Lesson Choose an unknown variable in a rational way for your problem

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Choose an unknown variable in a rational way for your problem


Problem 1

Tom had  4/5  as many stamps as  Michael.  After  Michael gave away  3/7  of his stamps,
Tom had  40  more stamps than  Michael.  How many stamps did  Tom have?

Solution 

Let 5x be the number of stamps Michael had originally.

Then Tom had  4x stamps originally.


After Michael gave away 3/7 of his stamps, you have THIS equation


    %284%2F7%29%2A%285x%29 = 4x - 40.


To solve, multiply both sides by 7.  You will get


    20x = 28x - 280

    280 = 28x - 20x

    280 =    8x

     x  = 280/8 = 35.


Tom had 4x = 4*35 = 140 stamps originally.    ANSWER

Notice that in my solution I introduced the unknown variable in a way to make my computations as simple as possible.


Problem 2

Mrs  Johnson and her  2  sons  Bernard and  Raymond shared a sum of money.
The ratio received by  Mrs  Johnson to that received by  Bernard was  5  to  2 .
The ratio of money received by  Bernard  to that received by Raymond was  3  to  4.
After  Mrs  Johnson gave  20%  of her share to  Raymond and B ernard donated  1/3  of his share to charity,
Raymond then had  $4270  more than  Bernard.
What was the sum of money shared by the three of them.

Solution 

From this statement, "The ratio received by Mrs Johnson to that received by Bernard was 5 to 2",
I choose to write 5x for Mrs. Johnson amount and 2x for the Bernard amount.

Here "x" is the common measure of these amounts, in dollars. 


From the second statement, "The ratio of money received by Bernard to that received by Raymond was 3 to 4"
we conclude that Raymond then received  %284%2F3%29%2A%282x%29 = %288x%29%2F3 dollars. 


After re-distribution and donation, Raymond has  %288x%29%2F3+%2B+%281%2F5%29%2A%285x%29 = %288x%29%2F3+%2B+x = %2811x%29%2F3 dollars;

                                    Bernard has  2x+-+%281%2F3%29%2A%282x%29 = %282%2F3%29%2A%282x%29 = %284x%29%2F3.


Now Raymond has $4270 more than Bernard

    %2811x%29%2F3 - %284x%29%2F3 = 4270  dollars.


The setup is complete. Now we should solve this equation and find x.

Multiply both sides by 3

    11x - 4x = 3*4270

        7x   =  12810

         x   =  12810/7 = 1830.


Thus the sum of money shared by the three of them was  5x + 2x + %288x%29%2F3 = 5%2A1830+%2B+2%2A1830+%2B+%288%2A1830%29%2F3 = 17690 dollars.


ANSWER.  The sum of money shared by the three of them was  17690 dollars.

Solved completely, using only one single unknown variable,
and in a way as it should be done and as it is expected to be done.


My other additional lessons on solving single linear equations and word problems in one unknown in this site (section 2) are
    - One more lesson on solving problems by the backward method
    - Entertainment problems on finding three unknowns using only one equation
    - Upper level word problems to solve using a single linear equation
    - OVERVIEW of my additional lessons on solving single linear equations and word problems in one unknown

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