Alright, let's go through these problems step by step, translating them into mathematical sentences and solving them.
### Problem 1
1. Given: [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{16}{21}\)[/tex]
2. Asked: The product of these two fractions.
3. Operation: Multiplication
4. Mathematical Sentence: [tex]\(\frac{3}{4} \times \frac{16}{21}\)[/tex]
5. Solution:
[tex]\[
\frac{3}{4} \times \frac{16}{21} = \frac{3 \times 16}{4 \times 21} = \frac{48}{84} = 0.5714285714285714
\][/tex]
### Problem 2
1. Given: [tex]\(5 \frac{7}{9}\)[/tex] and [tex]\(\frac{27}{56}\)[/tex]
2. Asked: The product of these two numbers.
3. Operation: Multiplication
4. Mathematical Sentence: [tex]\(\left( 5 + \frac{7}{9} \right) \times \frac{27}{56}\)[/tex]
Convert [tex]\(5 \frac{7}{9}\)[/tex] into an improper fraction:
[tex]\(
5 \frac{7}{9} = \frac{5 \times 9 + 7}{9} = \frac{45 + 7}{9} = \frac{52}{9}
\)[/tex]
Now the mathematical sentence is:
[tex]\[
\frac{52}{9} \times \frac{27}{56}
\][/tex]
5. Solution:
[tex]\[
\frac{52}{9} \times \frac{27}{56} = \frac{52 \times 27}{9 \times 56} = \frac{1404}{504} = 2.7857142857142856
\][/tex]
### Problem 3
1. Given: [tex]\(4 \frac{2}{5}\)[/tex] and [tex]\(7 \frac{1}{3}\)[/tex]
2. Asked: The product of these two numbers.
3. Operation: Multiplication
4. Mathematical Sentence: [tex]\(\left( 4 + \frac{2}{5} \right) \times \left( 7 + \frac{1}{3} \right)\)[/tex]
Convert into improper fractions:
[tex]\[
4 \frac{2}{5} = \frac{4 \times 5 + 2}{5} = \frac{20 + 2}{5} = \frac{22}{5}
\][/tex]
[tex]\[
7 \frac{1}{3} = \frac{7 \times 3 + 1}{3} = \frac{21 + 1}{3} = \frac{22}{3}
\][/tex]
Now the mathematical sentence is:
[tex]\[
\frac{22}{5} \times \frac{22}{3}
\][/tex]
5. Solution:
[tex]\[
\frac{22}{5} \times \frac{22}{3} = \frac{22 \times 22}{5 \times 3} = \frac{484}{15} = 32.266666666666666
\][/tex]
### Problem 4
1. Given: [tex]\(\frac{8}{15}\)[/tex] and [tex]\(2 \frac{4}{7}\)[/tex]
2. Asked: Twice the product of these two numbers.
3. Operation: Multiplication and then double the result.
4. Mathematical Sentence: [tex]\(2 \times \left( \frac{8}{15} \times \left(2 + \frac{4}{7}\right) \right)\)[/tex]
Convert [tex]\(2 \frac{4}{7}\)[/tex] into an improper fraction:
[tex]\[
2 \frac{4}{7} = \frac{2 \times 7 + 4}{7} = \frac{14 + 4}{7} = \frac{18}{7}
\][/tex]
Now the mathematical sentence is:
[tex]\[
2 \times \left( \frac{8}{15} \times \frac{18}{7} \right)
\][/tex]
5. Solution:
[tex]\[
\frac{8}{15} \times \frac{18}{7} = \frac{8 \times 18}{15 \times 7} = \frac{144}{105} = \frac{144}{105} = \frac{24}{35}
\][/tex]
Then, twice the product is:
[tex]\[
2 \times \frac{24}{35} = \frac{48}{35} = 2.742857142857143
\][/tex]
These steps bring us to the same numerical answers for each problem.
