To answer this question, we need to understand how to add repeating decimals and express the sum as a fraction in lowest terms. Let's go through each option to understand why it is correct or incorrect:
Option A) 2475 - This option is incorrect because the product of the numerator and denominator of the fraction is not equal to 2475.
Option B) 680 - This option is incorrect because the product of the numerator and denominator of the fraction is not equal to 680.
Option C) 670 - This option is incorrect because the product of the numerator and denominator of the fraction is not equal to 670.
Option D) 660 - This option is correct because the product of the numerator and denominator of the fraction is equal to 660.
To solve the problem, let's add the repeating decimals 0.353535... and 0.252525...
Step 1: Let x = 0.353535...
Step 2: Multiply x by 100 to shift the decimal point two places to the right: 100x = 35.353535...
Step 3: Subtract x from 100x to eliminate the decimals: 100x - x = 35.353535... - 0.353535...
99x = 35
Step 4: Divide both sides of the equation by 99 to solve for x: x = 35/99
Similarly,
Step 1: Let y = 0.252525...
Step 2: Multiply y by 100 to shift the decimal point two places to the right: 100y = 25.252525...
Step 3: Subtract y from 100y to eliminate the decimals: 100y - y = 25.252525... - 0.252525...
99y = 25
Step 4: Divide both sides of the equation by 99 to solve for y: y = 25/99
Now, let's add x and y:
x + y = 35/99 + 25/99 = (35 + 25)/99 = 60/99
To express 60/99 in lowest terms, we can divide both the numerator and denominator by their greatest common divisor, which is 3:
60/99 = (60 ÷ 3) / (99 ÷ 3) = 20/33
The product of the numerator and denominator is 20 * 33 = 660.
Therefore, the correct answer is option D) 660.