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Let's address each question step by step.
### Question 21
We need to determine which of the given mixed numbers has the same decimal value as [tex]\( 110 \frac{147}{168} \)[/tex].
1. Convert the mixed number [tex]\( 110 \frac{147}{168} \)[/tex] into a decimal:
- [tex]\( \frac{147}{168} \approx 0.875 \)[/tex]
- So, [tex]\( 110 \frac{147}{168} = 110 + 0.875 = 110.875 \)[/tex]
Now, let's convert each option to its decimal form and compare:
Option (A): [tex]\( 110 \frac{49}{56} \)[/tex]
- Convert [tex]\( \frac{49}{56} \approx 0.875 \)[/tex]
- Thus, [tex]\( 110 + 0.875 = 110.875 \)[/tex]
Option (B): [tex]\( 110 \frac{170}{180} \)[/tex]
- Convert [tex]\( \frac{170}{180} \approx 0.944444 \)[/tex]
- Thus, [tex]\( 110 + 0.944444 \approx 110.944444 \)[/tex]
Option (C): [tex]\( 110 \frac{56}{72} \)[/tex]
- Convert [tex]\( \frac{56}{72} \approx 0.777778 \)[/tex]
- Thus, [tex]\( 110 + 0.777778 \approx 110.777778 \)[/tex]
Option (D): [tex]\( 110 \frac{247}{268} \)[/tex]
- Convert [tex]\( \frac{247}{268} \approx 0.921642 \)[/tex]
- Thus, [tex]\( 110 + 0.921642 \approx 110.921642 \)[/tex]
Comparing these values:
- [tex]\( 110 \frac{49}{56} = 110.875 \)[/tex]
- [tex]\( 110 \frac{170}{180} \approx 110.944444 \)[/tex]
- [tex]\( 110 \frac{56}{72} \approx 110.777778 \)[/tex]
- [tex]\( 110 \frac{247}{268} \approx 110.921642 \)[/tex]
Thus, the correct answer is [tex]\( 110 \frac{49}{56} \)[/tex].
### Question 22
We need to analyze the statements about the negative fractions [tex]\( -\frac{4}{5} \)[/tex] and [tex]\( -\frac{5}{6} \)[/tex].
1. [tex]\( -\frac{4}{5} \)[/tex] expressed as a decimal:
- [tex]\( \frac{4}{5} = 0.8 \)[/tex]. It is a terminating decimal, not a repeating decimal.
2. [tex]\( -\frac{5}{6} \)[/tex] expressed as a decimal:
- [tex]\( \frac{5}{6} \approx 0.833333\ldots \)[/tex]. This is a repeating decimal where the digit 3 repeats indefinitely.
Statement Analysis:
- " [tex]\( -\frac{4}{5} \)[/tex] can be expressed as a repeating decimal.": False, because [tex]\( -\frac{4}{5} = -0.8 \)[/tex] is a terminating decimal.
- " [tex]\( -\frac{5}{6} \)[/tex] can be expressed as a repeating decimal.": True, because [tex]\( -\frac{5}{6} = -0.8333\ldots \)[/tex].
- "Both fractions can be expressed as repeating decimals.": False, as only [tex]\( -\frac{5}{6} \)[/tex] is a repeating decimal, not [tex]\( -\frac{4}{5} \)[/tex].
- "The digit that repeats is 3.": True for [tex]\( -\frac{5}{6} \)[/tex].
- "The digit that repeats is 8.": False, since 8 does not repeat in either fraction's decimal form.
So, the true statements are:
- [tex]\( -\frac{5}{6} \)[/tex] can be expressed as a repeating decimal.
- The digit that repeats is 3.
### Question 21
We need to determine which of the given mixed numbers has the same decimal value as [tex]\( 110 \frac{147}{168} \)[/tex].
1. Convert the mixed number [tex]\( 110 \frac{147}{168} \)[/tex] into a decimal:
- [tex]\( \frac{147}{168} \approx 0.875 \)[/tex]
- So, [tex]\( 110 \frac{147}{168} = 110 + 0.875 = 110.875 \)[/tex]
Now, let's convert each option to its decimal form and compare:
Option (A): [tex]\( 110 \frac{49}{56} \)[/tex]
- Convert [tex]\( \frac{49}{56} \approx 0.875 \)[/tex]
- Thus, [tex]\( 110 + 0.875 = 110.875 \)[/tex]
Option (B): [tex]\( 110 \frac{170}{180} \)[/tex]
- Convert [tex]\( \frac{170}{180} \approx 0.944444 \)[/tex]
- Thus, [tex]\( 110 + 0.944444 \approx 110.944444 \)[/tex]
Option (C): [tex]\( 110 \frac{56}{72} \)[/tex]
- Convert [tex]\( \frac{56}{72} \approx 0.777778 \)[/tex]
- Thus, [tex]\( 110 + 0.777778 \approx 110.777778 \)[/tex]
Option (D): [tex]\( 110 \frac{247}{268} \)[/tex]
- Convert [tex]\( \frac{247}{268} \approx 0.921642 \)[/tex]
- Thus, [tex]\( 110 + 0.921642 \approx 110.921642 \)[/tex]
Comparing these values:
- [tex]\( 110 \frac{49}{56} = 110.875 \)[/tex]
- [tex]\( 110 \frac{170}{180} \approx 110.944444 \)[/tex]
- [tex]\( 110 \frac{56}{72} \approx 110.777778 \)[/tex]
- [tex]\( 110 \frac{247}{268} \approx 110.921642 \)[/tex]
Thus, the correct answer is [tex]\( 110 \frac{49}{56} \)[/tex].
### Question 22
We need to analyze the statements about the negative fractions [tex]\( -\frac{4}{5} \)[/tex] and [tex]\( -\frac{5}{6} \)[/tex].
1. [tex]\( -\frac{4}{5} \)[/tex] expressed as a decimal:
- [tex]\( \frac{4}{5} = 0.8 \)[/tex]. It is a terminating decimal, not a repeating decimal.
2. [tex]\( -\frac{5}{6} \)[/tex] expressed as a decimal:
- [tex]\( \frac{5}{6} \approx 0.833333\ldots \)[/tex]. This is a repeating decimal where the digit 3 repeats indefinitely.
Statement Analysis:
- " [tex]\( -\frac{4}{5} \)[/tex] can be expressed as a repeating decimal.": False, because [tex]\( -\frac{4}{5} = -0.8 \)[/tex] is a terminating decimal.
- " [tex]\( -\frac{5}{6} \)[/tex] can be expressed as a repeating decimal.": True, because [tex]\( -\frac{5}{6} = -0.8333\ldots \)[/tex].
- "Both fractions can be expressed as repeating decimals.": False, as only [tex]\( -\frac{5}{6} \)[/tex] is a repeating decimal, not [tex]\( -\frac{4}{5} \)[/tex].
- "The digit that repeats is 3.": True for [tex]\( -\frac{5}{6} \)[/tex].
- "The digit that repeats is 8.": False, since 8 does not repeat in either fraction's decimal form.
So, the true statements are:
- [tex]\( -\frac{5}{6} \)[/tex] can be expressed as a repeating decimal.
- The digit that repeats is 3.
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