Show that 0.3333… = 0.3, can be expressed in the form of p/q

Before proceeding with the given question one must know what are rational numbers. Hence a brief definition of rational number is added below.

Rational Numbers

A number is said to be rational if it can be represented in the form of p/q, where p and q are integers and q ≠ 0. All rational numbers are real numbers. All positive integers, positive fractions, zero, negative integers, and negative fractions are part of the set of rational numbers.

Decimal Property of Rational Number

It is known that all rational numbers can be expressed in the form p/q where p and q are integers provided q ≠ 0. p divided q can result into an integer, terminating decimal, or repeating decimal.

Now, let’s solve the given problem statement,

Show that 0.3333… = 0 3, can be expressed in the form of rational number, i.e. p/q.

To show that 0.333… can be expressed in the form of rational number, i.e. p/q, we can use the concept of infinite geometric series.

Step 1: Let x = 0.333…. This means that is a repeating decimal where the digit 3 repeats infinitely.

Step 2: Multiply x by 10. This shifts the decimal point one place: 10x = 3.3333….

Step 3: Now, subtract x from 10x:

10x – x = 3.3333…. – 0.3333….

9x = 3

Step 4: Solve for x:

x = 3/9 = 1/3 (p/q where q ≠ 0)

So, 0.3333… = 1/3, which shows that 0.3333… can indeed be expressed in the form of a rational number p/q, where p = 1 and q = 3.

Similar Question

Q1: Express 0.40777777… in rational form.

Solution:

Let x = 0.40777777…     -(1)

Multiplying (1) by 100 we get

100x = 40.777777…     -(2)

Multiplying (2) by 10 we get

1000x = 407.777777..    -(3)

Now, subtracting (1) from (2) we get

900x = 367

⇒ x = 367/900

Hence, 0.40777777… in rational form is 367/900.

Q2: Express 1.0033333… in rational form.

Solution:

Let x = 1.0033333…     -(1)

Multiplying (1) by 100 we get

100x = 100.33333…     -(2)

Now, subtracting (2) from (1) we get

99x = 99.33

⇒ x = 99.33/99 = 9933/9900

Hence, 1.0033333… in rational form is 9933/9900.