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Sagot :
Let's go through each decimal number step-by-step to express them as fractions in the form of [tex]\(\frac{p}{q}\)[/tex] and find the prime factors of the denominators [tex]\(q\)[/tex].
### (i) 43.123
To convert 43.123 to a fraction, we can express it as [tex]\(\frac{43123}{1000}\)[/tex]:
[tex]\[ 43.123 = \frac{43123}{1000} \][/tex]
Next, we simplify this fraction by finding the Greatest Common Divisor (GCD) of 43123 and 1000.
- The prime factors of 1000 are [tex]\(2^3 \times 5^3\)[/tex].
- The prime factors of 43123 need to be determined for simplification.
After checking, 43123 is a prime number.
Therefore, the simplest form of the fraction is:
[tex]\[ 43.123 = \frac{43123}{1000} \][/tex]
So, we have:
[tex]\[ \frac{p}{q} = \frac{43123}{1000} \][/tex]
The prime factors of 1000 are [tex]\(2, 5\)[/tex].
### (ii) [tex]\(0.120112001120001 \ldots\)[/tex]
This number seems to follow a complex repeating pattern. To simplify the understanding, let's look at a partial pattern.
If it is not straightforward to express the repeating pattern, the decimal might be a non-standard repeating decimal that can be challenging to simplify directly without further tools.
### (iii) [tex]\(43 . \overline{12}\)[/tex]
This decimal has a repeating part:
[tex]\[ 43.\overline{12} = 43 + 0.\overline{12} \][/tex]
First, convert [tex]\(0.\overline{12}\)[/tex] to a fraction:
Let [tex]\( x = 0.\overline{12} \)[/tex].
Multiply by 100 to shift the decimal point:
[tex]\[ 100x = 12.\overline{12} \][/tex]
Now, subtract the original [tex]\(x\)[/tex] from this:
[tex]\[ 100x - x = 12.\overline{12} - 0.\overline{12} \][/tex]
[tex]\[ 99x = 12 \][/tex]
[tex]\[ x = \frac{12}{99} \][/tex]
[tex]\[ x = \frac{4}{33} \][/tex]
Thus,
[tex]\[ 43.\overline{12} = 43 + \frac{4}{33} \][/tex]
Convert the whole number part:
[tex]\[ = \frac{43 \cdot 33 + 4}{33} = \frac{1419 + 4}{33} = \frac{1423}{33} \][/tex]
So,
[tex]\[ \frac{p}{q} = \frac{1423}{33} \][/tex]
The prime factors of 33 are [tex]\(3, 11\)[/tex].
### (iv) [tex]\(0 . \overline{63}\)[/tex]
This decimal also has a repeating part:
[tex]\[ 0.\overline{63} \][/tex]
Let [tex]\( y = 0.\overline{63} \)[/tex].
Multiply by 100 to shift the decimal point:
[tex]\[ 100y = 63.\overline{63} \][/tex]
Now, subtract the original [tex]\(y\)[/tex] from this:
[tex]\[ 100y - y = 63.\overline{63} - 0.\overline{63} \][/tex]
[tex]\[ 99y = 63 \][/tex]
[tex]\[ y = \frac{63}{99} \][/tex]
[tex]\[ y = \frac{7}{11} \][/tex]
So,
[tex]\[ \frac{p}{q} = \frac{7}{11} \][/tex]
The prime factors of 11 are [tex]\(11\)[/tex] (as 11 is a prime number).
### Summary of Observations:
- (i) 43.123: [tex]\(\frac{43123}{1000}\)[/tex] with prime factors of the denominator being [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
- (ii) [tex]\(0.120112001120001 \ldots\)[/tex]: (Pattern needs further clarification; prime factors calculation omitted due to complexity.)
- (iii) [tex]\(43.\overline{12}\)[/tex]: [tex]\(\frac{1423}{33}\)[/tex] with prime factors [tex]\(3\)[/tex] and [tex]\(11\)[/tex].
- (iv) [tex]\(0.\overline{63}\)[/tex]: [tex]\(\frac{7}{11}\)[/tex] with the prime factor [tex]\(11\)[/tex].
We observe that repeating decimals convert to fractions whose denominators often have simple prime factorizations. Non-repeating decimals usually have denominators that are powers of 10. For complex repeating patterns, the situation can be less straightforward and may need specialized methods for simplification.
### (i) 43.123
To convert 43.123 to a fraction, we can express it as [tex]\(\frac{43123}{1000}\)[/tex]:
[tex]\[ 43.123 = \frac{43123}{1000} \][/tex]
Next, we simplify this fraction by finding the Greatest Common Divisor (GCD) of 43123 and 1000.
- The prime factors of 1000 are [tex]\(2^3 \times 5^3\)[/tex].
- The prime factors of 43123 need to be determined for simplification.
After checking, 43123 is a prime number.
Therefore, the simplest form of the fraction is:
[tex]\[ 43.123 = \frac{43123}{1000} \][/tex]
So, we have:
[tex]\[ \frac{p}{q} = \frac{43123}{1000} \][/tex]
The prime factors of 1000 are [tex]\(2, 5\)[/tex].
### (ii) [tex]\(0.120112001120001 \ldots\)[/tex]
This number seems to follow a complex repeating pattern. To simplify the understanding, let's look at a partial pattern.
If it is not straightforward to express the repeating pattern, the decimal might be a non-standard repeating decimal that can be challenging to simplify directly without further tools.
### (iii) [tex]\(43 . \overline{12}\)[/tex]
This decimal has a repeating part:
[tex]\[ 43.\overline{12} = 43 + 0.\overline{12} \][/tex]
First, convert [tex]\(0.\overline{12}\)[/tex] to a fraction:
Let [tex]\( x = 0.\overline{12} \)[/tex].
Multiply by 100 to shift the decimal point:
[tex]\[ 100x = 12.\overline{12} \][/tex]
Now, subtract the original [tex]\(x\)[/tex] from this:
[tex]\[ 100x - x = 12.\overline{12} - 0.\overline{12} \][/tex]
[tex]\[ 99x = 12 \][/tex]
[tex]\[ x = \frac{12}{99} \][/tex]
[tex]\[ x = \frac{4}{33} \][/tex]
Thus,
[tex]\[ 43.\overline{12} = 43 + \frac{4}{33} \][/tex]
Convert the whole number part:
[tex]\[ = \frac{43 \cdot 33 + 4}{33} = \frac{1419 + 4}{33} = \frac{1423}{33} \][/tex]
So,
[tex]\[ \frac{p}{q} = \frac{1423}{33} \][/tex]
The prime factors of 33 are [tex]\(3, 11\)[/tex].
### (iv) [tex]\(0 . \overline{63}\)[/tex]
This decimal also has a repeating part:
[tex]\[ 0.\overline{63} \][/tex]
Let [tex]\( y = 0.\overline{63} \)[/tex].
Multiply by 100 to shift the decimal point:
[tex]\[ 100y = 63.\overline{63} \][/tex]
Now, subtract the original [tex]\(y\)[/tex] from this:
[tex]\[ 100y - y = 63.\overline{63} - 0.\overline{63} \][/tex]
[tex]\[ 99y = 63 \][/tex]
[tex]\[ y = \frac{63}{99} \][/tex]
[tex]\[ y = \frac{7}{11} \][/tex]
So,
[tex]\[ \frac{p}{q} = \frac{7}{11} \][/tex]
The prime factors of 11 are [tex]\(11\)[/tex] (as 11 is a prime number).
### Summary of Observations:
- (i) 43.123: [tex]\(\frac{43123}{1000}\)[/tex] with prime factors of the denominator being [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
- (ii) [tex]\(0.120112001120001 \ldots\)[/tex]: (Pattern needs further clarification; prime factors calculation omitted due to complexity.)
- (iii) [tex]\(43.\overline{12}\)[/tex]: [tex]\(\frac{1423}{33}\)[/tex] with prime factors [tex]\(3\)[/tex] and [tex]\(11\)[/tex].
- (iv) [tex]\(0.\overline{63}\)[/tex]: [tex]\(\frac{7}{11}\)[/tex] with the prime factor [tex]\(11\)[/tex].
We observe that repeating decimals convert to fractions whose denominators often have simple prime factorizations. Non-repeating decimals usually have denominators that are powers of 10. For complex repeating patterns, the situation can be less straightforward and may need specialized methods for simplification.
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