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Sagot :
Sure! Let's go through each fraction step-by-step to find their decimal expansions and determine the type of decimal expansion they have.
### (i) [tex]\(\frac{42}{8}\)[/tex]
First, let's perform the division:
[tex]\[ \frac{42}{8} = 5.25 \][/tex]
The decimal expansion of [tex]\(\frac{42}{8}\)[/tex] is [tex]\( 5.25 \)[/tex].
- Type of Decimal Expansion: This is a Terminating decimal because the decimal terminates after 2 decimal places.
### (ii) [tex]\(\frac{5}{6}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{5}{6} = 0.8333333333\ldots \][/tex]
The decimal expansion of [tex]\(\frac{5}{6}\)[/tex] is [tex]\( 0.8333333333\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digit 3 repeats endlessly.
### (iii) [tex]\(\frac{9}{11}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{9}{11} = 0.8181818181\ldots \][/tex]
The decimal expansion of [tex]\(\frac{9}{11}\)[/tex] is [tex]\( 0.8181818181\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 81 repeat endlessly.
### (iv) [tex]\(\frac{37}{20}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{37}{20} = 1.85 \][/tex]
The decimal expansion of [tex]\(\frac{37}{20}\)[/tex] is [tex]\( 1.85 \)[/tex].
- Type of Decimal Expansion: This is a Terminating decimal because the decimal terminates after 2 decimal places.
### (v) [tex]\(\frac{4}{7}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{4}{7} = 0.5714285714\ldots \][/tex]
The decimal expansion of [tex]\(\frac{4}{7}\)[/tex] is [tex]\( 0.5714285714\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 571428 repeat endlessly.
### (vi) [tex]\(\frac{43}{13}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{43}{13} = 3.3076923076\ldots \][/tex]
The decimal expansion of [tex]\(\frac{43}{13}\)[/tex] is [tex]\( 3.3076923076\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 307692 repeat endlessly.
In summary, the results are:
- (i) [tex]\(\frac{42}{8} = 5.25\)[/tex] - Terminating
- (ii) [tex]\(\frac{5}{6} = 0.8333333333\ldots\)[/tex] - Non-Terminating, Repeating
- (iii) [tex]\(\frac{9}{11} = 0.8181818181\ldots\)[/tex] - Non-Terminating, Repeating
- (iv) [tex]\(\frac{37}{20} = 1.85\)[/tex] - Terminating
- (v) [tex]\(\frac{4}{7} = 0.5714285714\ldots\)[/tex] - Non-Terminating, Repeating
- (vi) [tex]\(\frac{43}{13} = 3.3076923076\ldots\)[/tex] - Non-Terminating, Repeating
Each fraction's decimal expansion and classification has been provided.
### (i) [tex]\(\frac{42}{8}\)[/tex]
First, let's perform the division:
[tex]\[ \frac{42}{8} = 5.25 \][/tex]
The decimal expansion of [tex]\(\frac{42}{8}\)[/tex] is [tex]\( 5.25 \)[/tex].
- Type of Decimal Expansion: This is a Terminating decimal because the decimal terminates after 2 decimal places.
### (ii) [tex]\(\frac{5}{6}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{5}{6} = 0.8333333333\ldots \][/tex]
The decimal expansion of [tex]\(\frac{5}{6}\)[/tex] is [tex]\( 0.8333333333\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digit 3 repeats endlessly.
### (iii) [tex]\(\frac{9}{11}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{9}{11} = 0.8181818181\ldots \][/tex]
The decimal expansion of [tex]\(\frac{9}{11}\)[/tex] is [tex]\( 0.8181818181\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 81 repeat endlessly.
### (iv) [tex]\(\frac{37}{20}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{37}{20} = 1.85 \][/tex]
The decimal expansion of [tex]\(\frac{37}{20}\)[/tex] is [tex]\( 1.85 \)[/tex].
- Type of Decimal Expansion: This is a Terminating decimal because the decimal terminates after 2 decimal places.
### (v) [tex]\(\frac{4}{7}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{4}{7} = 0.5714285714\ldots \][/tex]
The decimal expansion of [tex]\(\frac{4}{7}\)[/tex] is [tex]\( 0.5714285714\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 571428 repeat endlessly.
### (vi) [tex]\(\frac{43}{13}\)[/tex]
Next, let's perform the division:
[tex]\[ \frac{43}{13} = 3.3076923076\ldots \][/tex]
The decimal expansion of [tex]\(\frac{43}{13}\)[/tex] is [tex]\( 3.3076923076\ldots \)[/tex].
- Type of Decimal Expansion: This is a Non-Terminating, Repeating decimal because the digits 307692 repeat endlessly.
In summary, the results are:
- (i) [tex]\(\frac{42}{8} = 5.25\)[/tex] - Terminating
- (ii) [tex]\(\frac{5}{6} = 0.8333333333\ldots\)[/tex] - Non-Terminating, Repeating
- (iii) [tex]\(\frac{9}{11} = 0.8181818181\ldots\)[/tex] - Non-Terminating, Repeating
- (iv) [tex]\(\frac{37}{20} = 1.85\)[/tex] - Terminating
- (v) [tex]\(\frac{4}{7} = 0.5714285714\ldots\)[/tex] - Non-Terminating, Repeating
- (vi) [tex]\(\frac{43}{13} = 3.3076923076\ldots\)[/tex] - Non-Terminating, Repeating
Each fraction's decimal expansion and classification has been provided.
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