0.94 = 0.94 Express the given repeating decimal quotient of integers possible , reduce lowest terms
The Correct Answer and Explanation is:
✅ Step 1: Let x equal the repeating decimal
Let
x = 0.949494…
✅ Step 2: Multiply x by 100
Since the repeating part has two digits (“94”), we multiply both sides of the equation by 100 to move the decimal point two places to the right:
100x = 94.949494…
✅ Step 3: Subtract the original equation from this one
Now subtract the original equation (x = 0.949494…) from the one we just created: 100x=94.949494…− x= 0.949494…99x=94\begin{align*} 100x &= 94.949494… \\ – \ \ \ x &= \ \ \ 0.949494… \\ \hline 99x &= 94 \end{align*}100x− x99x=94.949494…= 0.949494…=94
✅ Step 4: Solve for x
x=9499x = \frac{94}{99}x=9994
✅ Step 5: Reduce the fraction to lowest terms
We now simplify 94/99.
Check if any common factors exist. The GCD of 94 and 99 is 1, so the fraction is already in lowest terms.
✅ Final Answer:
0.94‾=9499\boxed{0.\overline{94} = \frac{94}{99}}0.94=9994
📘 Textbook-style Explanation (300 words)
To convert a repeating decimal into a fraction, we use algebra to isolate and eliminate the repeating part. The decimal 0.94̅ means that the digits “94” repeat indefinitely, forming the sequence 0.949494…, and so on. Our goal is to express this as a fraction of two integers in lowest terms.
We begin by assigning a variable to the decimal: let x = 0.949494…. Since the repeating block is two digits long, we multiply both sides of the equation by 100 (a power of 10 with two zeros) to shift the decimal point two places to the right. This gives us 100x = 94.949494….
Now we subtract the original x = 0.949494… from 100x = 94.949494…, which eliminates the repeating part of the decimal:
100x – x = 94.949494… – 0.949494…, resulting in 99x = 94.
Next, we solve for x by dividing both sides by 99:
x = 94 / 99.
The final step is to reduce the fraction to its lowest terms. To do this, we find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 94 and 99 is 1, so the fraction is already in its simplest form.
Therefore, the repeating decimal 0.94̅ is exactly equal to the fraction 94/99.
