
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 


x_{1}
:
y_{1} =
x_{2}
:
y_{2}. 
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. 

¯ 
3^{rd}
floor 

¯ 
7
sec. 



18^{th}
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°?



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
x_{1}
:
x_{2} =
y_{2}
:
y_{1}. 
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.




Basic percent formulas 


Solution: 
From the proportion, 

part
: base (whole) = rate : 100
or y
: x =
p
: 100 

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 

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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