ALGEBRA - solved problems
Decimal representation of rational numbers or fractions
26.    Given fractions write as decimal numbers:
Solution:
27.    Given fractions write as decimal numbers:
Solution:
28.    Given decimal write as fraction:
Solution:
Terminating decimals
29.    Given fractions write as terminating decimals:
Solution:
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/(2n 5m).
Recurring decimals
30.    Given fractions write as purely recurring decimals:
Solution:
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:
Solution:
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:
Solution:
33.    Convert purely recurring decimals to fractions:
Solution:
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 9s as is the number of digits in the repeating pattern.
34.    Convert mixed recurring decimals to fractions:
Solution:
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 non-recurring pattern to its numerator.
To the denominator of the equivalent fraction write as much 9s as is the number of digits in the repeating pattern and add as much 0s as is the number of digits in the non-recurring pattern.
35.    Given decimals convert to scientific notation:
Solution:
a)   302 567 908 = 3.02567908 108      b)   0.000040635 = 4.0635 105.
36.    Convert from scientific to decimal notation:
Solution:
a)   2.09085 107 = 20908500               b)   7.81 105 = 0.0000781 
  Order of operations
37.    Solve given expression:
Solution:
- 8 -  3 (- 4) + 15 (- 5) = - 8 - (- 12) + (- 3) = - 8 + 12 - 3 = 1.
38.    Solve given expression:
Solution:
- 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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