Lesson Upper level word problems to solve using a single linear equation

Upper level word problems to solve using a single linear equation

Problem 1

Let x be the same number of lanterns sold in each shop.


Shop 1 sold 1/2 of its lanterns; this 1/2 is x; 
       hence, the original amount of lanterns in shop 1 was 2x.
       2x-x = x is the amount of lanterns remained in shop 1.



Shop 2 sold 2/3 of its lanterns; this 2/3 is x; 
       hence, the original amount of lanterns in shop 2 was .
        =  is the amount of lanterns remained in shop 2.



Shop 3 sold 3/4 of its lanterns; this 3/4 is x; 
       hence, the original amount of lanterns in shop 3 was .
        =  is the amount of lanterns remained in shop 3.



The three shops had a total of 330 lanterns left. It gives us this equation

    x +  +  = 330.


Simplify and find x

    x +  +  = 330

     +  +  = 330

     = 330

    x =  = 30*6 = 180.

 
Each shop sold 180 lanterns.    ANSWER

Problem 2

Let x be the price of one diary, in dollars.

Then the price of one book is (x+3.30) dollars, according to the problem.


Jamie spent 3/4 of her money and bought 3 books.

Hence, one book costs 1/4 of Jamie's money.

But 1/4 of Jamie's money is exactly the amount which left in Jamie's possession..

So, we conclude that Jamie left (x+3.30) dollars.


    +-----------------------------------------------------------------------------+
    |   OK, it is fine.  So, (x+3.30) dollars is the left side of the equation,   |
    |             which we are going to built to solve the problem.               |
    +-----------------------------------------------------------------------------+


Now let's calculate on the Grace side.


Grace spent 2x dollars, which is  5/9 of her starting money.  Hence, Grace's starting money was  

     = .


From it, we should take  4/9  of    to calculate the part of Grace's money which left.  So,

    Grace's money left =   =  = 


    +----------------------------------------------------+
    |   Thus, Grace's money left is  (8/5)*x  dollars.   |
    |   It is the right side of our equation for         |
    |                money left.                         |
    +----------------------------------------------------+


And now the equation for  money left  is

    x + 3.30 = .    ( ! ! ! - here three exclamation signs express my delight)


To solve, multiply by 5 both sides of this equation

    5x + 5*3.30 = 8x

    5*3.30 = 8x - 5x

    5*3.30 = 3x

    x =  = 5*1.10 = 5.50.


So, one diary costs 5.50 dollars and one book costs $3.30 more,  or  5.50+3.30 = 8.80 dollars.



        Now we are in the finish line.



Jamie spent 3*8.80 = 26.40 dollars.  It is 3/4 of her starting money.

Hence, Jamie's starting money was   = 35.20 dollars.



Grace spent 2*5.50 = 11 dollars.  It is 5/9 of her starting money.

Hence, Grace's starting money was   = 19.80 dollars.



Thus the starting money of two girls, altogether, was  35.20 + 19.80 = 55 dollars.


ANSWER.  Initially, the girls had  55 dollars, altogether.