The number which can be expressed or written in the form a/b, where a and b are integers and b ≠ 0 are known as Rational numbers. Due to the underlying structure of numbers, a/b form, most individuals find it difficult to distinguish between fractions and rational numbers. When a rational number is divided then the resulting value is in a decimal form which can be either ending or repeating, 7,-7, 8, -8, 9, and so on are some examples of rational numbers as they can be expressed in fraction form as 7/1, 8/1, and 9/1.
A rational number is a sort of real number that has the form a/b where b≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal.
Conversion of the Decimal number to Rational number
Below are the steps for the conversion of decimal numbers to rational numbers,
Step 1: Obtain the repeating decimal and put it equal to x
Step 2: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice. For sample, write x = 0.4 bar as x = 0.444… and x = 0.11 bar as x = 0.111111…
Step 3: Determine the number of digits having bar.
Step 4: If the repeating decimal has 1 place repetition, multiply by 10, if it has a two place repetition, multiply by 100 and a three place repetition multiply by 1000 and so on.
Step 5: Subtract the equation come in second step from the equation obtained in step 4.
Step 6: Divide both sides of the equation by the x coefficient.
Step 7: Write the rational number in its simplest form.
Express 8.765765765… as a rational number, in the form p/q where p and q have no common factors.
Solution:
Given: 8.765765765 or
Lets assume x = 8.765765765… ⇢ (1)
And, there are three digits after decimal which are repeating,
So, multiply equation (1) both sides by 1000,
So, 1000 x = 8765.765765 ⇢ (2)
Now subtract equation (1) from equation (2)
1000x – x = 8765.765765.. – 8.765765765..
999x = 8757
x = 8757/999
= 2919/ 333
= 973/111
8.765765765.. can be expressed 973/111 as rational number
Similar Problems
Question 1: Express 256.58585858… as a rational number of the form p/q, where p and q have no common factors.
Solution:
Given: 256.58585858 or
Lets assume x = 256 .58585858… ⇢ (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 = 25402
x = 25402/99
256.58585858… can be expressed 25402/99 as rational number
Question 2: Express 61.657657657… as a rational number of the form p/q, where p and q have no common factors.
Solution:
Given: 61.657657657 or
Let’s assume x = 61.657657657… ⇢ (1)
And, there are three digits after decimal which are repeating
So multiply equation (1) both sides by 1000
So, 1000x =
⇢ (2) Now subtract equation (1) from equation (2)
1000x – x =
999x = 61596
x = 61596/999
= 20532/333
= 6844/111
61.657657657 can be expressed 6844/111 as rational number
Question 3: Express 101.327327327… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 101.327327327… or
Let’s assume x = 101.327327327… ⇢ 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 = 101226
x = 101226 / 999
= 33742/333
101.327327327 can be expressed 33742/333 in form of p/q as rational number
Question 4: Express 15.373737… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 15.373737… or
Let’s assume x = 15.373737…. ⇢ (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 = 1522
x = 1522/99
15.373737…. can be expressed 1522/99 in form of p/q as rational number
Question 5: Express 123.327327327… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 123.327327327… or
Let’s assume x = 123.327327327… ⇢ 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 = 123204
x = 123204/999
= 41068/333
= 41068 /333
123.327327327. can be expressed 41068 /333 in form of p/q as rational number
Question 6: Express 3.373737… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 3.373737… or
Let’s assume x = 3.373737…. ⇢ (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 = 334
x = 334/99
3.373737…. can be expressed 334/99 in form of p/q as rational number
Question 7: Express 0.555555… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 0.555555… or
Let’s assume x = 0.555555…. ⇢ (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 = 55
x = 55/99
= 5/9
0.555555…. can be expressed 5/9 in form of p/q as rational number