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# Express 1.3212121… as a rational number

A rational number is a kind of real number with the formula m/n, where n≠ 0. When a rational number is divided, the outcome is a decimal number, which can be either ended or repeated. These numbers are written as p/q, where p and q are integers and q ≠ zero. Because of the underlying structure of numbers, which is p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. Whole numbers constitute fractions, while integers constitute the numerator and denominator of rational numbers.

### Conversion of the Decimal number to Rational number

Below are the steps for the conversion of decimal numbers to rational numbers,

• Step 1: Identify the repeating decimal and put it equal to x
• Step 2: Write it in decimal form by removing the bar from the top of repeating digits and listing repeating digits at least twice.

For Example, write x = 0.3 bar as x = 0.333… and x = 0.33 bar as x = 0.333333…

• Step 3: Examine the number of digits having a bar.
• Step 4: If the number having a repeating decimal has 1 place repetition then we will multiply it by 10, if it has a two-place repetition then it will be multiplied by 100, and a three-place repetition after the decimal will be multiplied by 1000, and so on.
• Step 5: After that Subtract the equation obtained in the second step from the equation obtained in step 4.
• Step 6: Whatever is left, Divide both sides of the equation by the x coefficient.
• Step 7: At last Write the rational number in its simplest form.

### Express = 1.3212121… as a rational number

Solution:

Given: 1.3212121…

Let’s assume x = 1.3212121…. ⇢  (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 99x = 130.8

x = 130.8/99

= 1308/990

=  436/330

1.3212121…. can be expressed 436/330  in form of p/q as rational number.

### Sample Questions

Question 1: Express 5.959595… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 5.959595…. or Let’s assume x = 5.959595… ⇢  (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 99x = 590

x = 590/99

5.959595… can be expressed 590/99 in form of p/q as rational number.

Question 2:  Express 26.333333… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 26.333333… or Let’s assume x = 26.333333…. ⇢  (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 99x = 2607

x = 2607 /99

26.333333… can be expressed 2607/99 in form of p/q as rational number.

Question 3: Express 9.969696… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given : 9.969696… or Let’s assume x = 9.969696… ⇢ (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 99x = 987

x = 987/99

=  329/33

9.969696…. can be expressed 329/33 in form of p/q as rational number.

Question 4: Express 10.65656565… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 10.65656565… or Let’s assume x = 10.65656565… ⇢  (1)

And, there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100,

So 100 x = ⇢ (2)

Now subtract equation (1) from equation (2)

100x – x = 99x = 1055

x = 1055/99

= 1055/99

10.656565…. can be expressed 10555/99 in form of p/q as rational number.

Question 5: Express 159.986986986… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 159.986986986… or Let’s assume x = 159.986986986… ⇢ 1

And, there are three digits after decimal which are repeating,

So multiply equation 1 both sides by 1000

So 1000 x = ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x = 999x = 159827

x = 159827 / 99

159.986986986 can be expressed 159827 / 99 in form of p/q as rational number.

Question 6: Express 56.55555.. as a rational number of the form p/q, where p and q have no common factors.

Solution:

Given: 56.55555 or Lets assume x = 56.55555… ⇢ (1)

And, there are one digit after decimal which are repeating,

So, multiply equation (1) both sides by 10,

So 10 x = ⇢ (2)

Now subtract equation (1) from equation (2)

10x – x = 9x = 509

x = 509 /9

56.55555… can be expressed 509/9 as rational number.

Question 7:  Express 0.99999… as a rational number, in form p/q where p and q have no common factors.

Solution:

Given: 0.99999… or Let’s assume x = 0.99999…. ⇢  (1)

And, there are one digit after decimal which are repeating, so multiply equation (1) both sides by 10,

So 10 x = ⇢ (2)

Now subtract equation (1) from equation (2)

10x – x = 99x = 9

x = 9/99

= 1/11

0.99999…. can be expressed 1/11 in form of p/q as rational number.

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