will the product of 0.6... and 7/8 be an irrational or rational number?

Answer 1

The product of 0.6... and 7/8 will be a rational number.

To understand this, let's consider the decimal representation 0.6.... The ellipsis (...) indicates an infinite number of 6's after the decimal point.

To express the decimal in fractional form, we can call it x:

x = 0.666...

Now, we multiply both sides of the equation by 10 to shift the decimal place one space to the right:

10x = 6.666...

At this point, we can observe that the left side of the equation is 10 times the value of x, while the right side of the equation is 6 more than the value of x. Therefore, we can subtract the first equation from the second equation to eliminate the repeating decimal:

10x - x = 6.666... - 0.666...

This simplifies to:

9x = 6

Now, divide both sides of the equation by 9:

x = 6/9

Simplifying further:

x = 2/3

We have successfully represented the repeating decimal 0.6... as the fraction 2/3, which proves that it is a rational number.

Now, let's calculate the product of 0.6... and 7/8:

0.6... * 7/8 = (2/3) * (7/8) = 14/24 = 7/12

Since 7/12 can be represented as a fraction, it is a rational number.