Answer :
Certainly! Let's go through each part of the question step-by-step.
1. Equivalent Fraction:
- We need to find the missing numerator for [tex]\(\frac{2}{3} = \frac{\square}{12}\)[/tex].
- To find the equivalent fraction, we set up the proportion:
[tex]\[
\frac{2}{3} = \frac{x}{12}
\][/tex]
- To find [tex]\(x\)[/tex], we cross-multiply:
[tex]\[
2 \times 12 = 3 \times x \quad \Rightarrow \quad 24 = 3x \quad \Rightarrow \quad x = \frac{24}{3} = 8
\][/tex]
- So, [tex]\(\frac{2}{3} = \frac{8}{12}\)[/tex].
2. Simplifying Fractions:
- For [tex]\(\frac{5}{10}\)[/tex], we find the greatest common divisor (GCD) of 5 and 10, which is 5. We divide both the numerator and denominator by 5:
[tex]\[
\frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2}
\][/tex]
3. Comparing Fractions:
- We are asked to compare [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex].
- To compare, convert both fractions to have a common denominator or convert each to decimals:
[tex]\[
\frac{5}{8} = 0.625 \quad \text{and} \quad \frac{2}{3} \approx 0.6667
\][/tex]
- Since [tex]\(0.625 < 0.6667\)[/tex], it follows that [tex]\(\frac{5}{8} < \frac{2}{3}\)[/tex].
4. Converting Mixed Numbers to Improper Fractions:
- For [tex]\(6 \frac{2}{3}\)[/tex], follow these steps:
- Multiply the whole number by the denominator: [tex]\(6 \times 3 = 18\)[/tex].
- Add the numerator: [tex]\(18 + 2 = 20\)[/tex].
- The improper fraction is [tex]\(\frac{20}{3}\)[/tex].
5. Dividing and Writing Quotient as a Mixed Number:
- For [tex]\(16 \div 740\)[/tex], the quotient is 0 because [tex]\(16\)[/tex] is less than [tex]\(740\)[/tex].
- The remainder is [tex]\(16\)[/tex].
- As a mixed number, it remains [tex]\(0\)[/tex] with a fraction of [tex]\(\frac{16}{740}\)[/tex] (a fraction of the remainder over the original divisor).
6. Renaming Improper Fractions:
- For [tex]\(\frac{15}{7}\)[/tex]:
- Divide 15 by 7, which gives 2 with a remainder of 1.
- So, the fraction is written as [tex]\(2 \frac{1}{7}\)[/tex].
Each step corresponds to a different task in the original question, and these steps provide a detailed breakdown of each problem's solution!
1. Equivalent Fraction:
- We need to find the missing numerator for [tex]\(\frac{2}{3} = \frac{\square}{12}\)[/tex].
- To find the equivalent fraction, we set up the proportion:
[tex]\[
\frac{2}{3} = \frac{x}{12}
\][/tex]
- To find [tex]\(x\)[/tex], we cross-multiply:
[tex]\[
2 \times 12 = 3 \times x \quad \Rightarrow \quad 24 = 3x \quad \Rightarrow \quad x = \frac{24}{3} = 8
\][/tex]
- So, [tex]\(\frac{2}{3} = \frac{8}{12}\)[/tex].
2. Simplifying Fractions:
- For [tex]\(\frac{5}{10}\)[/tex], we find the greatest common divisor (GCD) of 5 and 10, which is 5. We divide both the numerator and denominator by 5:
[tex]\[
\frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2}
\][/tex]
3. Comparing Fractions:
- We are asked to compare [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex].
- To compare, convert both fractions to have a common denominator or convert each to decimals:
[tex]\[
\frac{5}{8} = 0.625 \quad \text{and} \quad \frac{2}{3} \approx 0.6667
\][/tex]
- Since [tex]\(0.625 < 0.6667\)[/tex], it follows that [tex]\(\frac{5}{8} < \frac{2}{3}\)[/tex].
4. Converting Mixed Numbers to Improper Fractions:
- For [tex]\(6 \frac{2}{3}\)[/tex], follow these steps:
- Multiply the whole number by the denominator: [tex]\(6 \times 3 = 18\)[/tex].
- Add the numerator: [tex]\(18 + 2 = 20\)[/tex].
- The improper fraction is [tex]\(\frac{20}{3}\)[/tex].
5. Dividing and Writing Quotient as a Mixed Number:
- For [tex]\(16 \div 740\)[/tex], the quotient is 0 because [tex]\(16\)[/tex] is less than [tex]\(740\)[/tex].
- The remainder is [tex]\(16\)[/tex].
- As a mixed number, it remains [tex]\(0\)[/tex] with a fraction of [tex]\(\frac{16}{740}\)[/tex] (a fraction of the remainder over the original divisor).
6. Renaming Improper Fractions:
- For [tex]\(\frac{15}{7}\)[/tex]:
- Divide 15 by 7, which gives 2 with a remainder of 1.
- So, the fraction is written as [tex]\(2 \frac{1}{7}\)[/tex].
Each step corresponds to a different task in the original question, and these steps provide a detailed breakdown of each problem's solution!
