This is part of our
GRE Math Essentials project is the best complement to our
Official GRE Quant Book. It provides a cutting-edge, in-depth overview of all the math concepts to achieve 170 in the Quantitative Reasoning portion of the GRE. The book still remains our hallmark. However, the following chapters will give you a lot of tips, tricks, and shortcuts to make your quant preparation more robust and solid.
INTRODUCTION1.
What is a fraction?
A fraction consists of a numerator (part) on top of a denominator (total) separated by a horizontal line.
For example, the fraction of the circle which is shaded is:
\(\frac{2 (parts shaded)}{4 (total parts)}\)
In the square, the fraction shaded is \(\frac{3}{8}\) and the fraction unshaded is \(\frac{5}{8}\)
2.
Equivalent Fractions – MultiplyingThe three circles on the right each have equal parts shaded, yet are represented by different but equal fractions. These fractions, because they are equal, are called equivalent fractions.
Any fraction can be changed into an equivalent fraction by multiplying
both the numerator and denominator by the same number.
3.
Equivalent Fractions – Dividing (Reducing)Equivalent fractions can also be created if both the numerator and denominator can be divided by the same number (a factor) evenly. This process is called “reducing a fraction” by dividing a common factor (a number which divides into both the numerator and denominator evenly).
\(\frac{4}{8} \div \frac{4}{4} = \frac{1}{2}\)
\(\frac{27}{81} \div \frac{9}{9} = \frac{3}{9}\)
4.
Simplifying a Fraction (Reducing to its Lowest Terms)It is usual to reduce a fraction until it can’t be reduced any further. A simplified fraction has no common factors which will divide into both numerator and denominator. Notice that, since 27 and 81 have a common factor of 9, we find that \(\frac{3}{9}\) is an equivalent fraction. But this fraction has a factor of 3 common to both numerator and denominator. So, we must reduce this fraction again. It is difficult to see, but if we had known that 27 was a factor (divides into both parts of the fraction evenly), we could have arrived at the answer in one step
\(\frac{8}{24} \div \frac{8}{8} = \frac{1}{3}\)
TYPES OF FRACTIONS1.
Common FractionsA common fraction is one in which the numerator is less than the denominator (or a fraction which is less than the number 1). A common fraction can also be called a proper fraction.
\(\frac{88}{93},\frac{8}{15}\),..............are al common fractions
2.
Fractions that are Whole NumbersSome fractions, when reduced, are really whole numbers (1, 2, 3, 4… etc). Whole numbers occur if the denominator divides into the numerator evenly
\(\frac{8}{4}\) is the same as \(\frac{8}{4} \div \frac{4}{4}=\frac{2}{1}\) or \(2\)
3.
Mixed NumbersA mixed number is a combination of a whole number and a common fraction.
e.g. \(2 \frac{3}{5}\) (two and three-fifths)
4.
Improper FractionsAn improper fraction is one in which the numerator is larger than the denominator. From the circles on the right, we see that \(1 \frac{3}{4}\) (mixed number) is the same as \(\frac{7}{4}\) (improper fraction). An improper fraction, like \(\frac{7}{4 }\) can be changed to a mixed number by dividing the denominator into the numerator and expressing the remainder (3) as the numerator.
e.g. \(\frac{16}{5}=3 \frac{1}{5}\)
A mixed number can be changed to an improper fraction by changing the whole number to a fraction with the same denominator as the common fraction.
\(2 \frac{3}{5} = \frac{10}{5} and \frac{3}{5}=\frac{13}{5}\)
A simple way to do this is to multiply the whole number by the denominator, and then add the numerator.
e.g. 4 \(\frac{5}{9} = \frac{4 \times 9 + 5 }{9} = \frac{36+5}{9 }= \frac{41}{9}\)
5.
Simplifying fractions All types of fractions must always be simplified (reduced to lowest terms).
e.g. \(\frac{6}{9}=\frac{2}{3}\)
Note that many fractions can not be reduced since they have no common factors.
source:
Vancouver Island UniversityADDING/SUBTRACTING/MULTIPLYING/DIVIDINGAdding and Subtracting Fractions:
1. Common (like) denominators are necessary, so change all unlike fractions to equivalent fractions with like denominators. To make equivalent fractions, multiply the numerator and denominator by the same number.
2. Keep mixed numbers; DO NOT change mixed numbers into improper fractions.
3. Add (or subtract) the numerators, put the numerator answer over the common denominator. If any improper fractions arise in the answer, change the improper portion to a mixed number. (In Math 105, answers are often left in improper form)
Multiplying and Dividing Fractions:
1. Common denominators are NOT needed.
2. Always change mixed numbers to improper fractions.
3. CANCEL (reduce) between any numerator and any denominator if you can, but cancel only when a multiplication sign is present: Never cancel when you have a division sign.
4. TO MULTIPLY: Multiply numerator times numerator, denominator times denominator. Reduce answer to a mixed number in lowest terms. (In Math 105, answers are often left in improper form)
5. TO DIVIDE: Change the divide sign to a multiplication sign, then invert the second fraction and multiply as in Step 4.
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