write the repeating decimal as a the ratio of integers:0.463(63 repeating)?

0.463ᵣ₂ = 0.1 + 0.363ᵣ₂ = 0.1 + 0.36ᵣ₂ = 1/10 + 36/99 = 1/10 + 4/11 = (11 + 40)/110 = 51/110
One nice thing with fractions is they fracture.

x = 0.4636363...

Since the repeat is 2 digits, multiply both sides by 100:
100x = 46.3636363...

Subtract the first equation from the second.
99x = 45.9

Multiply both sides by 10 to get rid of decimals:
990x = 459

Divide both sides by 990:
x = 459/990

Reduce to lowest terms by dividing top and bottom by 9:
x = 51/110

Answer:
51/110 = 0.46363636...

.463 = 463/1000
.6363.... = 63/99
.0006363..... = 63/99000

Thus the decimal = 463/1000 + 63/9900
(463*99 + 630)/99000 = 4647/99000