Multiply To Find Both An Exact Answer And An Estimated Answer For The Expression Below:$4 \frac{3}{7} \cdot 1 \frac{5}{6}$1. The Exact Answer Of $4 \frac{3}{7} \cdot 1 \frac{5}{6}$ Is _______. (Simplify Your Answer. Type An

Introduction

In mathematics, mixed numbers are a combination of a whole number and a fraction. When we multiply mixed numbers, we need to follow a specific procedure to get the exact answer. In this article, we will learn how to multiply mixed numbers to find both an exact answer and an estimated answer for the expression $4 \frac{3}{7} \cdot 1 \frac{5}{6}$.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator. For example, $4 \frac{3}{7}$ is a mixed number where $4$ is the whole number, $3$ is the numerator, and $7$ is the denominator.

Multiplying Mixed Numbers: A Step-by-Step Guide

To multiply mixed numbers, we need to follow these steps:

1. Convert the mixed numbers to improper fractions: We need to convert the mixed numbers to improper fractions by multiplying the whole number by the denominator and adding the numerator. For example, $4 \frac{3}{7}$ can be converted to an improper fraction as follows:

$$4 \frac{3}{7} = \frac{(4 \times 7) + 3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$$

Similarly, $1 \frac{5}{6}$ can be converted to an improper fraction as follows:

$$1 \frac{5}{6} = \frac{(1 \times 6) + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6}$$

2. Multiply the numerators: We need to multiply the numerators of the two improper fractions. In this case, we need to multiply $31$ and $11$.

$$31 \times 11 = 341$$

3. Multiply the denominators: We need to multiply the denominators of the two improper fractions. In this case, we need to multiply $7$ and $6$.

$$7 \times 6 = 42$$

4. Write the product as an improper fraction: We need to write the product as an improper fraction by dividing the product of the numerators by the product of the denominators.

$$\frac{341}{42}$$

5. Simplify the improper fraction: We need to simplify the improper fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $341$ and $42$ is $1$, so the improper fraction cannot be simplified further.

The Exact Answer

The exact answer of $4 \frac{3}{7} \cdot 1 \frac{5}{6}$ is $\frac{341}{42}$.

Estimating the Answer

To estimate the answer, we need to multiply the whole numbers and the fractions separately. We can estimate the product of the whole numbers as follows:

$$4 \times 1 = 4$$

We can estimate the product of the fractions as follows:

$$\frac{3}{7} \times \frac{5}{6} = \frac{15}{42}$$

We can add the estimated product of the whole numbers and the estimated product of the fractions to get an estimated answer:

$$4 + \frac{15}{42} = 4 + 0.357 = 4.357$$

Conclusion

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Introduction

In our previous article, we learned how to multiply mixed numbers to find both an exact answer and an estimated answer for the expression $4 \frac{3}{7} \cdot 1 \frac{5}{6}$. In this article, we will answer some frequently asked questions about multiplying mixed numbers.

Q: What is the difference between multiplying mixed numbers and multiplying fractions?

A: Multiplying mixed numbers and multiplying fractions are similar, but not the same. When we multiply fractions, we simply multiply the numerators and denominators separately. However, when we multiply mixed numbers, we need to convert the mixed numbers to improper fractions first, and then multiply the numerators and denominators.

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

A: To convert a mixed number to an improper fraction, you need to multiply the whole number by the denominator and add the numerator. For example, to convert $4 \frac{3}{7}$ to an improper fraction, you would multiply $4$ by $7$ and add $3$:

$$4 \times 7 = 28$$

$$28 + 3 = 31$$

So, $4 \frac{3}{7}$ can be written as an improper fraction as follows:

$$\frac{31}{7}$$

Q: What is the greatest common divisor (GCD) and why is it important?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. When we multiply mixed numbers, we need to simplify the improper fraction by dividing the numerator and denominator by their GCD. This is important because it helps us to get the exact answer.

Q: How do I simplify an improper fraction?

A: To simplify an improper fraction, you need to divide the numerator and denominator by their GCD. For example, to simplify $\frac{31}{7}$, you would divide both numbers by $1$ (since $1$ is the GCD of $31$ and $7$):

$$\frac{31}{7} = \frac{31 \div 1}{7 \div 1} = \frac{31}{7}$$

So, $\frac{31}{7}$ cannot be simplified further.

Q: Can I estimate the answer when multiplying mixed numbers?

A: Yes, you can estimate the answer when multiplying mixed numbers. To estimate the answer, you can multiply the whole numbers and the fractions separately. For example, to estimate the answer for $4 \frac{3}{7} \cdot 1 \frac{5}{6}$, you would multiply the whole numbers and the fractions separately:

$$4 \times 1 = 4$$

$$\frac{3}{7} \times \frac{5}{6} = \frac{15}{42}$$

You can add the estimated product of the whole numbers and the estimated product of the fractions to get an estimated answer:

$$4 + \frac{15}{42} = 4 + 0.357 = 4.357$$

Q: What are some common mistakes to avoid when multiplying mixed numbers?

A: Some common mistakes to avoid when multiplying mixed numbers include:

• Not converting the mixed numbers to improper fractions
• Not multiplying the numerators and denominators correctly
• Not simplifying the improper fraction
• Not estimating the answer correctly

Conclusion

In this article, we answered some frequently asked questions about multiplying mixed numbers. We covered topics such as converting mixed numbers to improper fractions, simplifying improper fractions, estimating the answer, and common mistakes to avoid. We hope that this article has been helpful in clarifying any doubts you may have had about multiplying mixed numbers.