Let's find the product of the given fractions step-by-step:
We need to multiply the fractions:
[tex]\[
\frac{15}{4} \times \frac{35}{24} \times \frac{16}{35}
\][/tex]
First, multiply the numerators together:
[tex]\[
15 \times 35 \times 16
\][/tex]
Next, multiply the denominators together:
[tex]\[
4 \times 24 \times 35
\][/tex]
Now, let's write these products down in fraction form:
[tex]\[
\frac{15 \times 35 \times 16}{4 \times 24 \times 35}
\][/tex]
Notice that both the numerator and the denominator contain the factor 35, which can be cancelled out:
[tex]\[
\frac{15 \times \cancel{35} \times 16}{4 \times 24 \times \cancel{35}} = \frac{15 \times 16}{4 \times 24}
\][/tex]
Next, let's simplify the fraction [tex]\(\frac{15 \times 16}{4 \times 24}\)[/tex]:
[tex]\[
\frac{240}{96}
\][/tex]
To simplify [tex]\(\frac{240}{96}\)[/tex], we find the greatest common divisor (GCD) of 240 and 96, which is 48. Then, we divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{240 \div 48}{96 \div 48} = \frac{5}{2}
\][/tex]
Thus, the product of the fractions [tex]\(\frac{15}{4} \times \frac{35}{24} \times \frac{16}{35}\)[/tex] is:
[tex]\[
\boxed{\frac{5}{2}}
\][/tex]
