Answer :
Let's go through each part of the question step-by-step and solve them in detail.
### 1) Multiplying [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex]
First, we convert the mixed number [tex]\(3 \frac{4}{5}\)[/tex] to an improper fraction.
[tex]\[ 3 \frac{4}{5} = 3 + \frac{4}{5} = \frac{3 \times 5 + 4}{5} = \frac{15 + 4}{5} = \frac{19}{5} \][/tex]
Next, we multiply the two fractions:
[tex]\[ \frac{5}{4} \times \frac{19}{5} \][/tex]
When multiplying fractions, we multiply the numerators and the denominators:
[tex]\[ \frac{5 \times 19}{4 \times 5} = \frac{95}{20} \][/tex]
We can simplify [tex]\(\frac{95}{20}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \frac{95 \div 5}{20 \div 5} = \frac{19}{4} \][/tex]
Finally, we convert [tex]\(\frac{19}{4}\)[/tex] into a decimal:
[tex]\[ \frac{19}{4} = 4.75 \][/tex]
So, the product of [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex] is [tex]\(4.75\)[/tex].
### 2) Finding the value of [tex]\(N\)[/tex] in the statement [tex]\(\frac{4}{7} \times 6 \frac{3}{5} = 10\)[/tex]
First, we convert the mixed number [tex]\(6 \frac{3}{5}\)[/tex] to an improper fraction:
[tex]\[ 6 \frac{3}{5} = 6 + \frac{3}{5} = \frac{6 \times 5 + 3}{5} = \frac{30 + 3}{5} = \frac{33}{5} \][/tex]
Next, we use the given equation:
[tex]\[ \frac{4}{7} \times \frac{33}{5} = 10 \][/tex]
Let [tex]\(N\)[/tex] be the value we need to find. To isolate [tex]\(N\)[/tex], we rewrite the equation with [tex]\(N\)[/tex]:
[tex]\[ N = 10 \div \left(\frac{4}{7} \times \frac{33}{5}\right) \][/tex]
First, calculate the product [tex]\(\frac{4}{7} \times \frac{33}{5}\)[/tex]:
[tex]\[ \frac{4 \times 33}{7 \times 5} = \frac{132}{35} \][/tex]
Now, divide 10 by [tex]\(\frac{132}{35}\)[/tex]:
[tex]\[ N = 10 \div \left(\frac{132}{35}\right) = 10 \times \frac{35}{132} \][/tex]
[tex]\[ N = \frac{10 \times 35}{132} = \frac{350}{132} = 2.651515151515152 \][/tex]
So, the value of [tex]\(N\)[/tex] is approximately [tex]\(2.6515\)[/tex].
### 3) Multiplying [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex]
We simply multiply the numerators and the denominators:
[tex]\[ \frac{2}{7} \times \frac{5}{8} = \frac{2 \times 5}{7 \times 8} = \frac{10}{56} \][/tex]
We simplify [tex]\(\frac{10}{56}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[ \frac{10 \div 2}{56 \div 2} = \frac{5}{28} \][/tex]
Converting [tex]\(\frac{5}{28}\)[/tex] to a decimal:
[tex]\[ \frac{5}{28} = 0.17857142857142855 \][/tex]
So, the product of [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex] is [tex]\(0.17857142857142855\)[/tex].
### Summary of Results
1. The product of [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex] is [tex]\(4.75\)[/tex].
2. The value of [tex]\(N\)[/tex] in the statement [tex]\(\frac{4}{7} \times 6 \frac{3}{5} = 10\)[/tex] is approximately [tex]\(2.6515\)[/tex].
3. The product of [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex] is [tex]\(0.17857142857142855\)[/tex].
### 1) Multiplying [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex]
First, we convert the mixed number [tex]\(3 \frac{4}{5}\)[/tex] to an improper fraction.
[tex]\[ 3 \frac{4}{5} = 3 + \frac{4}{5} = \frac{3 \times 5 + 4}{5} = \frac{15 + 4}{5} = \frac{19}{5} \][/tex]
Next, we multiply the two fractions:
[tex]\[ \frac{5}{4} \times \frac{19}{5} \][/tex]
When multiplying fractions, we multiply the numerators and the denominators:
[tex]\[ \frac{5 \times 19}{4 \times 5} = \frac{95}{20} \][/tex]
We can simplify [tex]\(\frac{95}{20}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \frac{95 \div 5}{20 \div 5} = \frac{19}{4} \][/tex]
Finally, we convert [tex]\(\frac{19}{4}\)[/tex] into a decimal:
[tex]\[ \frac{19}{4} = 4.75 \][/tex]
So, the product of [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex] is [tex]\(4.75\)[/tex].
### 2) Finding the value of [tex]\(N\)[/tex] in the statement [tex]\(\frac{4}{7} \times 6 \frac{3}{5} = 10\)[/tex]
First, we convert the mixed number [tex]\(6 \frac{3}{5}\)[/tex] to an improper fraction:
[tex]\[ 6 \frac{3}{5} = 6 + \frac{3}{5} = \frac{6 \times 5 + 3}{5} = \frac{30 + 3}{5} = \frac{33}{5} \][/tex]
Next, we use the given equation:
[tex]\[ \frac{4}{7} \times \frac{33}{5} = 10 \][/tex]
Let [tex]\(N\)[/tex] be the value we need to find. To isolate [tex]\(N\)[/tex], we rewrite the equation with [tex]\(N\)[/tex]:
[tex]\[ N = 10 \div \left(\frac{4}{7} \times \frac{33}{5}\right) \][/tex]
First, calculate the product [tex]\(\frac{4}{7} \times \frac{33}{5}\)[/tex]:
[tex]\[ \frac{4 \times 33}{7 \times 5} = \frac{132}{35} \][/tex]
Now, divide 10 by [tex]\(\frac{132}{35}\)[/tex]:
[tex]\[ N = 10 \div \left(\frac{132}{35}\right) = 10 \times \frac{35}{132} \][/tex]
[tex]\[ N = \frac{10 \times 35}{132} = \frac{350}{132} = 2.651515151515152 \][/tex]
So, the value of [tex]\(N\)[/tex] is approximately [tex]\(2.6515\)[/tex].
### 3) Multiplying [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex]
We simply multiply the numerators and the denominators:
[tex]\[ \frac{2}{7} \times \frac{5}{8} = \frac{2 \times 5}{7 \times 8} = \frac{10}{56} \][/tex]
We simplify [tex]\(\frac{10}{56}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[ \frac{10 \div 2}{56 \div 2} = \frac{5}{28} \][/tex]
Converting [tex]\(\frac{5}{28}\)[/tex] to a decimal:
[tex]\[ \frac{5}{28} = 0.17857142857142855 \][/tex]
So, the product of [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex] is [tex]\(0.17857142857142855\)[/tex].
### Summary of Results
1. The product of [tex]\(\frac{5}{4}\)[/tex] and [tex]\(3 \frac{4}{5}\)[/tex] is [tex]\(4.75\)[/tex].
2. The value of [tex]\(N\)[/tex] in the statement [tex]\(\frac{4}{7} \times 6 \frac{3}{5} = 10\)[/tex] is approximately [tex]\(2.6515\)[/tex].
3. The product of [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex] is [tex]\(0.17857142857142855\)[/tex].
