SOLUTION: Some children share an amount of money equally and each of them gets $20. If the number of children sharing the same amount of money is reduced by 5, each child will get $40. Find

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Question 1169761: Some children share an amount of money equally and each of them gets $20. If the number of children sharing the same amount of money is reduced by 5, each child will get $40. Find the amount of money.
Found 4 solutions by ankor@dixie-net.com, MathTherapy, ikleyn, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Some children share an amount of money equally and each of them gets $20.
If the number of children sharing the same amount of money is reduced by 5, each child will get $40.
Find the amount of money.
:
let a = amt of money
let c = no. of children
:
Two equations
a%2Fc = 20
a = 20c
and
a%2F%28%28c-5%29%29 = 40
a = 40(c-5)
a = 40c - 200
:
a=a, therefore
40c - 200 = 20c
40c - 20c = 200
c = 200/20
c = 10 children
:
Find the amt
{{[a/10}} = 20
a = 10*20
a = $200 is the amt
:
Check
200%2F5 = 40

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Some children share an amount of money equally and each of them gets $20. If the number of children sharing the same amount of money is reduced by 5, each child will get $40. Find the amount of money. 
Let amount of money be A
Then number of children = A%2F20+
Reducing number of children by 5 makes the new number A%2F20+-+5
So, amount each child would get = A%2F%28A%2F20+-+5%29, and so, we get: matrix%281%2C3%2C+A%2F%28A%2F20+-+5%29%2C+%22=%22%2C+40%29
              
                                                            matrix%281%2C3%2C+A%2F%28%28A+-+100%29%2F20%29%2C+%22=%22%2C+40%29                    
                                                            
                                                            matrix%281%2C3%2C+A+%2A+%2820%2F%28A+-+100%29%29%2C+%22=%22%2C+40%29
                                                            matrix%281%2C3%2C+20A%2F%28A+-+100%29%2C+%22=%22%2C+40%29
                                                            
                                                            matrix%281%2C3%2C+A%2F%28A+-+100%29%2C+%22=%22%2C+2%29 ----- Factoring out LCD, 20, in numerator
                                                           
                                                            A = 2(A - 100) ---- Cross-multiplying
                                                            A = 2A - 200
                                                            A - 2A = - 200
                                                          - A = - 200 
                                       Amount of money, or

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

            For many students,  such problems perplex them.

            Surely, the most intriguing part is to setup them correctly.

            Here I propose my version of setup and solution. 


Let M be the amount of money: the unknown value under the problem's question.


In the basic scenario, the number of students is  M%2F20.

In the hypothetical scenario  ( " if " ), the number of students is  M%2F40.


The difference is 5, which gives you this equation

    M%2F20 - M%2F40 = 5.


At this point, the setup is just done and completed.


The solution of the equation is in couple of lines.  Multiply both sides by 40.  You will get

    2M     -  M  = 5*40

         M       = 200.


ANSWER.  The amount of money is 200 dollars.


CHECK.   200%2F20 = 10 students;   200%2F40 = 5 students;   10-5 = 5 students,   which is PRECISELY CORRECT !

Solved.

-------------

When you solve it in this way,  you feel a joy of getting mathematically beautiful solution  (!)


/\/\/\/\/\/\/\/

If you ask me for what purposes math teachers and textbooks give such problems to students, my answer would be


        "such problems play very important role teaching students to distinct really beautiful Math problems from routine ones

         and really nice solutions from all the others."


Therefore I so enthusiastically support and promote such problems at this forum.



Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


You have received, to this point, three algebraic solutions to the problem, showing a variety of different levels of complexity -- emphasizing, as noted in the solution from tutor @ikleyn, that the way you set up the problem can make a big difference in the amount of work required to solve it.

It is probable that a formal algebraic solution was wanted for this problem. But note that an informal solution using logical reasoning will give you some good brain exercise; and it will get you to the answer with relatively little effort.

When the number of children is reduced by 5, the amount of money each one gets is doubled, from $20 to $40.

That means when the number of children is reduced by 5, the number of children is cut in half.

That means the original number of children was 10 and the reduced number of children was 5.

And that means the amount of money was 10 times $20, which is $200; or, alternatively, it was 5 times $40, which of course is also $200.