Multiply 3 3/4*8/9*4 9/12

Solution:

Step:1 Transform $3 \frac{3}{4}$ into an improper fraction.

Step:1.1 A mixed number combines its integer and fractional parts. $(3 + \frac{3}{4}) \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:1.2 Sum $3$ and $\frac{3}{4}$.

Step:1.2.1 Express $3$ as a fraction with the same denominator by multiplying by $\frac{4}{4}$. $(3 \cdot \frac{4}{4} + \frac{3}{4}) \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:1.2.2 Merge $3$ and $\frac{4}{4}$. $(\frac{3 \cdot 4}{4} + \frac{3}{4}) \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:1.2.3 Combine the numerators over the same denominator. $\frac{3 \cdot 4 + 3}{4} \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:1.2.4 Simplify the numerator.

Step:1.2.4.1 Calculate $3$ times $4$. $\frac{12 + 3}{4} \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:1.2.4.2 Add $12$ to $3$. $\frac{15}{4} \cdot \frac{8}{9} \cdot 4 \frac{9}{12}$

Step:2 Transform $4 \frac{9}{12}$ into an improper fraction.

Step:2.1 A mixed number combines its integer and fractional parts. $\frac{15}{4} \cdot \frac{8}{9} \cdot (4 + \frac{9}{12})$

Step:2.2 Sum $4$ and $\frac{9}{12}$.

Step:2.2.1 Express $4$ as a fraction with the same denominator by multiplying by $\frac{12}{12}$. $\frac{15}{4} \cdot \frac{8}{9} \cdot (4 \cdot \frac{12}{12} + \frac{9}{12})$

Step:2.2.2 Merge $4$ and $\frac{12}{12}$. $\frac{15}{4} \cdot \frac{8}{9} \cdot (\frac{4 \cdot 12}{12} + \frac{9}{12})$

Step:2.2.3 Combine the numerators over the same denominator. $\frac{15}{4} \cdot \frac{8}{9} \cdot \frac{4 \cdot 12 + 9}{12}$

Step:2.2.4 Simplify the numerator.

Step:2.2.4.1 Calculate $4$ times $12$. $\frac{15}{4} \cdot \frac{8}{9} \cdot \frac{48 + 9}{12}$

Step:2.2.4.2 Add $48$ to $9$. $\frac{15}{4} \cdot \frac{8}{9} \cdot \frac{57}{12}$

Step:3 Eliminate the common factor of $3$.

Step:3.1 Extract $3$ from $15$. $\frac{3(5)}{4} \cdot \frac{8}{9} \cdot \frac{57}{12}$

Step:3.2 Extract $3$ from $9$. $\frac{3 \cdot 5}{4} \cdot \frac{8}{3 \cdot 3} \cdot \frac{57}{12}$

Step:3.3 Remove the common factor. $\frac{\cancel{3} \cdot 5}{4} \cdot \frac{8}{\cancel{3} \cdot 3} \cdot \frac{57}{12}$

Step:3.4 Reformulate the expression. $\frac{5}{4} \cdot \frac{8}{3} \cdot \frac{57}{12}$

Step:4 Eliminate the common factor of $4$.

Step:4.1 Extract $4$ from $8$. $\frac{5}{4} \cdot \frac{4(2)}{3} \cdot \frac{57}{12}$

Step:4.2 Remove the common factor. $\frac{5}{\cancel{4}} \cdot \frac{\cancel{4} \cdot 2}{3} \cdot \frac{57}{12}$

Step:4.3 Reformulate the expression. $5 \cdot \frac{2}{3} \cdot \frac{57}{12}$

Step:5 Merge $5$ and $\frac{2}{3}$. $\frac{5 \cdot 2}{3} \cdot \frac{57}{12}$

Step:6 Calculate $5$ times $2$. $\frac{10}{3} \cdot \frac{57}{12}$

Step:7 Eliminate the common factor of $2$.

Step:7.1 Extract $2$ from $10$. $\frac{2(5)}{3} \cdot \frac{57}{12}$

Step:7.2 Extract $2$ from $12$. $\frac{2 \cdot 5}{3} \cdot \frac{57}{2 \cdot 6}$

Step:7.3 Remove the common factor. $\frac{\cancel{2} \cdot 5}{3} \cdot \frac{57}{\cancel{2} \cdot 6}$

Step:7.4 Reformulate the expression. $\frac{5}{3} \cdot \frac{57}{6}$

Step:8 Eliminate the common factor of $3$.

Step:8.1 Extract $3$ from $57$. $\frac{5}{3} \cdot \frac{3(19)}{6}$

Step:8.2 Remove the common factor. $\frac{5}{\cancel{3}} \cdot \frac{\cancel{3} \cdot 19}{6}$

Step:8.3 Reformulate the expression. $5 \cdot \frac{19}{6}$

Step:9 Merge $5$ and $\frac{19}{6}$. $\frac{5 \cdot 19}{6}$

Step:10 Calculate $5$ times $19$. $\frac{95}{6}$

Step:11 The result can be represented in various forms.

Exact Form: $\frac{95}{6}$ Decimal Form: $15.8333...$ Mixed Number Form: $15 \frac{5}{6}$

Knowledge Notes:

1. Mixed Numbers to Improper Fractions: A mixed number can be converted to an improper fraction by multiplying the whole number by the denominator of the fractional part, adding the numerator, and placing the result over the original denominator.

2. Multiplying Fractions: To multiply fractions, multiply the numerators together and the denominators together. Simplify the resulting fraction if possible.

3. Simplifying Fractions: Fractions can be simplified by canceling common factors in the numerator and denominator.

4. Common Factors: When both the numerator and denominator share a common factor, it can be divided out to simplify the fraction.

5. Final Representation: The result of a multiplication of fractions can be expressed as an improper fraction, a decimal, or a mixed number. Converting between these forms requires division and understanding of place value for decimals, and for mixed numbers, finding the whole number part and the remainder as a fraction.

6. LaTeX Formatting: Mathematical expressions can be formatted using LaTeX, a typesetting system that uses commands to represent various mathematical symbols and structures, such as fractions and operations.