Sure, let's tackle each problem step-by-step.
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11. Convert [tex]\( \frac{3}{7} \)[/tex] into decimals:
To convert [tex]\( \frac{3}{7} \)[/tex] into a decimal, you divide 3 by 7.
The decimal form of [tex]\( \frac{3}{7} \)[/tex] is approximately [tex]\( 0.42857142857142855 \)[/tex].
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12. Simplify: [tex]\( 3.125 + 0.125 + 0.50 - 0.225 \)[/tex] and express the result as a rational number in its lowest form:
First, let's perform the arithmetic operations step-by-step.
1. Add [tex]\( 3.125 \)[/tex] and [tex]\( 0.125 \)[/tex]:
[tex]\[
3.125 + 0.125 = 3.250
\][/tex]
2. Then add [tex]\( 0.50 \)[/tex] to the result:
[tex]\[
3.250 + 0.50 = 3.750
\][/tex]
3. Finally, subtract [tex]\( 0.225 \)[/tex] from the result:
[tex]\[
3.750 - 0.225 = 3.525
\][/tex]
So, the simplified sum is [tex]\( 3.525 \)[/tex].
Next, we express [tex]\( 3.525 \)[/tex] as a rational number in its lowest form. The decimal [tex]\( 3.525 \)[/tex] is equivalent to the fraction [tex]\( \frac{3525}{1000} \)[/tex].
To simplify [tex]\( \frac{3525}{1000} \)[/tex] to its lowest terms, we divide both the numerator and the denominator by their greatest common divisor (GCD).
After simplification, [tex]\( 3.525 \)[/tex] can be expressed as the rational number [tex]\( \frac{141}{40} \)[/tex] in its lowest form.
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13. Without actual division, determine which of the following rational numbers have a terminating decimal representation:
To determine if a rational number has a terminating decimal representation, you only need to examine the prime factors of its denominator after simplification. A rational number [tex]\( \frac{p}{q} \)[/tex] has a terminating decimal if and only if, after simplification, the denominator [tex]\( q \)[/tex] has no prime factors other than 2 and/or 5.
Since no specific rational numbers are listed in the question, we can state the general rule:
- Check the simplified form of the rational number.
- Factorize the denominator.
- If the only prime factors are 2 and/or 5, the decimal representation will terminate.
For example:
- [tex]\( \frac{1}{4} \)[/tex] has a denominator of 4, which is [tex]\( 2^2 \)[/tex]. Since the only prime factor is 2, it has a terminating decimal.
- [tex]\( \frac{1}{6} \)[/tex] has a denominator of 6, which is [tex]\( 2 \times 3 \)[/tex]. Since 3 is not 2 or 5, it does not have a terminating decimal.
Applying this rule, you can determine the nature of the decimal representation of any given rational number.
