will the product of 0.6... and 7/8 be an irrational or rational number?
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.
To answer this question, let's break it down step by step:
Step 1: Convert the decimal form of 0.6... to a fraction.
Let's represent 0.6... as x.
So, x = 0.6...
To convert x to a fraction, we can multiply both sides of the equation by 10
10x = 6.6...
Now, we subtract x from 10x to eliminate the decimal part.
10x - x = 6.6... - 0.6...
9x = 6
Step 2: Solve for x.
Divide both sides of the equation by 9 to solve for x.
9x/9 = 6/9
x = 2/3
So, the decimal form of 0.6... is equivalent to the fraction 2/3.
Step 3: Multiply 2/3 by 7/8.
To multiply fractions, we simply multiply the numerators and multiply the denominators.
(2/3) * (7/8) = (2 * 7) / (3 * 8)
= 14/24
Step 4: Simplify the fraction.
We can simplify 14/24 by finding the greatest common divisor (GCD) of 14 and 24, which is 2.
Dividing both the numerator and denominator by 2 gives:
14/24 = 7/12
So, the product of 0.6... and 7/8 is the rational number 7/12.
To summarize, the product of 0.6... and 7/8 is a rational number.
To determine if the product of 0.6... (repeating decimal) and 7/8 is an irrational or rational number, we need to find the value of the repeating decimal.
First, let's convert 0.6... into a fraction. Let x = 0.6...
Multiplying both sides of the equation by 10, we get:
10x = 6.6...
Now, let's subtract x from 10x to eliminate the repeating part:
10x - x = 6.6... - 0.6...
9x = 6
Simplifying the equation, we find that:
x = 6/9
Next, we multiply x = 6/9 and 7/8:
(6/9) * (7/8) = (6 * 7) / (9 * 8)
42 / 72
To simplify the fraction, we need to find the greatest common divisor (GCD) of 42 and 72.
The GCD of 42 and 72 is 6.
Dividing both the numerator and denominator by 6, we get:
42/72 = (42/6) / (72/6) = (7/1) / (12/1) = 7/12
Therefore, the product of 0.6... and 7/8 is a rational number, specifically 7/12.