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 three-digits numbers divisible by 3 is?
Solution:   The three-digits numbers divisible by 3 are;
  102, 105, 108, . . . , 996, 999
and they represent the arithmetic progression
  a1, a2, a3. . . , an - 1, an,   where  a1 = 102, an = 999  and  d = 3,
since  an = a1 + (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. 
78.  
  If  v = g t + v0  and then t is equal?
Solution:
 
 
 
 
 
 
 
 
 
 
 
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