Simplify The Following Expression:$\[ 73 \frac{7}{10} - 25 \frac{5}{8} \\]

Introduction

Mixed numbers are a combination of a whole number and a fraction. They are commonly used in mathematics to represent quantities that have both a whole and a fractional part. In this article, we will focus on simplifying mixed numbers, specifically the expression $73 \frac{7}{10} - 25 \frac{5}{8}$.

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, $3 \frac{2}{5}$ is a mixed number that represents the quantity $3$ and $\frac{2}{5}$.

Simplifying the Expression

To simplify the expression $73 \frac{7}{10} - 25 \frac{5}{8}$, we need to first convert the mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we need to multiply the whole number by the denominator and add the numerator. The result is then written as a fraction with the denominator.

Step 1: Convert $73 \frac{7}{10}$ to an Improper Fraction

To convert $73 \frac{7}{10}$ to an improper fraction, we need to multiply the whole number $73$ by the denominator $10$ and add the numerator $7$.

$73 \frac{7}{10} = \frac{(73 \times 10) + 7}{10} = \frac{730 + 7}{10} = \frac{737}{10}$

Step 2: Convert $25 \frac{5}{8}$ to an Improper Fraction

To convert $25 \frac{5}{8}$ to an improper fraction, we need to multiply the whole number $25$ by the denominator $8$ and add the numerator $5$.

$25 \frac{5}{8} = \frac{(25 \times 8) + 5}{8} = \frac{200 + 5}{8} = \frac{205}{8}$

Subtracting the Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can subtract them.

$\frac{737}{10} - \frac{205}{8} = \frac{737 \times 8}{10 \times 8} - \frac{205 \times 10}{8 \times 10} = \frac{5896}{80} - \frac{2050}{80} = \frac{3846}{80}$

Simplifying the Result

The result $\frac{3846}{80}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of $3846$ and $80$ is $2$. Dividing both the numerator and the denominator by $2$ gives:

$\frac{3846}{80} = \frac{1923}{40}$

Conclusion

In this article, we have simplified the expression $73 \frac{7}{10} - 25 \frac{5}{8}$ by converting the mixed numbers to improper fractions and then subtracting them. The result is $\frac{1923}{40}$.

Tips and Tricks

• When simplifying mixed numbers, it is often helpful to convert them to improper fractions first.
• When subtracting improper fractions, make sure to have a common denominator.
• When simplifying the result, look for the greatest common divisor (GCD) of the numerator and the denominator.

Practice Problems

• Simplify the expression $42 \frac{3}{4} - 17 \frac{2}{3}$.
• Simplify the expression $25 \frac{1}{2} - 12 \frac{3}{4}$.

References

• [1] "Mixed Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/mixednumbers.html
• [2] "Improper Fractions" by Math Is Fun. Retrieved from https://www.mathisfun.com/improper-fractions.html
  Simplifying Mixed Numbers: A Q&A Guide =====================================

Introduction

In our previous article, we discussed how to simplify mixed numbers by converting them to improper fractions and then subtracting them. In this article, we will answer some frequently asked questions about simplifying mixed numbers.

Q: What is a mixed number?

A: 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.

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. The result is then written as a fraction with the denominator.

Example: Convert $73 \frac{7}{10}$ to an improper fraction.

$73 \frac{7}{10} = \frac{(73 \times 10) + 7}{10} = \frac{730 + 7}{10} = \frac{737}{10}$

Q: How do I subtract two mixed numbers?

A: To subtract two mixed numbers, you need to first convert them to improper fractions. Then, you can subtract the numerators and keep the same denominator.

Example: Subtract $73 \frac{7}{10}$ and $25 \frac{5}{8}$.

$73 \frac{7}{10} = \frac{737}{10}$

$25 \frac{5}{8} = \frac{205}{8}$

$\frac{737}{10} - \frac{205}{8} = \frac{737 \times 8}{10 \times 8} - \frac{205 \times 10}{8 \times 10} = \frac{5896}{80} - \frac{2050}{80} = \frac{3846}{80}$

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Then, you can divide both the numerator and the denominator by the GCD.

Example: Simplify $\frac{3846}{80}$.

The GCD of $3846$ and $80$ is $2$. Dividing both the numerator and the denominator by $2$ gives:

$\frac{3846}{80} = \frac{1923}{40}$

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

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

• Not converting the mixed numbers to improper fractions before subtracting them.
• Not finding the greatest common divisor (GCD) of the numerator and the denominator when simplifying the result.
• Not dividing both the numerator and the denominator by the GCD when simplifying the result.

Q: How can I practice simplifying mixed numbers?

A: You can practice simplifying mixed numbers by working through examples and exercises. You can also use online resources and math software to help you practice.

Conclusion

Simplifying mixed numbers can be a challenging task, but with practice and patience, you can become proficient in it. Remember to convert the mixed numbers to improper fractions before subtracting them, and to find the greatest common divisor (GCD) of the numerator and the denominator when simplifying the result.

Tips and Tricks

• When simplifying mixed numbers, it is often helpful to convert them to improper fractions first.
• When subtracting improper fractions, make sure to have a common denominator.
• When simplifying the result, look for the greatest common divisor (GCD) of the numerator and the denominator.

Practice Problems

• Simplify the expression $42 \frac{3}{4} - 17 \frac{2}{3}$.
• Simplify the expression $25 \frac{1}{2} - 12 \frac{3}{4}$.

References

• [1] "Mixed Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/mixednumbers.html
• [2] "Improper Fractions" by Math Is Fun. Retrieved from https://www.mathisfun.com/improper-fractions.html