2 1/4 • 16/27 • 4 1/3

To calculate the product of \( 2 \frac{1}{4} \), \( \frac{16}{27} \), and \( 4 \frac{1}{3} \), we first convert the mixed numbers into improper fractions.

1. Convert \( 2 \frac{1}{4} \): \[ 2 \frac{1}{4} = 2 \times 4 + 1 = 8 + 1 = 9 \Rightarrow \frac{9}{4} \]

2. Convert \( 4 \frac{1}{3} \): \[ 4 \frac{1}{3} = 4 \times 3 + 1 = 12 + 1 = 13 \Rightarrow \frac{13}{3} \]

Now we have: \[ \frac{9}{4} \times \frac{16}{27} \times \frac{13}{3} \]

Next, we multiply the fractions together: \[ = \frac{9 \times 16 \times 13}{4 \times 27 \times 3} \]

Calculating the numerator: \[ 9 \times 16 = 144 \] \[ 144 \times 13 = 1872 \] So the numerator is \( 1872 \).

Calculating the denominator: \[ 4 \times 27 = 108 \] \[ 108 \times 3 = 324 \] So the denominator is \( 324 \).

Now we have: \[ \frac{1872}{324} \]

To simplify \( \frac{1872}{324} \), we find the greatest common divisor (gcd) of 1872 and 324.

Calculating the gcd:

• The prime factorization of 1872 is \( 2^4 \times 3 \times 13 \).
• The prime factorization of 324 is \( 2^2 \times 3^4 \).

The lowest powers of the common factors:

• For \( 2 \), the minimum is \( 2^2 \).
• For \( 3 \), the minimum is \( 3^1 \).

Thus, \( \text{gcd} = 2^2 \times 3^1 = 4 \times 3 = 12 \).

Now we divide both the numerator and denominator by 12: \[ \frac{1872 \div 12}{324 \div 12} = \frac{156}{27} \]

Now we simplify \( \frac{156}{27} \):

• The gcd of 156 and 27 is 3.

Dividing by the gcd: \[ \frac{156 \div 3}{27 \div 3} = \frac{52}{9} \]

Thus, the simplified product of \( 2 \frac{1}{4} \times \frac{16}{27} \times 4 \frac{1}{3} \) is: \[ \frac{52}{9} \]

If you want to express it as a mixed number: \[ 52 \div 9 = 5 \quad \text{remainder } 7 \] So, \( \frac{52}{9} = 5 \frac{7}{9} \).

Therefore, the final answer is: \[ \frac{52}{9} \quad \text{or} \quad 5 \frac{7}{9} \]