Which Is The Best Estimate Of \left(-\frac{3}{5}\right)\left(17 \frac{5}{6}\right ]?A. { -18$}$B. { -9$}$C. 9D. 18

Mar 13, 2025 by ADMIN 115 views

**Estimating the Product of Fractions and Mixed Numbers**

Understanding the Problem

When dealing with fractions and mixed numbers, it's essential to understand the concept of multiplying them. In this problem, we are asked to find the best estimate of the product of $\left(-\frac{3}{5}\right)$ and $17 \frac{5}{6}$ . To solve this, we need to multiply the two numbers and simplify the result.

Multiplying Fractions and Mixed Numbers

To multiply fractions and mixed numbers, we can follow these steps:

Multiply the numerators (the numbers on top) together.
Multiply the denominators (the numbers on the bottom) together.
Simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Converting the Mixed Number to an Improper Fraction

Before we can multiply the two numbers, we need to convert the mixed number $17 \frac{5}{6}$ to an improper fraction. To do this, we multiply the whole number part (17) by the denominator (6) and add the result to the numerator (5).

$17 \frac{5}{6} = \frac{(17 \times 6) + 5}{6} = \frac{102 + 5}{6} = \frac{107}{6}$

Multiplying the Fractions

Now that we have both numbers in fraction form, we can multiply them together.

$\left(-\frac{3}{5}\right)\left(\frac{107}{6}\right) = \frac{(-3) \times 107}{5 \times 6} = \frac{-321}{30}$

Simplifying the Result

To simplify the result, we need to find the greatest common divisor (GCD) of the numerator (-321) and the denominator (30). The GCD of -321 and 30 is 3. We can divide both the numerator and the denominator by 3 to simplify the result.

$\frac{-321}{30} = \frac{(-321) \div 3}{30 \div 3} = \frac{-107}{10}$

Estimating the Product

Now that we have simplified the result, we can estimate the product. Since the numerator is negative and the denominator is positive, the product will be negative. The absolute value of the product is 10.7, which is close to 10.

Conclusion

Based on our calculations, the best estimate of the product of $\left(-\frac{3}{5}\right)$ and $17 \frac{5}{6}$ is $\boxed{-10.7}$ . However, since the answer choices are not in decimal form, we can round the result to the nearest whole number. The closest answer choice is -9.

Answer

The best estimate of the product of $\left(-\frac{3}{5}\right)$ and $17 \frac{5}{6}$ is $\boxed{-9}$ .

Key Takeaways

When multiplying fractions and mixed numbers, we need to convert the mixed number to an improper fraction first.
We can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Estimating the product involves finding the absolute value of the result and rounding it to the nearest whole number.

Practice Problems

Multiply $\left(\frac{2}{3}\right)$ and $8 \frac{1}{4}$ .
Multiply $\left(-\frac{5}{6}\right)$ and $3 \frac{2}{3}$ .
Multiply $\left(\frac{3}{4}\right)$ and $2 \frac{1}{2}$ .

Solutions

$\left(\frac{2}{3}\right)\left(8 \frac{1}{4}\right) = \frac{2 \times 33}{3 \times 4} = \frac{66}{12} = \frac{11}{2} = 5.5$
$\left(-\frac{5}{6}\right)\left(3 \frac{2}{3}\right) = \frac{(-5) \times 11}{6 \times 4} = \frac{-55}{24} = -2.29$
$\left(\frac{3}{4}\right)\left(2 \frac{1}{2}\right) = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} = 1.875$

Conclusion

In this article, we learned how to estimate the product of fractions and mixed numbers. We converted the mixed number to an improper fraction, multiplied the fractions, simplified the result, and estimated the product. We also practiced solving similar problems and provided solutions to help reinforce our understanding of the concept.

Q: What is the first step in estimating the product of fractions and mixed numbers?

A: The first step is to convert the mixed number to an improper fraction. This involves multiplying the whole number part by the denominator and adding the result to the numerator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, follow these steps:

Multiply the whole number part by the denominator.
Add the result to the numerator.
Write the result as a fraction with the denominator.

Q: What is the next step after converting the mixed number to an improper fraction?

A: After converting the mixed number to an improper fraction, you can multiply the two fractions together. This involves multiplying the numerators and denominators separately.

Q: How do I multiply fractions?

A: To multiply fractions, follow these steps:

Multiply the numerators together.
Multiply the denominators together.
Simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, follow these steps:

Find the greatest common divisor (GCD) of the numerator and the denominator.
Divide both the numerator and the denominator by the GCD.
Write the result as a simplified fraction.

Q: What is the final step in estimating the product of fractions and mixed numbers?

A: The final step is to estimate the product by finding the absolute value of the result and rounding it to the nearest whole number.

Q: Why is it important to estimate the product?

A: Estimating the product is important because it helps you to understand the magnitude of the result. It also helps you to check your work and make sure that your answer is reasonable.

Q: Can I use a calculator to estimate the product?

A: Yes, you can use a calculator to estimate the product. However, it's always a good idea to check your work by hand to make sure that your answer is correct.

Q: What are some common mistakes to avoid when estimating the product of fractions and mixed numbers?

A: Some common mistakes to avoid include:

Forgetting to convert the mixed number to an improper fraction
Multiplying the numerators and denominators incorrectly
Not simplifying the result
Not estimating the product correctly

Q: How can I practice estimating the product of fractions and mixed numbers?

A: You can practice estimating the product of fractions and mixed numbers by working through examples and exercises. You can also use online resources and practice tests to help you prepare.

Q: What are some real-world applications of estimating the product of fractions and mixed numbers?

A: Estimating the product of fractions and mixed numbers has many real-world applications, including:

Cooking and baking
Building and construction
Science and engineering
Finance and economics

Q: Can I use this method to estimate the product of other types of numbers?

A: Yes, you can use this method to estimate the product of other types of numbers, including decimals and percentages. However, you may need to modify the method slightly to accommodate the different types of numbers.