
ALGEBRA
 solved problems 





Linear
equations
in one variable


Miscellaneous word
problems


70. 
If fresh grapes contain 90% water and dried 12%, how much dry grapes we get from 22 kg of fresh


grapes?

Solution:
Fresh grapes contain 90% water and 10% dry substance.

Dry grapes contain 12% water and 88% dry substance.

22 kg of fresh grapes = x kg
of dry grapes, so 



71. 
An amount decreased 20% and then increased 50%, what is the total increase in relation to initial


value.

Solution: An initial amount
x decreased by 20%, 


The obtained amount increased by 50%, 


The difference in relation to the initial value x,



shows increase by 20%.


72. 
From a total deducted is 5% for expenses and the remainder is equally divided to three persons.


What was the total
if each person gets $190?

Solution: If
x
denotes the total then 




73. 
Into 10 liters of the liquid
A poured is 4 liters of the liquid
B
and 6 liters of the liquid
C.


From obtained mixture D
poured is out 3 liters, how many liters of the liquid C
remains in the mixture
D?

Solution:
A
+ B
+ C
= D => 10 l + 4 l + 6 l = 20 l



Variable expressions
(expressions involving variables) and formula problems 
Solving
a formula for a specified variable, transposition of a
formula (changing the subject of
a formula) 

74. 
Into a cistern are poured 37.4 tonnes of petrol and remains 6.5% of the cistern not filled.


How many tonnes of the petrol will fit in the cistern to be full.

Solution:
Applying the
proportion B :
A
=
100% :
p,

where
A = 37.4 tonnes or 93.5%
denotes the amount (that
correspond to percentage), B
is base (correspond 
to the whole or 100%) and p
is percent, it follows that 


75. 
Fresh figs contain 90%
of water and dry figs contain 12%. How many kilos of dry figs are obtained


by drying 264 kg of fresh
figs?

Solution:
Using the
proportion B
:
(B
± A)
= 100 :
(100
± p),

As fresh figs contain 10% of the dry substance and dry figs contain 88% of the dry
substance it follows that

264 kg of fresh figs contains 26.4 kg of the dry substance.

Therefore, 26.4 kg is 88% of quantity of dry figs or it is
weight of the dry figs decreased by 12% of water, 
so
B 
A = 26.4 kg,
and by using the above formula 

obtained
is the
weight of the dry figs. 

76. 
Sum of all threedigits numbers divisible by 3 is?


Solution: The
threedigits numbers divisible by 3
are;

102, 105, 108, .
. . , 996, 999

and
they represent the arithmetic
progression 
a_{1},
a_{2},
a_{3}, .
. . , a_{n
}_{ }_{1},
a_{n},
where a_{1
}
= 102, a_{n}
= 999 and d
= 3, 
since
a_{n}
= a_{1} + (n  1)
· d then 999
= 102 + (n  1)
· 3

3
· (n  1)
= 897  ¸3

n
 1
= 299
=> n
=
300 
Using the
sum formula, 



77. 
How many minutes pass while watch hands coincide again?


Solution: Suppose watch hands last coincide at noon. They will
coincide again 5 to 6 minutes after 13 hours.

To reach 13 hours, the minute hand traveled full circle 360° or 60 minutes,
and the hour hand traveled 30°.

Therefore, the minute hand travels at speed of 

degrees per minute, 

while the hour hand travels at speed of 

degree per minute. 

Hands will coincide again the minute hand passes additional distance of
30° +
x° while, at the same time, the hour hand passes the distance of
x°, i.e.,


Thus, watch hands will coincide again after
60 min + 5 min + 5/11 min = 65 and 5/11 min. 


If v = g t + v_{0}
and 

then t is
equal?













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