During the conversion of repeating decimals to fractions, the following steps are followed:
Step 1. Let y be equal to the decimal that is repeating and that which you wish to convert to become a fraction.
Step 2. Examine the repeating decimal carefully to determine the repeating digits. It is important to note that it may be one repeating digit or they might be more repeating digits.
Step 3. Place the digit or digits that are repeating to the left-hand side of the decimal point.
Step 4. Place the digit or digits that repeat to the right-hand side of the decimal point.
Step 5. Subtract the left-hand sides of the two equations. Also, subtract the right-hand sides of both the equations. Ensure that there is a positive difference for both sides as you subtract.
Example 1. Convert the following decimal to a fraction, 0.55555555555,
Step 1. Y = 0.55555555555
Step 2. Here, you are supposed to examine what the repeating digit is or digits if it is more than one. In this case, the digit is 5.
Step 3. In order for you to put the repeating digit to the left-hand side of the decimal point, move the decimal point to the right by one place. In other words, this can be said to multiply by ten since it would lead to the same result, a shift of the decimal point to the right by one place. Once you multiply a side by a number, make sure to multiply the opposite side with the same number, this is to maintain the balance of the equation. Therefore, the result of this will be, 10y = 5.5555555555.
Step 4. Place the digits that repeat to the right-hand side of the decimal point. In this case, the digit that repeats is already on the right-hand side therefore, we live it at that. y = 0.55555555555.
Step 5. You now have two equations which are, 10y = 5.5555555555 and y = 0.55555555555. subtract, therefore, 10y − y = 5.5555555555 − 0.55555555555. this results to 9x = 5. So the value of x is 5/9.
Another example, what fraction equals 1.04242424242?
Step 1. y = 1.042424242
Step 2. The repeating digit, in this case, is 42.
Step 3. In order for you to move the digit that repeats to the left-hand side of the decimal point, move the decimal point to the right by three places. In other words, it can be said to multiply by 1,000 since it would bring the same result as moving the decimal three places to the right. Remember to multiply the opposite side by the same number (1,000). This is done for the purpose of maintaining the balance of the equation. Therefore, 1,000y = 1042.42424242.
Step 4. Place the digits that repeat to the right-hand side of the decimal point. In this equation, this is accomplished by moving the decimal point to the right by one place. Multiply both sides by ten. Therefore, 10y = 10.4242424242.
Step 5. The two equations that result are, 1000y = 1042.42424242 and 10y = 10.42424242. subtract, 1000y − 10y = 1042.42424242 −10.42424242. this results to 990y = 1032. Therefore, the value of y is 1032/900.
