

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 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, 



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 + v_{0}
and 

then t is
equal?


Solution:










Intermediate
algebra contents 



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