
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 Q_{a}
liters of vine, of C_{a}
dollars per liter, with Q_{b}
liters of vine, of C_{b}
dollars per liter, how


much will cost one liter of the
mixture

Where; Q_{a}= quantity
of a,
C_{a}= cost of a,

Q_{b}=
quantity of b,
C_{b}= cost of b.

Solution:
If C_{M}
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,
Q_{a} · C_{a} +
Q_{b} · C_{b} =
(Q_{a} +
Q_{b})
· C_{M}

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














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