4. Calculați:a) \[$\left(\frac{1}{2}+\frac{1}{3}\right) \cdot \frac{6}{25}\$\]b) \[$\left(\frac{2}{5}+\frac{1}{10}\right) \cdot \frac{2}{15}\$\]c) \[$\frac{5}{4} \cdot\left(\frac{5}{9}-\frac{1}{3}\right)\$\]d)

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Introduction

Fractions are an essential part of mathematics, and learning to calculate expressions with fractions is a crucial skill for students of all ages. In this article, we will explore four different expressions involving fractions and provide step-by-step solutions to each one. We will also discuss the importance of understanding fractions and how they are used in real-life applications.

Calculating Expression a)

Expression a)

(12+13)⋅625\left(\frac{1}{2}+\frac{1}{3}\right) \cdot \frac{6}{25}

Solution

To calculate this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 12+13\frac{1}{2}+\frac{1}{3}
  2. Find a common denominator for the two fractions: 6
  3. Convert each fraction to have a common denominator: 36+26\frac{3}{6}+\frac{2}{6}
  4. Add the two fractions: 36+26=56\frac{3}{6}+\frac{2}{6}=\frac{5}{6}
  5. Multiply the result by 625\frac{6}{25}: 56⋅625\frac{5}{6} \cdot \frac{6}{25}
  6. Cancel out the common factors: 525\frac{5}{25}
  7. Simplify the fraction: 15\frac{1}{5}

Therefore, the final answer to expression a) is 15\frac{1}{5}.

Calculating Expression b)

Expression b)

(25+110)⋅215\left(\frac{2}{5}+\frac{1}{10}\right) \cdot \frac{2}{15}

Solution

To calculate this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 25+110\frac{2}{5}+\frac{1}{10}
  2. Find a common denominator for the two fractions: 10
  3. Convert each fraction to have a common denominator: 410+110\frac{4}{10}+\frac{1}{10}
  4. Add the two fractions: 410+110=510\frac{4}{10}+\frac{1}{10}=\frac{5}{10}
  5. Simplify the fraction: 12\frac{1}{2}
  6. Multiply the result by 215\frac{2}{15}: 12⋅215\frac{1}{2} \cdot \frac{2}{15}
  7. Cancel out the common factors: 115\frac{1}{15}

Therefore, the final answer to expression b) is 115\frac{1}{15}.

Calculating Expression c)

Expression c)

54⋅(59−13)\frac{5}{4} \cdot\left(\frac{5}{9}-\frac{1}{3}\right)

Solution

To calculate this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 59−13\frac{5}{9}-\frac{1}{3}
  2. Find a common denominator for the two fractions: 9
  3. Convert each fraction to have a common denominator: 59−39\frac{5}{9}-\frac{3}{9}
  4. Subtract the two fractions: 59−39=29\frac{5}{9}-\frac{3}{9}=\frac{2}{9}
  5. Multiply the result by 54\frac{5}{4}: 29⋅54\frac{2}{9} \cdot \frac{5}{4}
  6. Cancel out the common factors: 1036\frac{10}{36}
  7. Simplify the fraction: 518\frac{5}{18}

Therefore, the final answer to expression c) is 518\frac{5}{18}.

Calculating Expression d)

Expression d)

Unfortunately, expression d) is not provided. However, we can use the same steps as above to calculate any given expression.

Conclusion

Calculating expressions with fractions requires a clear understanding of the order of operations and the ability to simplify fractions. By following the steps outlined in this article, students can become proficient in calculating expressions with fractions and apply this skill to real-life problems. Remember to always evaluate expressions inside parentheses first, find a common denominator for fractions, and cancel out common factors to simplify the expression.

Real-Life Applications of Fractions

Fractions are used in a variety of real-life applications, including:

  • Cooking: Recipes often require fractions of ingredients, such as 1/4 cup of sugar or 3/4 teaspoon of salt.
  • Building: Architects use fractions to calculate the dimensions of buildings and structures.
  • Science: Scientists use fractions to measure the concentration of solutions and the amount of a substance.
  • Finance: Investors use fractions to calculate interest rates and investment returns.

By understanding fractions and how to calculate expressions with them, students can develop a strong foundation in mathematics and apply this skill to a wide range of real-life problems.

Practice Problems

Try calculating the following expressions:

  • (34+16)⋅1225\left(\frac{3}{4}+\frac{1}{6}\right) \cdot \frac{12}{25}
  • (23+14)⋅310\left(\frac{2}{3}+\frac{1}{4}\right) \cdot \frac{3}{10}
  • 35⋅(38−14)\frac{3}{5} \cdot\left(\frac{3}{8}-\frac{1}{4}\right)

Q: What is the order of operations for calculating expressions with fractions?

A: The order of operations for calculating expressions with fractions is:

  1. Evaluate the expression inside the parentheses
  2. Find a common denominator for the fractions
  3. Add or subtract the fractions
  4. Multiply or divide the fractions
  5. Simplify the fraction

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

A: To find a common denominator for two fractions, you need to find the least common multiple (LCM) of the two denominators. For example, if you have two fractions with denominators 4 and 6, the LCM of 4 and 6 is 12. You can then convert each fraction to have a denominator of 12.

Q: What is the difference between adding and subtracting fractions?

A: When adding fractions, you need to find a common denominator and then add the numerators. When subtracting fractions, you need to find a common denominator and then subtract the numerators.

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 denominator and then divide both numbers by the GCD.

Q: What is the difference between multiplying and dividing fractions?

A: When multiplying fractions, you need to multiply the numerators and denominators separately. When dividing fractions, you need to invert the second fraction and then multiply.

Q: Can I use a calculator to calculate expressions with fractions?

A: Yes, you can use a calculator to calculate expressions with fractions. However, it's always a good idea to check your work by hand to make sure you understand the steps involved.

Q: How do I apply the order of operations to real-life problems?

A: When applying the order of operations to real-life problems, you need to follow the same steps as above. For example, if you're calculating the cost of a recipe, you need to evaluate the expression inside the parentheses, find a common denominator, add or subtract the fractions, and then multiply or divide the fractions.

Q: What are some common mistakes to avoid when calculating expressions with fractions?

A: Some common mistakes to avoid when calculating expressions with fractions include:

  • Not following the order of operations
  • Not finding a common denominator
  • Not simplifying the fraction
  • Not checking your work by hand

Q: How can I practice calculating expressions with fractions?

A: You can practice calculating expressions with fractions by working through practice problems, such as those found in this article. You can also try creating your own practice problems or using online resources to generate random expressions.

Q: What are some real-life applications of calculating expressions with fractions?

A: Some real-life applications of calculating expressions with fractions include:

  • Cooking: Recipes often require fractions of ingredients, such as 1/4 cup of sugar or 3/4 teaspoon of salt.
  • Building: Architects use fractions to calculate the dimensions of buildings and structures.
  • Science: Scientists use fractions to measure the concentration of solutions and the amount of a substance.
  • Finance: Investors use fractions to calculate interest rates and investment returns.

By understanding the order of operations and how to calculate expressions with fractions, you can develop a strong foundation in mathematics and apply this skill to a wide range of real-life problems.