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

Understanding the Problem

When dealing with multiplication of fractions and mixed numbers, it's essential to understand the concept of equivalent ratios and how to convert between different forms of numbers. In this problem, we are given the expression $\left(-\frac{3}{5}\right)\left(17 \frac{5}{6}\right)$ and asked to find the best estimate of its value.

Converting the Mixed Number to an Improper Fraction

To simplify the multiplication, we need to convert the mixed number $17 \frac{5}{6}$ to an improper fraction. This can be done by multiplying the whole number part (17) by the denominator (6) and then adding the numerator (5). The result is:

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

Multiplying the Fractions

Now that we have the mixed number in improper fraction form, we can multiply it by the given fraction $\left(-\frac{3}{5}\right)$. To do this, we multiply the numerators and denominators separately:

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

Simplifying the Result

We can simplify the result by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 321 and 30 is 3. Dividing both numbers by 3 gives us:

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

Estimating the Value

To estimate the value of $\frac{-107}{10}$, we can divide the numerator by the denominator:

$$\frac{-107}{10} \approx -10.7$$

Comparing with the Answer Choices

Now that we have an estimate of the value, we can compare it with the answer choices:

A. -18 B. -9 C. 9 D. 18

Based on our estimate, the correct answer is:

B. -9

Conclusion

In this problem, we estimated the product of a fraction and a mixed number by converting the mixed number to an improper fraction, multiplying the fractions, simplifying the result, and estimating the value. Our final estimate was -10.7, which corresponds to answer choice B. -9.

Q: What is the first step in estimating the product of a fraction and a mixed number?

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 then adding 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:

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

For example, to convert 17 5/6 to an improper fraction, you would:

1. Multiply 17 by 6: 17 × 6 = 102
2. Add 5 to the result: 102 + 5 = 107
3. Write the result as a fraction with the denominator: 107/6

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, the next step is to multiply the fractions. This involves multiplying the numerators and denominators separately.

Q: How do I multiply fractions?

A: To multiply fractions, follow these steps:

1. Multiply the numerators: multiply the top numbers together.
2. Multiply the denominators: multiply the bottom numbers together.
3. Write the result as a fraction.

For example, to multiply -3/5 and 107/6, you would:

1. Multiply the numerators: (-3) × 107 = -321
2. Multiply the denominators: 5 × 6 = 30
3. Write the result as a fraction: -321/30

Q: How do I simplify the result of multiplying fractions?

A: To simplify the result of multiplying fractions, follow these steps:

1. Find the greatest common divisor (GCD) of the numerator and denominator.
2. Divide both the numerator and denominator by the GCD.

For example, to simplify -321/30, you would:

1. Find the GCD of 321 and 30: 3
2. Divide both the numerator and denominator by 3: -321 ÷ 3 = -107, 30 ÷ 3 = 10
3. Write the result as a fraction: -107/10

Q: How do I estimate the value of a fraction?

A: To estimate the value of a fraction, divide the numerator by the denominator.

For example, to estimate the value of -107/10, you would:

1. Divide the numerator by the denominator: -107 ÷ 10 = -10.7

Q: What is the final step in estimating the product of a fraction and a mixed number?

A: The final step is to compare the estimated value with the answer choices.

Q: How do I compare the estimated value with the answer choices?

A: To compare the estimated value with the answer choices, simply look for the answer that is closest to the estimated value.

For example, if the estimated value is -10.7 and the answer choices are -18, -9, 9, and 18, the correct answer would be -9.

Conclusion

Estimating the product of a fraction and a mixed number involves converting the mixed number to an improper fraction, multiplying the fractions, simplifying the result, estimating the value, and comparing the estimated value with the answer choices. By following these steps, you can accurately estimate the product of a fraction and a mixed number.