Understanding the Number
To express the number 0.6 07777. 07777. 477777 in the form of p/q, we first need to break it down into its components.
Identifying the Components
- The number consists of three parts:
- 0.6 (which is a terminating decimal)
- 0.07777... (which is a repeating decimal)
- 0.477777... (which is also a repeating decimal)
Converting Each Part
- Converting 0.6:
- This can be expressed as 6/10, which simplifies to 3/5.
- Converting 0.07777...:
- Let x = 0.07777...
- Then, 10x = 0.7777...
- Subtracting these, we get: 10x - x = 0.7777... - 0.07777...
- This simplifies to 9x = 0.7, which means x = 0.7/9 = 7/90.
- Converting 0.477777...:
- Let y = 0.477777...
- Then, 10y = 4.77777...
- Subtracting gives: 10y - y = 4.77777... - 0.477777...
- This simplifies to 9y = 4.3, thus y = 4.3/9 = 43/90.
Combining the Parts
- Now combine all parts:
- 0.6 + 0.07777... + 0.477777...
- This becomes (3/5) + (7/90) + (43/90).
Finding a Common Denominator
- The common denominator is 90:
- (3/5) = (54/90)
- Therefore, the sum is: 54/90 + 7/90 + 43/90 = 104/90.
Simplifying the Fraction
- Finally, we simplify 104/90:
- Both numerator and denominator can be divided by 2.
- This gives us 52/45.
Final Result
Thus, the number 0.6 07777. 07777. 477777 can be expressed as 52/45 in the form of p/q, where p = 52 and q = 45.