ALGEBRA - solved problems
     Proper and improper fractions, mixed numbers
39.     Convert given mixed number to improper fraction.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, add the numerator of the fraction to the product and write the sum over the given denominator.
To convert an improper fraction to a mixed number divide the fraction's numerator by its denominator. The integer part of the division is the integer part of the mixed number. The remainder of the division is the  numerator of the resulting fraction, as is shown in the above example.
Converting and reducing fractions
40.    Write equivalent fraction and reduce given fraction.
Solution: Equivalent fractions,  and 
Fractions that represent the same number are called equivalent fractions.
For any fraction, multiplying the numerator and denominator by the same nonzero number gives an
equivalent fraction.
A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. 
To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common  factor.
Comparing fractions
41.    Compare given fractions.
To compare fractions with different denominators, e.g. 4/5 and 7/9, multiply the numerator  and the denominator of both fractions, the first fraction by the denominator of the second fraction and the  second fraction by the denominator of the first fraction. We get equivalent fractions with the same  denominators, thus
Solution: since then
To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator.
If the numerators of two fractions are the same, the fraction with the smaller denominator is the larger fraction.
Decimal fractions
42.    Given decimal fraction write as a vulgar fraction.
Decimal fractions are vulgar fractions whose denominator is a power of ten. Or, it is a fraction written in the place-value notation with base 10 preceded by dot, after which each successive digit indicates a multiple of the successive negative powers of 10.
Adding and subtracting like fractions
43.    Add and subtract given fractions.
To add or subtract two fractions with the same denominator, add or subtract the numerators and write the sum over the common denominator.
Adding and subtracting fractions
44.    Add and subtract given fractions.
Solutions: a)
To add or subtract fractions with different denominators:
First find the least common denominator (the smallest number that can be divided by each denominator). Write equivalent fractions using this denominator.  Then add or subtract the fractions.
Multiplying fractions
45.    Multiply given fractions.
To multiply a whole number by a fraction means to multiply it by a numerator and to divide a product by a denominator.
When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators.
Dividing fractions, reciprocal fractions
46.    Divide given fractions.
To divide a number by a fraction, multiply the number by the reciprocal of the fraction.
To divide mixed numbers, you should always convert them to improper fractions, then multiply the first number by the reciprocal of the second.
To divide a fraction by a whole number, write the whole number as an improper fraction with a denominator of 1, then multiply as fractions.
Simplifying complex fractions
47.    Simplify given complex fractions.
Complex fractions or compound fractions are fractions in which the numerator and/or denominator contain fractions.
To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator by the denominator.
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