Express the repeating decimal as the ratio of two integers. 1.29456 = 1.29456456456. The ratio of two integers is (Type an integer or a fraction
The Correct Answer and Explanation is:
To express the repeating decimal 1.29456456456…1.29456456456\ldots1.29456456456… as a fraction, we can follow these steps:
Step 1: Define the repeating decimal as a variable.
Let x=1.29456456456…x = 1.29456456456\ldotsx=1.29456456456…, where the “45” repeats.
Step 2: Eliminate the repeating part.
Since the repeating part is two digits long (“45”), we multiply both sides of the equation by 100 to shift the decimal point two places to the right:100x=129.456456456…100x = 129.456456456\ldots100x=129.456456456…
Step 3: Subtract the original equation from this new equation.
Now subtract the original equation x=1.29456456456…x = 1.29456456456\ldotsx=1.29456456456… from 100x=129.456456456…100x = 129.456456456\ldots100x=129.456456456…:100x−x=129.456456456…−1.29456456456…100x – x = 129.456456456\ldots – 1.29456456456\ldots100x−x=129.456456456…−1.29456456456…
Simplifying both sides:99x=128.161892892…99x = 128.161892892\ldots99x=128.161892892…
Step 4: Separate the integer part and the decimal part.
Now separate the integer part from the decimal. The equation becomes:99x=128+0.161892892…99x = 128 + 0.161892892\ldots99x=128+0.161892892…
Let’s focus on the repeating decimal part. To handle this, we can now express the repeating part as a fraction.
Step 5: Express the repeating decimal as a fraction.
Let y=0.161892892…y = 0.161892892\ldotsy=0.161892892…. To convert yyy into a fraction, follow similar steps as we did for the original decimal.
Multiply yyy by 1000 to shift the decimal three places:1000y=161.892892892…1000y = 161.892892892\ldots1000y=161.892892892…
Subtract the original equation y=0.161892892…y = 0.161892892\ldotsy=0.161892892… from this new equation:1000y−y=161.892892892…−0.161892892…1000y – y = 161.892892892\ldots – 0.161892892\ldots1000y−y=161.892892892…−0.161892892…
Simplifying:999y=161.731999y = 161.731999y=161.731
Now, solve for yyy:y=161.731999y = \frac{161.731}{999}y=999161.731
Step 6: Substitute back into the original equation.
Now we substitute the value of yyy into our equation for xxx:99x=128+161.73199999x = 128 + \frac{161.731}{999}99x=128+999161.731
Now, solve for xxx by dividing both sides by 99:x=128+161.73199999x = \frac{128 + \frac{161.731}{999}}{99}x=99128+999161.731
After simplifying, we get:x=127389900x = \frac{12738}{9900}x=990012738
This fraction simplifies to the ratio of two integers.
Final Answer:
The repeating decimal 1.29456456456…1.29456456456\ldots1.29456456456… can be expressed as the fraction 127389900\frac{12738}{9900}990012738.
