Topic 8.4 (Problem Solving)

When solving thought problems make sure you start with an x or other variable for the first unknown.

All other unknowns must be set up using this variable. These thought problem will have fractional  equations

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Sample problems.

The ratio of jellybeans to gummy bears in a bag of candy is 7 to 2. If there are 459 pieces of candy in the bag, how many jellybeans are there?

Let x = the number of jellybeans in the bag that has 459 pieces of candy.

9 = the total number of candy in a bag of 7 to 2 jellybeans to gummy bears.

Equation (proportion)   7/9   =   x/459   If there are 7 jellybeans out of 9 pieces of candy, then there has to be x jellybeans out of 459 pieces of candy, (ratio & proportion).

To solve - cross multiply       7(459)   =   9x           x   =   357      Ans.

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To empty their swimming pool, the Johnsons decided to use both the regular drain and a pump. If it takes 15 hours for the pool to empty using the drain alone and 7 hours for the pool to empty using the pump alone, how long will it take for the pool to empty using both drain and the pump?

                                 Total time                               Rate

              drain            15 hrs.                              1/15 per hour

             pump             7 hrs.                                 1/7 per hour

                       x = time together

The rate of the drain and the pump is the reciprocal of  each total time it takes to fill the swimming pool.

The equation is derived from the fact that the rate times the time  of the drain plus the rate times the time  of the pump is equal to the full swimming pool.  (1 which is 100% changed to a decimal)

(1/15)(x)   +   (1/7)(x)   =   1 Multiply both sides of the equation by the LCD(105)

(105)(1/15)(x)   +   (105)(1/7)(x)   =   105(1)

           7x              +              15x         =      105

                             22x                         =       105

         x   =   105/22   approx.        4.8 hours       Ans.

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One cyclist can ride 2 miles per hour faster than another cyclist. If it takes the first cyclist 2 hours and 20 minutes to ride as far as the second cyclist rides in 2 hours, how fast can each go?

and 20 minutes to ride as far as the second cyclist rides in 2 hours, how fast can each go?

Set up a chart as follows.

                            Rate       Time       Distance     The first time is calculated as

     1^st cyclist         x          7/3 hrs.       x(7/3)        2 hr. 20 min.(20 min is 1/3 hr)

     2^nd cyclist    x + 2          2               (x + 2)2     2 1/3 changed to 7/3 hrs.

      To get the distance it is the rate times the time.

Equation:      (the distances are the same)      x(7/3)    =    (x + 2)2

7x/3   =   2x + 4,     multiply by 3     7x    =    6x   +   12.  

                x = 12,              1^st cyclist              12 mph            Ans.

            x  +  2                    2^nd cyclist            14 mph            Ans.