GRE Quantitative Arithmetic – Fractions

The practice of working with fractions that too without using a calculator was long forgotten since grade school. GRE demands dexterity in manipulation of fractions. Usually, the questions asked in the GRE Quantitative Reasoning section are pretty straight forward. However, due to loss of practice over many years, it turns out that these simple questions often morph into difficult problems for aspirants. So, let us first understand what fractions actually mean.

Any fraction can be represented in the form of xy; where x is called as numerator and y is the denominator, xy can also be called as ‘x is divided by y’.

Properties of fractions:

  • Dividing or multiplying both the numerator and the denominator by same number does not create any change in the value of the fraction, so for a single fraction, there exists many equivalent fractions. For example: \(\large \frac{1}{7} = \frac{3}{21}\), these two fractions are equivalent.
  • A fraction having a negative sign in either denominator or numerator can be written with a negative sign in front of the fraction. For example:\(\large \frac{1}{-5} = -\frac{1}{5}\)
  • Fractions can be expressed in form of mixed values as well, which is a combination of integer and fraction. For example:\(\large \frac{22}{7} =\frac{7}{7} + \frac{7}{7} + \frac{7}{7} + \frac{1}{7} = 3\frac{1}{7}\)
  • Reducing Fractions: If both numerator and denominator have common factors, then the numerator and denominator can be simplified and reduced into an equivalent fraction. For example:\(\large \frac{20}{12} =\frac{5}{3}\);as both numerator and denominator has a common factor of 4.
  • Addition and Subtraction of fractions having same denominator: If the denominator is same, then the numerators will be either simply added or subtracted according to the need. For example
    Addition :
    \(\large \frac{2}{9} + \frac{5}{9} + \frac{1}{9} = \frac{8}{9}\)

Subtraction:\(\large \frac{7}{5} – \frac{3}{5} = \frac{2}{5}\)

Mixed question: \(\large \frac{17}{7} + \frac{5}{7} – \frac{21}{7} = \frac{17+5-21}{7} = \frac{22-21}{7} = \frac{1}{7}\)

Addition and Subtraction of fractions having different denominators: If the denominators are different then find the LCM of all the denominators, so that it can be converted into fractions having same denominator and then it can be solved as the fractions having same denominators. For example: \(\large \frac{17}{3} + \frac{5}{7} – \frac{21}{9}\) L.C.M of 3, 7 and 9=63. Hence, fractions can be converted to;\(\large \frac{357}{63} + \frac{45}{63} – \frac{147}{63} = \frac{255}{63} = \frac{85}{21}\)

Let’s solve a few questions and prepare ourselves well for the GRE Exam :

Question: The numerator of a fraction is 3 less than the denominator. If both the numerator and denominator are increased by 4, then the fraction is increased by the fraction1277. Find the fraction:

Solution: Let the numerator of fraction be x, then the denominator will x+3.

The fraction will \(\large \frac{x}{x+3}\)

According to the question, if 4 are added to both numerator and the denominator, then the fraction becomes

\(\large \frac{x+4}{x+7}\)

\(\large \frac{x+4}{x+7} – \frac{x}{x+3} = \frac{12}{77}\)
\(\large So, \; 77(x+3)(x+4) – 77x(x+7) = 12 (x+7)(x+3)\)
\(\large \Rightarrow 77x^{2} + 539 x + 924 – 77x^{2} – 539x = 12x^{2} + 120x + 252\)
\(\large \Rightarrow 12x^{2} + 120 x – 672 = 0\)
\(\large \Rightarrow x^{2} + 10 x – 56 = 0\)
\(\large \Rightarrow x^{2} + 14 x -4x – 56 = 0\)
\(\large \Rightarrow (x+14)(x-4) = 0\)
\(\large \Rightarrow x = -140 \; or \; 4\)
Since, negative value is not applicable for this given question
\(\large Hence, \; x = 4\)
Hence, the fraction is \(\large \frac{4}{7}\).

Hence, the fraction is 47.

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