How to turn 1.27 repeating into a fraction
The correct answer and explanation is:
The decimal 1.27 repeating means the number is 1.27272727…, where “27” repeats indefinitely.
Step 1: Set the repeating decimal equal to a variable
Let x=1.27272727…x = 1.27272727….
Step 2: Multiply to shift the decimal point past one full repeat cycle
Since “27” has two digits, multiply by 100 to move the decimal two places to the right:
100x=127.27272727…100x = 127.27272727….
Step 3: Subtract the original number from this new equation
Subtract x=1.27272727…x = 1.27272727… from 100x=127.27272727…100x = 127.27272727…:
100x−x=127.27272727…−1.27272727…100x – x = 127.27272727… – 1.27272727…
This simplifies to:
99x=12699x = 126.
Step 4: Solve for xx
Divide both sides by 99:
x=12699x = \frac{126}{99}.
Step 5: Simplify the fraction
Both numerator and denominator can be divided by 9:
126÷999÷9=1411\frac{126 \div 9}{99 \div 9} = \frac{14}{11}.
Final answer:
1.2727‾=14111.27\overline{27} = \frac{14}{11}.
Explanation
Repeating decimals can be converted to fractions by using algebra to eliminate the repeating part. The key is to multiply the decimal by a power of 10 that shifts the decimal point right past one complete cycle of the repeating digits. For 1.27 repeating, the repeating part is two digits (“27”), so multiply by 100.
After multiplying, subtract the original decimal from this new number. This subtraction cancels out the repeating parts, leaving a simple equation involving xx. Solving for xx gives a fraction that represents the repeating decimal.
Simplifying the fraction reduces it to lowest terms, making the fraction cleaner and easier to understand.
This process works for any repeating decimal. The number of digits in the repeating sequence determines the power of 10 to multiply by. For example, if three digits repeat, multiply by 1000; for one digit, multiply by 10.
Converting repeating decimals to fractions is useful in math because fractions represent exact values, while decimals, especially repeating ones, are often approximations. Fractions can be easier to use in algebra, comparisons, and exact calculations. Understanding this conversion also strengthens algebra skills and helps in understanding number properties.
So, the repeating decimal 1.272727… is exactly equal to the fraction 14/11.