ALGEBRA - solved problems
  Linear equations in one variable
Word problems that lead to simple linear equations
Number problems
The general procedure to solve a word problem is:     1. Set the unknown.
   2. Write equation from the text of the problem.       3. Solve the equation for the unknown.
51.    A number multiplied by 5 and divided by 4 equals twice the number decreased by 15.

Solution:   Let x denotes the number, then

52.    Split the number 61 into two parts so that quarter of the first part increased by sixth of the other
part gives 12. What are these parts?
Solution:  Let x denotes the first part of the number, then
Age problems
53.    A father is 48 years old and his son is 14.
                      a)  In how many years will the father be three times older then his son?
                      b)  How many years before was the father seven times older then his son?
Solution:   After x years     a)    48 + x = 3 (14 + x)
                                                48 + x = 42 + 3x
                                                x - 3x = 42 - 48
                                                    -2x = -6 | (-2)
                                                        x = 3 years.
               Before x years     b)    48 - x = 7 (14 - x)
                                                48 - x = 98 - 7x
                                                7x - x = 98 - 48
                                                       6x = 50
                                                        x = 25/3 = 8 years and 4 months.
Mixture problems
54.    If we mix Qa liters of vine, of Ca dollars per liter, with Qb liters of vine, of Cb dollars per liter, how 
much will cost one liter of the mixture
Where;    Qa= quantity of a,    Ca= cost of a,
              Qb= quantity of b,    Cb= cost of b
Solution:   If CM  denotes the cost of one liter of the mixture, then
55.    How much liters of vine of $8 per liter should mix with 15 liters of vine of $14 per liter to get
mixture that will cost $10 per liter.

Solution:   Using the above equation,    Qa Ca + Qb Cb = (Qa + Qb) CM

                                                                 x 8 + 15 14 = (x + 15) 10
                                                                         8x + 210 = 10x + 150
                                                                                   2x = 60
                                                                                     x = 30 liters
56.    By mixing alcohol of 50% strength with alcohol of 80% we get 12 hectoliters of alcohol of 70%
strength. How much of each kind of alcohol was combined?

Solution:    If x denotes amount of alcohol of 50% strength, then

                                                                 x 50 + (12 - x) 80 = 12 70
                                                                        50x + 960 - 80x = 840
                                                                                             30x = 120
                                                                                                 x = 4 hectoliters
Therefore, combined are 4 hectoliters of alcohol of 50% strength and 8 hectoliters of alcohol of 80% strength.
Work problems
57.    One of two workers can finish a job for 15 days, and other for 10 days, how long will it take to
finish the job working together.

Solution:  The first worker does 1/15 of the job per day, and the second worker does 1/10 of the job per day.

If working together they complete the job in x days, then
                                                              10x  + 15x = 150
                                                                         25x = 150
                                                                             x = 6 days.
58.    Suppose a person A can finish a job in 12 days. He worked three days when a person B joins him
to help. Suppose the person B can finish the same job in 15 days. How long will they take to finish the job?

Solution:   If working together they complete the job in x days, then

59.    A tank can be filled with three pipes:
                 - first pipe (alone) takes 10 hours (a = 10) to fill the tank,
                 - second pipe takes 12 hours (b = 12),
                 - third pipe takes 15 hours (c = 15)  to fill the tank.
How long would it take to fill the tank with water,
               a)  if all pipes are opened at the same time,
               b)  if second and third pipes fill the tank while at the same time water leaking from the tank
through the first pipe?

Solution:  a) first pipe fills 1/a part of the tank per hour, second pipe fills 1/b and third pipe fills 1/c part of the

tank per hour. It will take x hours to fill the thank, thus
Solved problems contents
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