in 20 minutes. If this pattern continues, how much time will pass before 5,000 people have heard the story?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! A little complex, but here goes with a method of arriving at a solution. This first part is formal way of arriving at the answer. After we get through this, I'll discuss a little easier way (at least I think it is easier to understand) that uses more number sense than relying on a formal equation.
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First, let's look at the equation for the sum of a geometric progression to n terms. The general form of the equation is (according to a math textbook):
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a + ar^1 + ar^2 + ar^3 + ... ar^(n-1) = (a[r^n - 1])/(r -1)
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Note that the right side of this equation gives the sum of this series.
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If you look at the problem you were given you will note that it progresses as follows:
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1 person tells 4. Then those 4 each tell 4 more and this adds 16 more people who know. Then those 16 each tell 4 more for a increase of 16 times 4 = 64. Then those 64 each tell 4 more for an increase of 256 who know.
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When you compare this to a geometric progression you can see that it is of that form ... namely let a = 1 and r = 4 and you have:
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1 + 1*4^1 + 1*4^2 + 1*4^3 + ... 1*4^(n-1)
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Therefore, the sum will be (a[r^n - 1])/(r -1) if a = 1, r = 4, and n is the unknown number of terms you are trying to determine. Substitute a = 1 and r = 4 into this sum and you have:
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(1[4^n - 1]/(4-1))
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The denominator (4 -1) is 3 and the 1 times a quantity is just the quantity. So the sum formula reduces to:
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[4^n - 1]/3
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But the problem wants the sum to be 5000, the total number of people who know the story. So set the sum equal to 5000 as follows:
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[4^n - 1]/3 = 5000
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Get rid of the denominator of 3 by multiplying both sides by 3 and the equation becomes:
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[4^n - 1] = 15000
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The brackets can be removed and then get rid of the -1 by adding 1 to both sides of the equation and you have:
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4^n = 15001
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We need to solve for n, an exponent. Let's take the logarithm of both sides. Use base 10 logs:
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log(4^n) = log(15001)
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The rules of logs tell you that you can bring the exponent out as a multiplier of the log. So bring the n out as a multiplier and you have:
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n*log(4) = log(15001)
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Use a scientific calculator to find that log(4) is 0.602059991 and log(15001) is 4.176120211. Substitute these values into the equation and you have:
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n*0.602059991 = 4.176120211
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Solve for n by dividing the right side by the multiplier of n which is 0.602059991:
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n = 4.176120211/0.602059991
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Do the division and you get the answer than n = 6.936385528
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This value of n would give you an answer of 5000 exactly. However, it would involve telling fractions of persons instead of whole people. So let's just say that n = 7 and we'll have a little more than 5000 people who know the story. As a matter of fact the sum will be:
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(a[r^n - 1])/(r -1) in which a = 1, r = 4, and n = 7.
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Substitute the values for a, r, and n and the sum becomes:
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(1*(4^7 - 1))/(4-1) = (4^7 - 1)/3 = (16384 - 1)/3 =16383/3 = 5461 people
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Now return to the left side of the equation for a geometric series. With a = 1 and r = 4 the left side becomes:
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1 + 1*4^1 + 1*4^2 + 1*4^3 + ... 1*4^(n-1)
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Remember that we said n was 7. This being the case our last term in the progression should have the exponent (n-1) which would be 6. Notice also that the a = 1 multiplier can be dropped because multiplying each term by 1 doesn't change anything. So the progression applicable to this problem would be:
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1 + 4^1 + 4^2 + 4^3 + 4^4 + 4^5 + 4^6
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Think about it. It took 20 minutes to get from the first term to the second term. Then it took another 20 minutes to get from the second term to the third term. An additional 20 minutes was needed to get from the third term to the fourth term. An so on. To get all the way to the last term in this progression from the very first term will require 6 transitions of 20 minutes each, for a total of 120 minutes. That means beginning with a single person knowing the story, 120 minutes later (2 hours) a total of 5461 persons know the story.
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Sort of complex, but that's a formal way of getting the answer.
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In a less formal way, you could have just written down the terms:
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1 + 4 + 16 + 64 + 256 + 1024 + 4096 + 16384 + ...
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which recognizes the fact that each 20 minutes increases the number of people who know the story by a factor of 4. To get from the first term to the second term, multiply the first term by 4. To get from the second term (4) to the third term, multiply the second term (4) by 4. To get from the third term (16) to the fourth term multiply the third term (16) by 4. To get from the fourth term (64) to the 5th term, multiply the fourth term (64) by 4. And so on.
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Since the last calculation is 16384 you have obviously exceeded 5000 by quite a bit. So stop and now use a calculator to begin adding terms. Adding all the numbers up from 1 through 4096 will give you the total of 5461 persons. It's a method that is probably faster and easier to understand, but a little less elegant than using the equation for the sum of a geometric progression of n terms. You still get the answer of 120 minutes (or 2 hours) to get from the 1 person knowing the story to the 5461 people knowing it. (There are 6 of the 20 minute jumps to get from the first term (1) to the last term (4096).
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Hope that I didn't confuse you with the math. Maybe you gained enough insight into the problem to see another logical way of getting the answer. In any case, check my work. It's very late and I'm out of coffee. So I may have made some dumb mistakes. Hope not. Good luck.
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