Simplify The Following Expression:$\[ \left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{2}{10} + \frac{1}{3} \\]

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Introduction

In this article, we will simplify the given mathematical expression, which involves fractions and mixed numbers. The expression is as follows:

(214−138)÷210+13\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{2}{10} + \frac{1}{3}

We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.

Convert Mixed Numbers to Improper Fractions

The first step is to convert the mixed numbers to improper fractions. A mixed number is a combination of a whole number and a proper fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

Let's convert the mixed numbers in the given expression:

  • 214=(2×4)+14=8+14=942 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}
  • 138=(1×8)+38=8+38=1181 \frac{3}{8} = \frac{(1 \times 8) + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8}

Now, the expression becomes:

(94−118)÷210+13\left(\frac{9}{4} - \frac{11}{8}\right) \div \frac{2}{10} + \frac{1}{3}

Find a Common Denominator

To subtract the fractions, we need to find a common denominator. The least common multiple (LCM) of 4 and 8 is 8. So, we will convert 94\frac{9}{4} to have a denominator of 8:

94=9×24×2=188\frac{9}{4} = \frac{9 \times 2}{4 \times 2} = \frac{18}{8}

Now, the expression becomes:

(188−118)÷210+13\left(\frac{18}{8} - \frac{11}{8}\right) \div \frac{2}{10} + \frac{1}{3}

Subtract the Fractions

Now that we have a common denominator, we can subtract the fractions:

188−118=18−118=78\frac{18}{8} - \frac{11}{8} = \frac{18 - 11}{8} = \frac{7}{8}

So, the expression becomes:

78÷210+13\frac{7}{8} \div \frac{2}{10} + \frac{1}{3}

Divide the Fractions

To divide the fractions, we need to invert the second fraction and multiply:

78÷210=78×102=7×108×2=7016\frac{7}{8} \div \frac{2}{10} = \frac{7}{8} \times \frac{10}{2} = \frac{7 \times 10}{8 \times 2} = \frac{70}{16}

Now, the expression becomes:

7016+13\frac{70}{16} + \frac{1}{3}

Find a Common Denominator

To add the fractions, we need to find a common denominator. The LCM of 16 and 3 is 48. So, we will convert 7016\frac{70}{16} and 13\frac{1}{3} to have a denominator of 48:

7016=70×316×3=21048\frac{70}{16} = \frac{70 \times 3}{16 \times 3} = \frac{210}{48} 13=1×163×16=1648\frac{1}{3} = \frac{1 \times 16}{3 \times 16} = \frac{16}{48}

Now, the expression becomes:

21048+1648\frac{210}{48} + \frac{16}{48}

Add the Fractions

Now that we have a common denominator, we can add the fractions:

21048+1648=210+1648=22648\frac{210}{48} + \frac{16}{48} = \frac{210 + 16}{48} = \frac{226}{48}

Simplify the Result

We can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 226 and 48 is 2. So, we will divide both the numerator and the denominator by 2:

22648=226÷248÷2=11324\frac{226}{48} = \frac{226 \div 2}{48 \div 2} = \frac{113}{24}

The final result is 11324\frac{113}{24}.

Conclusion

In this article, we simplified the given mathematical expression, which involved fractions and mixed numbers. We broke down the expression into smaller parts, simplified each part, and then combined them to get the final result. The final result is 11324\frac{113}{24}.

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Q: What is the first step in simplifying a mathematical expression that involves fractions and mixed numbers?

A: The first step is to convert the mixed numbers to improper fractions. This involves multiplying the whole number by the denominator and 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, you multiply the whole number by the denominator and add the numerator. For example, to convert 2142 \frac{1}{4} to an improper fraction, you would multiply 2 by 4 and add 1, resulting in 94\frac{9}{4}.

Q: What is the next step after converting the mixed numbers to improper fractions?

A: After converting the mixed numbers to improper fractions, the next step is to find a common denominator for the fractions. This involves finding the least common multiple (LCM) of the denominators.

Q: How do I find a common denominator for fractions?

A: To find a common denominator for fractions, you need to find the least common multiple (LCM) of the denominators. For example, if you have fractions with denominators of 4 and 8, the LCM is 8.

Q: What is the next step after finding a common denominator?

A: After finding a common denominator, the next step is to subtract or add the fractions as needed. This involves using the common denominator to combine the fractions.

Q: How do I subtract or add fractions with a common denominator?

A: To subtract or add fractions with a common denominator, you simply subtract or add the numerators and keep the common denominator. For example, if you have 78−38\frac{7}{8} - \frac{3}{8}, you would subtract the numerators and keep the common denominator, resulting in 48\frac{4}{8}.

Q: What is the next step after subtracting or adding the fractions?

A: After subtracting or adding the fractions, the next step is to simplify the result. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

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 and divide both by the GCD. For example, if you have 1218\frac{12}{18}, the GCD is 6, so you would divide both the numerator and the denominator by 6, resulting in 23\frac{2}{3}.

Q: What is the final step in simplifying a mathematical expression?

A: The final step is to check your work and make sure that the expression is simplified correctly. This involves reviewing the steps you took to simplify the expression and making sure that you did not make any mistakes.

Q: What are some common mistakes to avoid when simplifying mathematical expressions?

A: Some common mistakes to avoid when simplifying mathematical expressions include:

  • Forgetting to convert mixed numbers to improper fractions
  • Not finding a common denominator for fractions
  • Subtracting or adding fractions incorrectly
  • Not simplifying the result
  • Not checking your work

Q: How can I practice simplifying mathematical expressions?

A: You can practice simplifying mathematical expressions by working through examples and exercises in a textbook or online resource. You can also try simplifying expressions on your own and checking your work to make sure that you are doing it correctly.

Q: What are some real-world applications of simplifying mathematical expressions?

A: Simplifying mathematical expressions has many real-world applications, including:

  • Calculating costs and prices in business and finance
  • Determining the area and perimeter of shapes in architecture and engineering
  • Calculating the speed and distance of objects in physics and engineering
  • Determining the probability of events in statistics and data analysis

Q: Why is it important to simplify mathematical expressions?

A: Simplifying mathematical expressions is important because it helps to:

  • Make calculations easier and faster
  • Reduce errors and mistakes
  • Improve understanding and comprehension of mathematical concepts
  • Enhance problem-solving skills and critical thinking
  • Prepare for more advanced mathematical concepts and applications.