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1 INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The denominator tells us how many parts there are in the whole. The numerator tells us how many parts of the whole we have. A proper fraction is a fraction in which the numerator is smaller than the denominator. are proper fractions. An improper fraction is a fraction in which the numerator is greater than, or the same as, the denominator. are improper fractions. A mixed number (sometimes called a mixed numeral) has two parts: a whole number part, followed by a fractional part. has a whole part of 2 and a fractional part of. + + Like fractions are fractions which have the same denominator. = are like fractions because they both have a denominator of 6.
2 CONVERSION BETWEEN MIXED NUMBERS AND INPROPER FRACTIONS: An improper fraction may be written as a mixed number, or a mixed number may be written as an improper fraction. To write an improper fraction as a mixed number: Divide the numerator by the denominator. Write the remainder as a fraction of the original denominator. = Hence (an improper fraction) can be written as (a mixed number). To write a mixed number as an improper fraction: The denominator of the improper fraction will be the denominator of the fractional part of the mixed numeral. To find the numerator of the improper fraction, multiply the whole part of the mixed number by the denominator of the fractional part and add this to the numerator of the fractional part. Hence (a mixed number) can be written as (an improper fraction). REDUCING FRACTIONS TO LOWERS TERMS, OR SIMPLIFYING A FRACTION: A fraction is in lowest terms when the largest number that will divide into both the numerator and the denominator is. Unless you are told otherwise, a fraction should always be expressed in lowest terms. To reduce a fraction to lowest terms: Find a number that can be divided evenly into both the numerator and the denominator. Divide. Continue this process until no number other than one will divide into both the numerator and the denominator, Simplify Both the numerator and the denominator can be divided by 5. Hence We say that and are equivalent fractions. They have the same value but different forms. 2
3 WRITING EQUIVALENT FRACTIONS: Only fractions which have the same denominator may be added or subtracted. Hence we often have to convert a fraction into an equivalent fraction which has a different denominator, We need to multiply both the numerator and the denominator by the same number to from an equivalent fraction. As you can see, this process is the reverse of the process of reducing a fraction to lowest terms. If we wanted to rewrite with a denominator of 6 we would need to multiply both the numerator and the denominator by 9. OTHER IMPORTANT POINTS: The fraction bar () means divide. For example, means 2 divided by. The number can be made into any number which has the same numerator and denominator. Remember that as a general rule, the larger the denominator, the smaller that fraction is. The lowest common multiple, LCM, of two or more numbers is the smallest number that all these numbers will divide into evenly. The LCM of,, and 6 is 2; the LCM of 7 and 9 is 6. We use the least common multiple to fine the lowest (or least) common denominator, LCD of a group of fractions. The least common denominator of a group of fractions is the least common multiple of the denominators of the fraction. The LCD of The LCD of
4 ADDING FRACTIONS: To add fractions, you must have like fractions (common denominators). To find the sum of two fractions which have the same denominator add the numerators, keeping the same denominator, and reduce to lowest terms. To find the sum of two fractions with different denominators we must first change then to like fractions, That is, we must find a common denominator. A common denominator is 6 x = 2 Hence we can rewrite the problem and solve it: this of course must be reduced When adding fractions it is often more expedient to find the lowest (or least) common denominator, usually abbreviated ass the LCD. Convert the fractions to equivalent fractions with the least common denominator. Convert mixed numbers into improper fractions. (Continued)
5 Now that we have a common denominator we can add the fractions: this must be reduced or the answer could be written as a mixed number: SUBTRACTING FRACTIONS: To subtract fractions, you must also have common denominators. When we have like fractions (those with the same denominator), we simply subtract the numerators and reduce to lowest terms. To subtract fractions involving mixed numbers, first convert the mixed number to an improper fraction. Manipulate the fractions to obtain common denominators. Subtract. Simplify the fraction if necessary. Alternate solution: Borrow from the 5 and rewrite. 5
6 MULTIPLYING FRACTIONS: You do not need common denominators for multiplication of fractions simply multiply the numerators and denominators and reduce to lowest terms. A faster way to get the same answer is to cancel before multiplying. When we cancel, we are dividing the numerator and the denominator by the same number in a crosswise direction. In one direction we divide by 2, and in the other we divide by : 2 To multiply mixed numbers, first change them to improper fractions, then multiply. Be sure to reduce or cancel first if possible. 2 DIVIDING FRACTIONS: When we divide fractions, we invert the second fraction and multiply. To invert a fraction, swap the numerator for the denominator, and the denominator for the numerator. For instance, we invert, we get. To divide mixed numbers, change the mixed numbers to improper fractions. Then proceed as usual. Cancel whenever possible. 6
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