Express the number as a ratio of integers.
0.26 = 0.26262626 . ..
The Correct Answer and Explanation is :
To express the repeating decimal 0.26262626…0.26262626… as a ratio of integers, we can follow a systematic method.
Step 1: Represent the repeating decimal as a variable.
Let x=0.26262626…x = 0.26262626…. Notice that the decimal repeats every two digits, “26”. We want to eliminate the repeating part by using algebraic manipulation.
Step 2: Multiply both sides by 100.
Since the decimal repeats every two digits, we multiply both sides of the equation by 100 to shift the decimal point two places to the right: 100x=26.26262626…100x = 26.26262626…
Step 3: Subtract the original equation from this new equation.
Now, subtract the original equation x=0.26262626…x = 0.26262626… from the equation 100x=26.26262626…100x = 26.26262626…: 100x−x=26.26262626…−0.26262626…100x – x = 26.26262626… – 0.26262626…
This simplifies to: 99x=2699x = 26
Step 4: Solve for xx.
Now that we have 99x=2699x = 26, divide both sides by 99 to solve for xx: x=2699x = \frac{26}{99}
Thus, 0.26262626…0.26262626… can be written as the fraction 2699\frac{26}{99}.
Verification:
To verify, divide 2626 by 9999: 2699≈0.26262626…\frac{26}{99} \approx 0.26262626…
This confirms that the repeating decimal 0.26262626…0.26262626… is equivalent to 2699\frac{26}{99}.
Conclusion:
The repeating decimal 0.26262626…0.26262626… is equal to the ratio of integers 2699\frac{26}{99}. This method, using algebra to eliminate the repeating decimal, is a standard approach for converting repeating decimals into fractions.