Variable expressions and formula problems
       Solving a formula for a specified variable, transposition of a formula (changing the subject of a formula)
Variable expressions (expressions involving variables) and formula problems
When letters of the alphabet are used in a mathematical expression, they are considered symbols standing for unknown quantities in order to determine the value of the mathematical expression.
Letters used in a mathematical expression may take different values and this is why they are called variables, i.e., their values can vary. We use letters whose values may vary to write formulas for various problems.
A formula is a formal expression (an equation) of some rule which specifies how variables are related to one another.
Formulas are written so that a single variable, the subject of the formula, is on the left hand side of the equation. Everything else goes on the right hand side of the equation.
Formulas are used to calculate (evaluate) the value of the subject when values of all of the other variables are known.
Solving a formula for a specified variable, transposition of a formula (changing the subject of a formula)
Example:  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
Example:  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.
Example:  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,
Example:  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
Example:   If  v = g t + v0  and   then t is equal?
Solution:
Intermediate algebra contents
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