Express .123123123… as a rational number
Rational numbers are numbers that may be expressed or written as m/n, where m and n are integers and n is not equal to zero (n ≠0). Because of the fundamental structure of numbers, the m/n form, most individuals have difficulties differentiating between fractions and rational numbers. When a rational number is divided, it gives a decimal value that might be either ending or repeating. Rational numbers include 11, -11, 5, -5, 9, and so on, which may be expressed in fraction form as 11/1, -5/1, and 7/1, respectively. A rational number is a kind of a real number having the formula m/n, where n is not equal to zero(n≠0). When you divide a rational number, you get a decimal number that may be terminated or repeated.
Steps for converting decimal values to rational numbers
- Step 1: Find the repeating decimal and keep it equal to x.
- Step 2: Write it in decimal form by eliminating the bar at the top of the repeated numbers and listing them at least twice.
For example, x = 0.4 bar is written as x = 0.444.. while x = 0.44 bar is written as x = 0.444444…
- Step 3: Check the number of digits with a bar.
- Step 4: If the number with a repeating decimal has a one-place repetition, we multiply it by 10, a two-place repetition by 100, a three-place repetition by 1000, and so on.
- Step 5: Following that, the Subtraction of the equation is obtained in step 2 from the equation obtained in step 4.
- Step 6: Divide the rest of the equation by the x coefficient.
- Step 7: Lastly, write the rational number in the most basic form.
Express 0.123123123… as a rational number
Solution:
Given: 0 .123123123. or
Lets assume x = 0 .123123123…. ⇢ (1)
As we can see there are three digits after decimal which are repeating,
So, multiply equation (1) both sides by 1000,
So, 1000 x = 123.123123 ⇢ (2)
Now subtract equation (1) from equation (2)
1000x – x = 123.123123.. – 0.123123123…
999x = 123
x = 123/999
= 41/ 333
0 .123123123. can be expressed 41/333 as rational number.
Similar Questions
Question 1: Express 26.588588… as a rational number of the form p/q, where p and q have no common factors.
Solution:
Given: 26.588588 or
Lets assume x = 26 .588588… ⇢ (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)
999x = 26562
x = 26562/999
x = 8854/333
26.588588… can be expressed 8854/333 as rational number.
Question 2: Express 3.272727… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 3.272727… or
Let’s assume x = 3.272727…. ⇢ (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 = 324
x = 324/99
= 108/33
= 36/11
3.272727…. can be expressed 36/11 in form of p/q as rational number.
Question 3: Express 65.232323… as a rational number, in form p/q where p and q have no common factors?
Solution:
Given: 65.232323… or
Let’s assume x = 65.232323…. ⇢ (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 = 6458
x = 6458/99
65.232323…. can be expressed 6458/99 in form of p/q as rational number.
Question 4: Express 11.777… as a rational number, in form p/q where p and q have no common factors?
Solution:
Given: 11 .777777… or
Let’s assume x = 11.777777…. ⇢ (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)
99x = 1166
x = 1166 /99
11.777777…. can be expressed 1166/99 in form of p/q as rational number.
Question 5: Express 14.555555… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 14.555555… or
Let’s assume x = 14.555555…. ⇢ (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 = 14541
x = 14541 /999
= 4847/333
14.555555…. can be expressed 4847/333 in form of p/q as rational number.
Question 6: Express 2.5050… as a rational number, in form p/q where p and q have no common factors.
Solution:
Given: 2.5050….
Let’s assume x = 2.5050….. ⇢ (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 = 248
x = 248/99
2.5050…. can be expressed 248/99 in form of p/q as rational number.
Last Updated :
30 Dec, 2023
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