
ALGEBRA
 solved problems 






Decimal representation of rational
numbers or fractions


26. 
Given
fractions write as decimal numbers:




27. 
Given
fractions write as decimal numbers:




28. 
Given
decimal write as fraction:




Terminating
decimals 

29. 
Given
fractions write as terminating decimals:




The irreducible fractions, i.e., the vulgar fractions in lowest terms
whose the only prime factors in a denominator are 2 and/or 5
can be converted to terminating decimals.

That is, the terminating
decimals represent rational numbers whose fractions in the lowest
terms are of the form
a/(2^{n}
· 5^{m}).


Recurring
decimals 

30. 
Given
fractions write as
purely recurring
decimals:




The irreducible fractions, i.e., the vulgar fractions in lowest terms
whose prime factors in the denominator are other than 2 or 5, that is,
the prime numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the
purely recurring decimals, i.e., the decimals which
start their recurring cycle immediately after the decimal point.


Mixed
recurring decimals 

31. 
Given
fractions write as
mixed recurring
decimals:




The irreducible fractions, i.e., the vulgar fractions in lowest terms
whose denominator is a product of 2's and/or 5's besides the prime
numbers from the sequence (3, 7, 11, 13, 17, 19, ...) convert to the
mixed recurring decimals, i.e., the decimals that have some extra
digits before the repeating sequence of digits. 
The repeating
sequence may consist of just one digit or of any finite number of
digits. The number of digits in the repeating pattern is called the
period. All recurring decimals are infinite decimals. 
All fractions can be written either as terminating decimals or as
recurring/repeating decimals. 

32. 
Given
terminating decimals write as fractions:





33. 
Convert
purely recurring decimals to fractions:



When converting the purely recurring decimal less than one to
fraction, write the group of repeating digits to the numerator, and to
the denominator of the equivalent fraction write as much 9’s as is
the number of digits in the repeating
pattern. 

34. 
Convert
mixed recurring decimals to fractions:




When converting the mixed recurring decimal less than one to
fraction, write the difference between the number formed by the
entire sequence of digits, including the digits of the recurring part,
and the number formed only by the digits of the nonrecurring pattern to its numerator.

To the denominator of the equivalent
fraction write as much 9’s as is the number of digits in the repeating
pattern and add as much 0’s as is the number of digits in the nonrecurring pattern.



35. 
Given
decimals convert to scientific notation:



a)
302 567 908 = 3.02567908 · 10^{8} b)
0.000040635 = 4.0635 · 10^{−}^{5}. 


36. 
Convert
from scientific to decimal notation:



a)
2.09085 · 10^{7} = 20908500
b)
7.81 · 10^{−}^{5}
= 0.0000781 


Order of operations 

37. 
Solve
given expression:




8 
3 ·
(
4)
+ 15
¸ (
5) =

8 
(
12)
+
(
3) =

8
+ 12

3
= 1. 


38. 
Solve
given expression:




3
+ 2
· [
2 
3 ·
(
4 ¸
2
+ 1)]
=

3 + 2
· [
2 
3 ·
(
2 +
1)] =

3
+ 2
· [
2 
3 ·
(
1)] 

=

3
+ 2
· [
2 +
3]
=

3
+ 2
=

1. 











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