Which Expression Will Best Estimate The Actual Product Of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right$\]?A. $(-1)\left(\frac{1}{4}\right)(-1)(-1$\]B.

**Which Expression Will Best Estimate the Actual Product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$?**

Introduction When dealing with fractions, it's essential to understand how to multiply them together to get the actual product. However, in some cases, we might need to estimate the product instead of finding the exact value. In this article, we will explore which expression will best estimate the actual product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$.

Understanding the Problem To solve this problem, we need to multiply the given fractions together. However, before we do that, let's break down the problem and understand what's being asked. We are given four fractions: $\left(-\frac{4}{5}\right)$, $\left(\frac{3}{5}\right)$, $\left(-\frac{6}{7}\right)$, and $\left(\frac{5}{6}\right)$. We need to find the product of these fractions.

Step 1: Multiply the Numerators To multiply fractions, we need to multiply the numerators together. The numerators are the numbers on top of the fractions. In this case, the numerators are $-4$, $3$, $-6$, and $5$. Let's multiply them together:

$(-4) \times (3) \times (-6) \times (5) = (-4) \times (3) \times (-6) \times (5) = 180$

Step 2: Multiply the Denominators Next, we need to multiply the denominators together. The denominators are the numbers on the bottom of the fractions. In this case, the denominators are $5$, $5$, $7$, and $6$. Let's multiply them together:

$5 \times 5 \times 7 \times 6 = 1050$

Step 3: Write the Product as a Fraction Now that we have the product of the numerators and the product of the denominators, we can write the product as a fraction:

$\frac{180}{1050}$

Simplifying the Fraction To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of $180$ and $1050$ is $30$. Let's divide both the numerator and the denominator by $30$:

$\frac{180}{1050} = \frac{180 \div 30}{1050 \div 30} = \frac{6}{35}$

Estimating the Product Now that we have the simplified fraction, we can estimate the product. To estimate the product, we can round the numerator and the denominator to the nearest whole number. Let's round $6$ to $10$ and $35$ to $40$:

$\frac{6}{35} \approx \frac{10}{40}$

Conclusion In conclusion, the expression that will best estimate the actual product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$ is $\frac{10}{40}$.

Q&A

Q: What is the product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$? A: The product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$ is $\frac{180}{1050}$.

Q: How do I simplify the fraction $\frac{180}{1050}$? A: To simplify the fraction $\frac{180}{1050}$, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of $180$ and $1050$ is $30$. Let's divide both the numerator and the denominator by $30$:

$\frac{180}{1050} = \frac{180 \div 30}{1050 \div 30} = \frac{6}{35}$

Q: How do I estimate the product of $\frac{6}{35}$? A: To estimate the product of $\frac{6}{35}$, you can round the numerator and the denominator to the nearest whole number. Let's round $6$ to $10$ and $35$ to $40$:

$\frac{6}{35} \approx \frac{10}{40}$

Q: What is the expression that will best estimate the actual product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$? A: The expression that will best estimate the actual product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$ is $\frac{10}{40}$.

Frequently Asked Questions

Q: What is the greatest common divisor (GCD) of $180$ and $1050$? A: The greatest common divisor (GCD) of $180$ and $1050$ is $30$.

Q: How do I multiply fractions? A: To multiply fractions, you need to multiply the numerators together and multiply the denominators together.

Q: How do I estimate the product of a fraction? A: To estimate the product of a fraction, you can round the numerator and the denominator to the nearest whole number.

Q: What is the product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$? A: The product of $\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)\left(-\frac{6}{7}\right)\left(\frac{5}{6}\right)$ is $\frac{180}{1050}$.

Q: How do I simplify the fraction $\frac{180}{1050}$? A: To simplify the fraction $\frac{180}{1050}$, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of $180$ and $1050$ is $30$. Let's divide both the numerator and the denominator by $30$:

$\frac{180}{1050} = \frac{180 \div 30}{1050 \div 30} = \frac{6}{35}$