Use the properties of operations and rules for multiplying and dividing to solve:

(−3/5)×(5/16)×(7/12)÷(−3/16)×(−5/7)

Question

Use the properties of operations and rules for multiplying and dividing to solve:

(−3/5)*(5/16)×(7/12) divided by(−3/16)*(-5/7)

GPT-4o mini · Answer

To solve the expression $(-\frac{3}{5}) \cdot (\frac{5}{16}) \cdot (\frac{7}{12}) \div (-\frac{3}{16}) \cdot (-\frac{5}{7})$, we can break it down step by step: **Step 1: Simplify the expression before dividing.** First, we can rewrite the division as multiplication by the reciprocal: $$ (-\frac{3}{5}) \cdot (\frac{5}{16}) \cdot (\frac{7}{12}) \cdot \frac{1}{(-\frac{3}{16}) \cdot (-\frac{5}{7})} $$ **Step 2: Simplify the denominator.** The denominator can be simplified: $$ (-\frac{3}{16}) \cdot (-\frac{5}{7}) = \frac{3}{16} \cdot \frac{5}{7} = \frac{3 \cdot 5}{16 \cdot 7} = \frac{15}{112} $$ **Step 3: Rewrite the entire expression.** The expression now looks like: $$ (-\frac{3}{5}) \cdot (\frac{5}{16}) \cdot (\frac{7}{12}) \cdot \frac{112}{15} $$ **Step 4: Start multiplying the fractions in the numerator:** $$ =\frac{-3 \cdot 5 \cdot 7 \cdot 112}{5 \cdot 16 \cdot 12 \cdot 15} $$ **Step 5: Cancel out common factors.** - The $5$ in the numerator and denominator cancels: $$ =\frac{-3 \cdot 7 \cdot 112}{16 \cdot 12 \cdot 15} $$ **Step 6: Calculate the numerator:** $$ -3 \cdot 7 = -21 $$ $$ -21 \cdot 112 = -2352 $$ **Step 7: Calculate the denominator:** $$ 16 \cdot 12 = 192 $$ $$ 192 \cdot 15 = 2880 $$ **Step 8: Put it all together:** Now we have: $$ \frac{-2352}{2880} $$ **Step 9: Simplify the fraction:** To simplify $-2352/2880$, find the greatest common divisor (GCD) of $2352$ and $2880$. The GCD is $48$. Dividing both the numerator and denominator by $48$: $$ -2352 \div 48 = -49 $$ $$ 2880 \div 48 = 60 $$ Thus, the simplified fraction is: $$ \frac{-49}{60} $$ **Final Answer:** $-\frac{49}{60}$

help · Answer

Use the properties of operations and rules for multiplying and dividing to solve: (−3/5)×(5/16)×(7/12)÷(−3/16)×(−5/7)

GPT-4o mini · Answer

To solve the expression $$ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \div \left(-\frac{3}{16}\right) \times \left(-\frac{5}{7}\right), $$ we will follow the order of operations and make use of multiplication and division properties. ### Step 1: Rewrite the Division as Multiplication We can rewrite the division as multiplication by the reciprocal: $$ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \times \left(-\frac{16}{3}\right) \times \left(-\frac{7}{5}\right). $$ ### Step 2: Combine the Negatives Since we have two negative signs from $-\frac{3}{16}$ and $-\frac{5}{7}$, the product of negatives gives a positive: Now, the expression simplifies to: $$ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \times \left(\frac{16}{3}\right) \times \left(\frac{7}{5}\right). $$ ### Step 3: Rearrange and Group the Fractions We can now multiply all the fractions together: $$ = \frac{-3 \times 5 \times 7 \times 16 \times 7}{5 \times 16 \times 12 \times 3 \times 5} $$ ### Step 4: Analyze and Simplify by Cancelling Common Factors In the fraction: - The $5$ in the numerator cancels with one of the $5$s in the denominator. - The $16$ in the numerator cancels with $16$ in the denominator. - The $3$ in the numerator cancels with $3$ in the denominator. This simplifies our expression to: $$ = \frac{-7 \times 7}{12} = \frac{-49}{12}. $$ ### Final Answer: Thus, the final answer is $$ -\frac{49}{12}. $$

help · Answer

so witch one is right

GPT-4o mini · Answer

Let's clarify the solution step-by-step to ensure we have an accurate answer for the expression:

Expression to solve:

\[ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \div \left(-\frac{3}{16}\right) \times \left(-\frac{5}{7}\right) \]

Step 1: Rewrite the Division as Multiplication

We can rewrite the division as multiplication by the reciprocal:

\[ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \times \left(-\frac{16}{3}\right) \times \left(-\frac{7}{5}\right) \]

Step 2: Analyze Negatives

The two negative signs in the multiplication will cancel, resulting in a positive product:

\[ \left(-\frac{3}{5}\right) \times \left(\frac{5}{16}\right) \times \left(\frac{7}{12}\right) \times \left(\frac{16}{3}\right) \times \left(\frac{7}{5}\right) \]

Step 3: Combine the Fractions

Multiply the numerators and denominators:

\[ \frac{(-3) \times 5 \times 7 \times 16 \times 7}{5 \times 16 \times 12 \times 3 \times 5} \]

Step 4: Cancel Common Factors

The $5$ in the numerator cancels with one $5$ in the denominator.
The $16$ in the numerator cancels with $16$ in the denominator.
The $3$ in the numerator cancels with $3$ in the denominator.

After canceling, we have:

\[ = \frac{(-7) \times 7}{12} \]

So the expression simplifies to:

\[ = \frac{-49}{12} \]

Conclusion

The correct and final answer to the expression is:

\[ -\frac{49}{12}. \]

This matches the calculations presented in my second response, confirming that the previous answer of $-\frac{49}{60}$ was incorrect. Thank you for your patience while resolving this!