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Find the decimal expansions of [tex][tex]$\frac{10}{3}$[/tex][/tex], [tex][tex]$\frac{7}{8}$[/tex][/tex], and [tex][tex]$\frac{1}{7}$[/tex][/tex].

Sagot :

GinnyAnswer
Certainly! Let's find the decimal expansions for each of the fractions given: [tex]\( \frac{10}{3} \)[/tex], [tex]\( \frac{7}{8} \)[/tex], and [tex]\( \frac{1}{7} \)[/tex].

### 1. Decimal Expansion of [tex]\( \frac{10}{3} \)[/tex]

First, we need to divide 10 by 3:

- 10 divided by 3 gives us 3 with a remainder of 1.
- Bring down a zero to make it 10 again.
- 10 divided by 3 is still 3, with a remainder of 1.

This process repeats indefinitely, giving us a repeating decimal.

So, the decimal expansion of [tex]\( \frac{10}{3} \)[/tex] is:
[tex]\[ 3.3333333333333335\][/tex]

### 2. Decimal Expansion of [tex]\( \frac{7}{8} \)[/tex]

Next, let's divide 7 by 8:

- 7 divided by 8 is 0 with a remainder of 7.
- Bring down a zero to make it 70.
- 70 divided by 8 is 8 with a remainder of 6.
- Bring down a zero to make it 60.
- 60 divided by 8 is 7 with a remainder of 4.
- Bring down a zero to make it 40.
- 40 divided by 8 is 5 with a remainder of 0.

Since we reached a remainder of 0, the division stops.

So, the decimal expansion of [tex]\( \frac{7}{8} \)[/tex] is:
[tex]\[ 0.875 \][/tex]

### 3. Decimal Expansion of [tex]\( \frac{1}{7} \)[/tex]

Lastly, we divide 1 by 7:

- 1 divided by 7 is 0 with a remainder of 1.
- Bring down a zero to make it 10.
- 10 divided by 7 is 1 with a remainder of 3.
- Bring down a zero to make it 30.
- 30 divided by 7 is 4 with a remainder of 2.
- Bring down a zero to make it 20.
- 20 divided by 7 is 2 with a remainder of 6.
- Bring down a zero to make it 60.
- 60 divided by 7 is 8 with a remainder of 4.
- Bring down a zero to make it 40.
- 40 divided by 7 is 5 with a remainder of 5.
- Bring down a zero to make it 50.
- 50 divided by 7 is 7 with a remainder of 1.

You can see that the remainder 1 brings us back to the starting point, creating a repeating cycle.

So, the decimal expansion of [tex]\( \frac{1}{7} \)[/tex] is:
[tex]\[ 0.14285714285714285\][/tex]

### Summary

Summarizing the decimal expansions we have calculated:

- The decimal expansion of [tex]\( \frac{10}{3} \)[/tex] is: [tex]\( 3.3333333333333335 \)[/tex].
- The decimal expansion of [tex]\( \frac{7}{8} \)[/tex] is: [tex]\( 0.875 \)[/tex].
- The decimal expansion of [tex]\( \frac{1}{7} \)[/tex] is: [tex]\( 0.14285714285714285 \)[/tex].
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