Converting 0.345 with a bar on 45 into p/q form:

To convert 0.345 with a bar on 45 into p/q form, we need to follow the given steps:

1. Let x = 0.3454545... (with a bar on 45)
2. Multiply x by 1000 to eliminate the bar: 1000x = 345.4545...
3. Subtract x from 1000x to eliminate the repeating decimal: 1000x - x = 345.4545... - 0.3454...
4. Simplify the equation by combining like terms: 999x = 345.1091...
5. Solve for x: x = 345.1091... / 999

Now, we need to simplify this fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

6. Find the GCF of 3451091 and 999 using the Euclidean algorithm:
- 999 divides into 3451 with a remainder of 242
- 3451 divides into 999 three times with a remainder of 464
- 999 divides into 2424 twice with a remainder of 846
- 2424 divides into 999 twice with a remainder of 426
- 999 divides into 426 four times with a remainder of 243
- 426 divides into 243 once with a remainder of 183
- 243 divides into 183 once with a remainder of 60
- 183 divides into 60 three times with a remainder of 3
- 60 divides into 3 zero times with a remainder of 3
- Therefore, the GCF of 3451091 and 999 is 3.

7. Divide both the numerator and denominator by 3 to simplify the fraction:
- 3451091 / 999 = 1150363 / 333

Therefore, the p/q form of 0.345 with a bar on 45 is 1150363/333.