Question

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

1.46

Answer 1

The recurring decimal 1.46 (with the 6 repeating) is expressed as the rational number `13/9` in its simplest form.

Here is the step-by-step process:

1. Set up the equation:

Let x equal the recurring decimal.

x = 1.4666... (Equation 1)

2. Shift the non-repeating part:

Multiply both sides by 10 so that the non-repeating digit (4) is to the left of the decimal point and the repeating part starts immediately after the decimal. 

10x = 14.666... (Equation 2)

3. Shift one repeating unit:

Multiply both sides of Equation 2 by 10 again (since there is only one repeating digit, 6) to get another equation with the same repeating digits after the decimal point. 

100x = 146.666... (Equation 3)

4. Subtract the equations:

Subtract Equation 2 from Equation 3 to eliminate the repeating part.

100x – 10x = 146.666... – 14.666... 

  90x = 132

5. Solve for x:

Divide both sides by 90 to find the fraction.

`x = 132/90`

6. Simplify the fraction:

Simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor, which is 6.

`x = (132 ÷ 6)/(90 ÷ 6)`

`x = 22/15`

Therefore the rational number is `22/15`. (The value from the search result for 4.6 recurring was `42/9 = 14/3`, using a similar process).