Question

Express 711
as a repeating decimal. Mark the cycle of repeating digits with a bar.

Note: You will find the option to add a bar above text in the Geometry ( ∠
) keyboard.

Show your work.

(1 point)

Answer 1

To express 711 as a repeating decimal, we start by dividing 1 by 711.

1. Perform long division of 1 by 711:

- 1.0000000 (we can add decimal places to help with the division)
- 711 does not go into 1. It goes into 10 zero times.
- Next, we consider 100 (0.001 in decimal). 711 still does not fit into it. It goes 0 times.
- Now consider 1000. 711 fits into 1000 only once.

\[
1 \times 711 = 711
\]

Subtract 711 from 1000:

\[
1000 - 711 = 289
\]

- Bring down another 0 to make 2890.

- 711 goes into 2890 four times:

\[
4 \times 711 = 2844
\]

Subtract 2844 from 2890:

\[
2890 - 2844 = 46
\]

- Bring down another 0 to make 460.

- 711 goes into 460 zero times.

Bring down another 0 to make 4600.

- 711 goes into 4600 six times:

\[
6 \times 711 = 4266
\]

Subtract 4266 from 4600:

\[
4600 - 4266 = 334
\]

- Bring down another 0 to make 3340.

- 711 goes into 3340 four times:

\[
4 \times 711 = 2844
\]

Subtract 2844 from 3340:

\[
3340 - 2844 = 496
\]

- Bring down another 0 to make 4960.

- 711 goes into 4960 six times:

\[
6 \times 711 = 4266
\]

Subtract 4266 from 4960:

\[
4960 - 4266 = 694
\]

- Bring down another 0 to make 6940.

- 711 goes into 6940 nine times:

\[
9 \times 711 = 6399
\]

Subtract 6399 from 6940:

\[
6940 - 6399 = 541
\]

- Bring down another 0 to make 5410.

- 711 goes into 5410 seven times:

\[
7 \times 711 = 4977
\]

Subtract 4977 from 5410:

\[
5410 - 4977 = 433
\]

- Bring down another 0 to make 4330.

- 711 goes into 4330 six times:

\[
6 \times 711 = 4266
\]

Subtract 4266 from 4330:

\[
4330 - 4266 = 64
\]

- Bring down another 0 to make 640.

- 711 goes into 640 only 0 times.

Bring down another 0 to make 6400.

- 711 goes into 6400 nine times:

\[
9 \times 711 = 6399
\]

Subtract 6399 from 6400:

\[
6400 - 6399 = 1
\]

This gives us a repeating cycle because we've returned to our original value, leading us back to 1.

From the calculations above, the decimal representation of **1/711** is:

\[
0.001404 \overline{004}
\]

Thus, the repeating decimal is:

\[
0.001404 \overline{004}
\]

with **004** repeating.