Lesson I do not have enough savings now

I do not have enough savings now

Problem 1

Let x represents the cost of one sundae in dollars.  Andrew is $2 short, so he has (x-2) dollars.  
Likewise, Michael has  (x-1) dollars.  Since together they have exactly enough to buy one sundae, x dollars, we can write:


    (x-2) + (x-1) = x.


Solve it for x

    2x - 3 = x,

    2x - x = 3,

       x   = 3.


ANSWER.  Each sundae costs 3 dollars.

Problem 2

Let x represents the cost of one sundae in dollars.  Lola is $4 short, so she must have (x-4) dollars.  
Likewise, Michael has  (x-3) dollars and Joaquin has (x-1).  Since together they have exactly enough to buy one sundae, x dollars, we can write:


  (x-4) + (x-3) + (x-1) = x.


Solve it for x


    3x - 8 = x

    3x - x = 8

       2x  = 8

        x  = 8/2 = 4.


ANSWER.  Each sundae costs 4 dollars.

Problem 3

In accordance with the context of the problem, all the shirts have the same price.


Let x be the price for one shirt.


Then Mr. Lee has, from one part of the condition,  3x + 6 dollars;

                  from the other part of the condition, he has  5x - 10 dollars.


It gives you THIS EQUATION


    3x + 6 = 5x - 10.


Simplify and find x


    6 + 10 = 5x - 3x

    16     = 2x

     x     = 16/2 = 8 dollars.


One shirt costs 8 dollars.


Hence, Mr Lee has  3x + 6 = 3*8 + 6 = 30 dollars.    ANSWER

Problem 4

Let x = How much money you have in your savings now.

Let y = the cost of the pair of shes.


Then from the condition you have these two equations


    x + 0.15x = y -  9     dollars     (1)

    x + 0.35x = y + 17     dollars     (2)


Subtract equation (1) from equation (2) to eliminate x. You will get

       0.35x - 0.15x = 17 - (-9)

       0.2x          = 26

       x             = 26/0.2 = 130 dollars.


ANSWER.  Your savings now is/(are) 130 dollars.

Problem 5

Let X be the sum of money they need to collect.

And let "n" be the number of people.


According to the first scenario,  0.35*n + 4.40 = X  dollars.


According to the second scenario,  0.40*n - 4.40 = X  dollars.


It gives you an equation


    0.35n + 4.40 = 0.40n - 4.40,    or


    4.40 + 4.40 = 0.40n - 0.35n,

    8.80        = 0.05n

       n        =  =  = 176.     ANSWER

Problem 6

Let F be the price for each file;  P be the price for each pen, and

let X be the amount of money that Bob had initially.


Then, based on the problem's description, we can write these THREE equations for three unknowns


    4F + 6P = 900     cents     (1)     (he spent $9 on 4F and 6P)

    5F + 6P = X + 30  cents     (2)     (he would be short 30 cents)

    4F + 7P = X - 70  cents     (3)     (he would be 70 cents left)



        The setup is just completed. 
        As I said at the beginning, it is tricky.
        But the solution of equations is simple.



To solve them, first subtract equation (3) from equation ((2).  You will get

    F - P   =     100  cents.   (4)


Next, express  F = P + 100 from (4), and substitute it into equation (1).  You will get

    4(P + 100) + 6P = 900

        10P         = 900 - 400 = 500

          P                     = 500/10 = 50.


Next, from equation (1),  4F + 6*50 = 900,  4F = 900 - 300 = 600,  F = 600/4 = 150.


Thus, the price for one file is $1.50;  the price for one pen is  P = F - 100 = 150-100 = 50 = $0.5  (from equation (4))

and  X = 5F + 6P - 30 = 5*150 + 6*50 - 30 = 1020 cents = $10.20.  (from equation (2) )


ANSWER.  The price for each file is  $1.50.  Bob had  $10.20  originally.