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FRACTIONS
Definition
Fractional numbers are ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.
Fraction can be expressed in two forms fractional representation \((\frac{m}{n})\) and decimal representation \((a.bcd)\).
Fractional representation
Fractional representation is a way to express numbers that fall in between integers (note that integers can also be expressed in fractional form). A fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole).
• The number on top of the fraction is called numerator or nominator. The number on bottom of the fraction is called denominator. In the fraction, \(\frac{9}{7}\), 9 is the numerator and 7 is denominator.
• Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator. \(\frac{1}{3}\) is a proper fraction.
• Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number. \(\frac{5}{2}\) is improper fraction.
• An integer combined with a proper fraction is called mixed number. \(4\frac{3}{5}\) is a mixed number. This can also be written as an improper fraction: \(\frac{23}{5}\)
Converting Improper Fractions
• Converting Improper Fractions to Mixed Fractions:
1. Divide the numerator by the denominator
2. Write down the whole number answer
3. Then write down any remainder above the denominator
Example #1: Convert \(\frac{11}{4}\) to a mixed fraction.
Solution: Divide \(\frac{11}{4} = 2\) with a remainder of \(3\). Write down the \(2\) and then write down the remainder \(3\) above the denominator \(4\), like this: \(2\frac{3}{4}\)
• Converting Mixed Fractions to Improper Fractions:
1. Multiply the whole number part by the fraction's denominator
2. Add that to the numerator
3. Then write the result on top of the denominator
Example #2: Convert \(3\frac{2}{5}\) to an improper fraction.
Solution: Multiply the whole number by the denominator: \(3*5=15\). Add the numerator to that: \(15 + 2 = 17\). Then write that down above the denominator, like this: \(\frac{17}{5}\)
Reciprocal
Reciprocal for a number \(x\), denoted by \(\frac{1}{x}\) or \(x^{-1}\), is a number which when multiplied by \(x\) yields \(1\). The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). To get the reciprocal of a number, divide 1 by the number. For example reciprocal of \(3\) is \(\frac{1}{3}\), reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\).
Operation on Fractions
• Adding/Subtracting fractions:
To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.
To add/subtract fractions with the different denominator, find the Least Common Denominator (LCD) of the fractions, rename the fractions to have the LCD and add/subtract the numerators of the fractions
• Multiplying fractions: To multiply fractions just place the product of the numerators over the product of the denominators.
• Dividing fractions: Change the divisor into its reciprocal and then multiply.
Example #1: \(\frac{3}{7}+\frac{2}{3}=\frac{9}{21}+\frac{14}{21}=\frac{23}{21}\)
Example #2: Given \(\frac{\frac{3}{5}}{2}\), take the reciprocal of \(2\). The reciprocal is \(\frac{1}{2}\). Now multiply: \(\frac{3}{5}*\frac{1}{2}=\frac{3}{10}\).
Decimal Representation
The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating (recuring) decimals (such as 0.333333....).
Reduced fraction \(\frac{a}{b}\) (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only \(b\) (denominator) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as \(250\) (denominator) equals to \(2*5^3\). Fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and denominator \(10=2*5\).
Converting Decimals to Fractions
• To convert a terminating decimal to fraction:
1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms
Example: Convert \(0.56\) to a fraction.
1: Total number after decimal point is 2.
2 and 3: \(\frac{56}{100}\).
4: Reducing it to lowest terms: \(\frac{56}{100}=\frac{14}{25}\)
• To convert a recurring decimal to fraction:
1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms
Example #1: Convert \(0.393939...\) to a fraction.
1: The recurring number is \(39\).
2: \(\frac{39}{99}\), the number \(39\) is of length \(2\) so we have added two nines.
3: Reducing it to lowest terms: \(\frac{39}{99}=\frac{13}{33}\).
• To convert a mixed-recurring decimal to fraction:
1. Write down the number consisting with non-repeating digits and repeating digits.
2. Subtract non-repeating number from above.
3. Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non-repeating digit write down a zero after 9's.
Example #2: Convert \(0.2512(12)\) to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.
Rounding
Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.
Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.
Ratios and Proportions
Given that \(\frac{a}{b}=\frac{c}{d}\), where a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred to by the names mentioned along each of the properties obtained.
\(\frac{b}{a}=\frac{d}{c}\) - invertendo
\(\frac{a}{c}=\frac{b}{d}\) - alternendo
\(\frac{a+b}{b}=\frac{c+d}{d}\) - componendo
\(\frac{a-b}{b}=\frac{c-d}{d}\) - dividendo
\(\frac{a+b}{a-b}=\frac{c+d}{c-d}\) - componendo & dividendo
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