Question

Express $3.\overline{7}$ as a common fraction.

Answer 1

To express the repeating decimal $3.\overline{7}$ as a fraction, we set it equal to x, then use algebraic manipulation to solve for x, resulting in the fraction 34/9.

Step-by-step explanation:

The student has asked to express the repeating decimal $3.\overline{7}$ as a common fraction. To convert a repeating decimal to a fraction, we can use algebraic techniques. Let's set x equal to the repeating decimal: x = 3.\overline{7}. Notice that the number 7 repeats indefinitely. If we multiply x by 10 to shift the decimal point to the right, we get 10x = 37.\overline{7}. Now we subtract the original equation (1) from this new equation (2) to get rid of the repeating part:

(2) 10x = 37.\overline{7}

(1) - x = 3.\overline{7}

This leaves us with:

9x = 34

Dividing both sides of the equation by 9 gives us x = 34/9, which is the fraction that represents $3.\overline{7}$.