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].
