ALGEBRA - solved problems
Ratios and proportions
Ratios and proportion properties
99.    Write a proportion by given ratios.
Solution: 9 : 6 = 3/2   and,   21 : 14 = 3/2,  follows,  9 : 6 = 21 : 14.
A proportion is a statement of the equality of two ratios. If the fractions both reduce to the same value, the proportion is true.
100.    Use ratios and proportion properties to rewrite given expressions.
Solutions: Since, a : b = (a · c) : (b · c) then a)
and b)
Direct and inverse proportion
101.    A lift reaches the third floor in seven seconds. When will reach the 18th floor?
Solution: Given quantities make the proportion
  x1 : y1 = x2 : y2. That is, 3 : 7 = 18 : x => seconds.  
We could obtain the same result by setting up the corresponding values of the proportional quantities in a table where arrows show the sequence of the terms of the proportion.
  ¯ 3rd floor   ¯ 7 sec.    
  18th floor   x sec. or 3 : 18 = 7 : x
     
=> seconds.
102.    To the central angle a = 30°, of a circle of radius r, corresponds the arc length equal  p/6 r.
What arc length corresponds to the central angle a = 320°?
Solution:
 
103.    If 12 workers can do a job in 15 hours, then how many workers would be needed to do the job in 9
hours?
Solution: It’s obvious that more workers will do a job in shorter time. That means, the time it takes to do
the job is inversely proportional to the number of workers.
Therefore, use the proportion or set up the table where the corresponding values make the colons and the arrows show the sequence of the terms of the proportion.
  ß 12 workers   Ý 15 hours    
Thus, x workers   9 hours or 12 : x = 9 : 15
     
=> workers.
104.    A distance d from place A to place B a car covers in 75 minutes traveling at the rate of 120 km/h.
At what rate it should drive so that it will travel the same distance in 60 minutes.
Solution: By driving two times faster, the same distance it will cover in half less time. 
Since, rate r and time t are inversely proportional quantities use the proportion  x1 : x2 = y2 : y1.
As the quantities we are comparing must always be measured in the same units, we’ll convert minutes to hours,  so   75 min = 75 min · 1hour / 60 min = 1.25 h.
  ß 1.25 h   Ý 120 km/h    
Thus, 1 h   x km/h or 1.25 : 1 = x : 120
     
=> km/h.
Percent or percentage
Percent, decimal number and fraction conversions
105.    As appropriate, given values convert to, a percent, a decimal number and a fraction.
Solution:
 
 
Basic percent formulas
106.    What is 25% of 80?
Solution: From the proportion,    
part : base (whole) = rate : 100    or    y : x = p : 100
then,   ,  and     
we calculate, the part or amount  y, the rate p and the base value x. 
Since given,  x = 80,   p% = 25%,   y = ?
Therefore,  
107.    What percent of 80 is 20?
Solution: Given,    y = 20,   x = 80,   p% = ?
Therefore,  
108.    What is the base value if  25% of it is 20?
Solution: Given,    p% = p/100 = 25%,   y = 20,   x = ?
Therefore,  
  The base value therefore   represents 1% of the base value.  
Percent increase or decrease,  base (x), amount (y), percent (p)
109.    In the price of $33 included is tax of 10%.  What is net price?
Solution: From the proportion,    
    x : 100 = ( x ± y ) : ( 100 ± p) then, y : p = ( x ± y ) : ( 100 ± p)  
 where   x ± y   denotes original or base value x increased or decreased by amount y
then,  , and  
Since given,  x +  y = 33 ,   p% = 10%,   x = ?
Therefore,  
110.    Price reduced by 20% amounts to $320. How much is reduced?
Solution:    Given,  x -  y = 320 ,   p% = 20%,   y = ?
Therefore,  
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