I need some help with this question. I am getting a different answer (the actual answer should be 259).

Question:

When the repeating decimal 0.476 is converted to a rational in simplest form a/b the value of b-a is? Show your work.

Does the entire 476 repeat?

For example .476476476...

If so then in fraction form it is (476)/999 which does not reduce.

So B = 999 and A=476
B-A = ...

But the answer is 259. How do you get that?

You did not answer Ms Pi when she asked which digits repeat

She took it that all the digits repeat, and got the correct answer of 476/999
and the difference in the digits was not 259

So, again, which digits repeat ?
.476476476... , does not meet your condition

if .476666.... = (476-47)/900 = 143/300 , difference in digits ≠ 259

if
.4767676... = (476-4)/990 = 236/495 , difference in digits = 259
ahhh!, but next time ask the question correctly

To convert the repeating decimal 0.476 into a rational number, we first need to identify the repeating part. In this case, the digits 476 repeat continuously.

To convert the repeating decimal into a rational number, we can use the concept of an infinite geometric series.

Let x represent the repeating decimal 0.476. Multiplying x by 1000 will move the decimal point three places to the right:

1000x = 476 + 0.476 + 0.476 + 0.476 + ...

Now, subtract the two equations:

1000x - x = 476 + 0.476 + 0.476 + 0.476 + ... - 0.476 - 0.476 - 0.476 - ...

Simplifying this equation, we get:

999x = 476

To find the value of x, divide both sides of the equation by 999:

x = 476/999

Now, we have a fraction 476/999, which can be simplified to simplest form by finding the greatest common divisor (GCD) of the numerator and denominator.

The GCD of 476 and 999 is 1. Thus, we can't simplify the fraction further.

Now, we can evaluate b-a, where the numerator (a) is 476 and the denominator (b) is 999:

b - a = 999 - 476 = 523

So, the value of b − a is 523, not 259 as you mentioned.