SOLUTION: For Christmas, each member of a class sends the other classmates card. If 992 cards are exchanged, find the number of pupils in the class

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Question 1198805: For Christmas, each member of a class sends the other classmates card. If 992 cards are exchanged, find the number of pupils in the class
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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For Christmas, each member of a class sends the other classmates card.
If 992 cards are exchanged, find the number of pupils in the class
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Let the number of members be n.

Then the number of cards is n*(n-1)   (each member sends (n-1) cards to other members).


Thus an equation to find n is

    n*(n-1) = 992.


From this equation

    n^2 -n - 992 = 0

    (n-32)*(n+31) = 0

    n = 32  or  n = -31, and we choose the positive root.


ANSWER.  There are 32 pupils in the class.

Solved.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Each of n students sends (n-1) cards; the total number of cards is n(n-1). The equation for solving the problem using formal algebra is then

n%28n-1%29=992

If you solve that using formal algebra, then

n%5E2-n=992
n%5E2-n-992=0

To solve that by the formal mathematical method of factoring, you need to find two whole numbers whose difference is 1 and whose product is 992.

But that is exactly what the original equation tells you to do -- so the formal algebra gets you no closer to the solution than you were at the start.

So if formal algebra is not required, use logical reasoning, a bit of estimation, and common sense to find the answer.

One way to do that is to reason that n(n-1) is very close to n(n) = n^2; so solve n^2=992 to get an estimate of the answer.

sqrt(992) = 31.496 to 3 decimal places, so n and (n-1) have to be 32 and 31; that means the number of students in the class is n=32.

Another way to get the answer using logical reasoning is to see that 30*30 = 900 and 40*40 = 1600. Since 992 is much closer to 900 than 1600, the numbers n and n-1 have to be consecutive whole numbers between 30 and 40 and much closer to 30; and the units digit of the product of the two whole numbers is 2. That makes 31 and 32 the clear choices, so the number of students in the class is 32.