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Number Theory

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CAT 2025 Lesson : Number Theory - Fractions

2.3 Fractions

Numbers expressed in the form

\dfrac{p}{q}

, where

p

and

q

are integers and

q \ne 0

. Examples are

\dfrac{5}{1}

\dfrac{-15}{4}

\dfrac{465}{41}

2.3.1 Elements of a Fraction

1) Denominator :
It is the part of a fraction that is below the line of a fraction. Like in the above case

1

4

41

are the denominators of the fraction.

2) Numerator :
It is the part of a fraction that is above the line of a fraction. Like in the above case

5

-15

465

are the numerators of the fraction.

2.3.2 Types of Fractions

1) Proper Fraction:

A fraction in which the the numerator is less than the denominator, i.e. the fraction is less than one. Eg.,

\dfrac{4}{5}

\dfrac{6}{10}

\dfrac{8}{19}

2) Improper fraction:

A fraction in which the the numerator is equal to or greater than the denominator, i.e. the fraction is more than one. Eg.,

\dfrac{5}{4}

\dfrac{10}{6}

\dfrac{8}{7}

3) Mixed fraction:

A fraction which is a combination of a whole number and a proper fraction. Eg.,

1\dfrac{1}{4}

1\dfrac{4}{6}

2\dfrac{4}{7}

You will notice that these examples are the same as the improper fractions. Therefore, every improper fraction can be expressed as a mixed fraction.

4) Equivalent fraction:

Fractions with the same value are called equivalent fractions. In other words the ratio of fractions which are same are called equivalent fractions. Eg.,

\dfrac{3}{4} = \dfrac{6}{8} = \dfrac{9}{12} = \dfrac{12}{16} = \dfrac{15}{20}

Note that we can reduce all of these fractions to

\dfrac{3}{4}

Also, the simplest form of the fraction is called the reduced fraction. Eg.,

\dfrac{3}{4}

is the reduced fraction of

\dfrac{15}{20}

5) Reciprocal:

The numerator and denominator of a fraction are interchanged to create its reciprocal. For example, if

a

is a number then

\dfrac{1}{a}

is the reciprocal of

a

and its also the other way around i.e.,

a

is the reciprocal of

\dfrac{1}{a}

. Eg.,

5

is the reciprocal of

\dfrac{1}{5}

and

\dfrac{1}{5}

is the reciprocal of

5

Example 6

x=\dfrac{4}{5}

y=\dfrac{5}{6}

and

z=8

, then what are

\dfrac{x}{y}

\dfrac{y}{z}

and

\dfrac{z}{x}

Solution

\dfrac{x}{y} = \dfrac{4}{5} \div \dfrac{5}{6} = \dfrac{4}{5} \times \dfrac{6}{5} = \dfrac{24}{25}

;

\dfrac{y}{z} = \dfrac{5}{6} \div \dfrac{8}{1} = \dfrac{5}{6} \times \dfrac{1}{8} = \dfrac{5}{48}

;

\dfrac{z}{x} = \dfrac{8}{1} \div \dfrac{4}{5} = \dfrac{8}{1} \times \dfrac{5}{4} = 10

Example 7

x = \dfrac{3}{5}

y = \dfrac{5}{6}

z = \dfrac{7}{9}

, then which of the following is true?

(1)

x \gt y \gt z

(2)

z \gt x \gt y

(3)

y \gt z \gt x

(4)

x \gt z \gt y

Solution

Converting all these to the nearest decimal form, we get

x = \dfrac{3}{5}=0.6

;

y = \dfrac{5}{6}=0.83

;

z = \dfrac{7}{9}=0.78

As the denominator for all the decimals is the same, which is

1

, we can directly compare them.

y

z

x

Alternatively

Common denominators will help us compare. Lowest Common Multiple (LCM) of the denominators

5

6

and

9

90

x = \dfrac{3}{5} \times \dfrac{18}{18} = \dfrac{54}{90}

;

y = \dfrac{5}{6} \times \dfrac{15}{15} = \dfrac{75}{90}

;

z = \dfrac{7}{9} \times \dfrac{10}{10} = \dfrac{70}{90}

With common denominators, we can compare the numerators ⇒

75 \gt 70 \gt 54

\therefore y \gt z \gt x

Alternatively

For two fractions

\dfrac{a}{b}

and

\dfrac{c}{d}

,if

ad\gt bc

, then

\dfrac{a}{b} \gt \dfrac{c}{d}

3 \times 6 \lt 5 \times 5, \dfrac{3}{5} \lt \dfrac{5}{6}

⇒

x \lt y

5 \times 9 \gt 6 \times 7, \dfrac{5}{6} \gt \dfrac{7}{9}

⇒

y \gt z

3 \times 9 \lt 7 \times 5, \dfrac{3}{5} \lt \dfrac{7}{9}

⇒

x \lt z

\therefore y \gt z \gt x

Answer: (3)

y \gt z \gt x

Note: LCM is explained in detail in the next lesson

-

Factors & Remainder.

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