1. Kerem is three times as old as Semih. In four years, he will be twice as old as Semih. How old are Kerem and Semih?
2. Murat’s age is four less than twice Kemal’s age. In ten years, Kemal’s age will be of Murat’s age. How old are Murat and Kemal?
3. The sum of Kerim’s age and twice Serdar’s age is 34. The difference between Serdar’s age and twice Kerim’s age is 7. How old are Kerim and Serdar?
4. In 1990, Ayþe was three times as old as her son. That year the difference of their ages was 22 years. In what year was each born?
5. Selim is six, Salih is ten and Ömer is thirteen years old. How old will Selim, Salih and Ömer be in eight years?
6. A father’s age is 10 more than the sum of the ages of his two sons. Eight years ago his age was three times the sum of the ages of his sons. How old is the father now?
7. The sum of the ages of Fatma and Levent is 46. If we subtract 2 from three times Levent’s age we get Fatma’s age. How old are Fatma and Levent?
8. Five years ago, Ahmet was three years older than twice Fatih’s age. In seven years Ahmet’s age will be 9 less than two times Fatih’s age. How old are Ahmet and Fatih?
Answers
1. Semih: 4 2. Kemal: 18 3. Kerim: 4 4. Ayþe: 1957 5. Selim: 14 6. Father: 35 7.Levent: 12 8.Ahmet: 20 Kerem: 12 Murta: 32 Serdar: 15
Her Son: 1979 Salih: 18 Fatma: 34 Fatih: 11 Ömer: 21
2 3
3. Work Problems
We can use algebra to calculate how long it takes for a number of workers to complete a par-ticular job. For example, we might want to know how long it will take for six men to build a wall. In problems like this, we suppose that each man works at the same speed or rate, for example, that each man alone can build 5m2of wall in one day.
We will use the formula
or .
For example, in the problem above, the formula for six men would use r = 5m2× 6 = 30m2of wall
t = number of days, and w = area of wall produced.
The following calculations are useful for solving work problems.
r t = w (work rate) (working time) = (amount of work done)
1. If a number of workers can complete a job in t hours, then the same number of workers can complete of the job in one hour.
2. Suppose two workers can complete a job in x and y hours respectively. If they work together, they will complete the job in t hours, where t is given by 1 1 1.
x y t 1
t
Solution
EXAMPLE
99
Ahmet can wash the family car in 45 minutes. Mehmet can wash it in 30 minutes. How long will it take them to wash the car if they work together?(2) (3)
(3) (2)
1 1 1
45 30
2 3
1 5 90
90
18 minutes.
We can also write
1 1 1 5 1 90 18.
30 45 90 5
t t
t t
t t
t t
Ahmet 1
45 t
rate time
1 45t
Mehmet 1
30 t
1 30t
work
Solution
EXAMPLE
100
Mustafa can paint a house in eighteen hours. Murat can paint the same house in twelve hours. If Mustafa works alone for six hours and then stops, how long will it take Murat to finish the job?First way:
Second way:Since Mustafa can paint a house in 18 hours, after six hours he will have fin-ished one third of the house. So of the job will remain.
Murat can paint the whole house in 12 hours, so he can complete painting two thirds of the house in 12 2 8 hours.
EXAMPLE
101
Two pipes can fill a pool in six hours. The larger pipe can fill the pool twice as fast as the smaller one. How long does it take the smaller pipe to fill the pool alone?EXAMPLE
102
Two pipes A and B can fill a storage tank in four and six hours respectively. A drain C can empty the full tank in three hours. How long will it take to fill the tank if both pipes and the drain are open?(2)
EXAMPLE
103
Two pipes A and B can fill an empty pool in six and eight hours respectively. A drain C can empty the full pool in twelve hours. For two hours, the pipes and the drain are left open.Then pipe A and pipe B are closed. How long will it take the drain C to empty the water in the pool?
Solution
of the pool will be filled in two hours.
After this,
So it will take drain C five hours to empty the pool.
1 5
EXAMPLE
104
Erkin can plough a garden in 20 hours and Asým can do the same job in 32 hours. They work together for eight hours, then Asým stops working. How long will it take Erkin to finish the job?EXAMPLE
105
In the storage tank shown in the figure, the height of drain C from the base of the tank is of the height of the tank. Pipes A and B can fill the the tank in 18 hours and 24 hours respectively. Drain C can empty of the full tank in 36 hours. If all the pipes and the drain are working, how many hours will it take the pipes to fill the tank?2
Solution
Notice that the drain C only affects the rate of work when the tank is over one-third full. So there are two rates of work: one rate to fill one-third of the tank, and another rate to fill the other two-thirds of the tank. It takes
to fill one-third of the storage
tank with pipes A and B.
Then, it takes hours
to fill the storage tank if pipes A and B and the drain C are working.
Therefore the total time is hours.
(5) (7)