Certainly! Let's go through each part of the question and work them out step-by-step.
### Part (a)
[tex]\[ \frac{3}{5} \times \frac{1}{6} \][/tex]
To multiply two fractions, you multiply the numerators together and the denominators together:
Numerator: [tex]\(3 \times 1 = 3\)[/tex]
Denominator: [tex]\(5 \times 6 = 30\)[/tex]
So, the product is:
[tex]\[ \frac{3}{30} \][/tex]
Now, simplify this fraction by finding the greatest common divisor (GCD) of 3 and 30, which is 3:
[tex]\[ \frac{3 \div 3}{30 \div 3} = \frac{1}{10} \][/tex]
Thus, the answer in its lowest terms is:
[tex]\[ \frac{1}{10} \][/tex]
### Part (b)
[tex]\[ 3 \frac{3}{4} \times \frac{2}{5} \][/tex]
First, convert the mixed number [tex]\(3 \frac{3}{4}\)[/tex] to an improper fraction:
[tex]\[ 3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \][/tex]
Now multiply it by [tex]\(\frac{2}{5}\)[/tex]:
Numerator: [tex]\(15 \times 2 = 30\)[/tex]
Denominator: [tex]\(4 \times 5 = 20\)[/tex]
So, the product is:
[tex]\[ \frac{30}{20} \][/tex]
Simplify the fraction by finding the GCD of 30 and 20, which is 10:
[tex]\[ \frac{30 \div 10}{20 \div 10} = \frac{3}{2} \][/tex]
Thus, the answer in its lowest terms is:
[tex]\[ \frac{3}{2} \][/tex]
### Part (c)
[tex]\[ \frac{2}{5} \div 3 \][/tex]
Dividing by a whole number is equivalent to multiplying by its reciprocal. The reciprocal of 3 is [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{2}{5} \times \frac{1}{3} \][/tex]
Numerator: [tex]\(2 \times 1 = 2\)[/tex]
Denominator: [tex]\(5 \times 3 = 15\)[/tex]
So, the product is:
[tex]\[ \frac{2}{15} \][/tex]
Thus, the answer in its lowest terms is:
[tex]\[ \frac{2}{15} \][/tex]
### Part (d)
[tex]\[ 4 \frac{4}{9} \div 6 \][/tex]
First, convert the mixed number [tex]\(4 \frac{4}{9}\)[/tex] to an improper fraction:
[tex]\[ 4 \frac{4}{9} = \frac{4 \times 9 + 4}{9} = \frac{36 + 4}{9} = \frac{40}{9} \][/tex]
Now divide it by 6. Dividing by 6 is the same as multiplying by the reciprocal of 6, which is [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \frac{40}{9} \times \frac{1}{6} \][/tex]
Numerator: [tex]\(40 \times 1 = 40\)[/tex]
Denominator: [tex]\(9 \times 6 = 54\)[/tex]
So, the product is:
[tex]\[ \frac{40}{54} \][/tex]
Simplify the fraction by finding the GCD of 40 and 54, which is 2:
[tex]\[ \frac{40 \div 2}{54 \div 2} = \frac{20}{27} \][/tex]
Thus, the answer in its lowest terms is:
[tex]\[ \frac{20}{27} \][/tex]
Hence, the final answers are:
[tex]\[
\text{a)} \frac{1}{10} \quad \text{b)} \frac{3}{2} \quad \text{c)} \frac{2}{15} \quad \text{d)} \frac{20}{27}
\][/tex]
