Calculate The Following Expression:${ \frac{3}{7} \times \frac{8}{15} }$

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Introduction

In this article, we will delve into the world of mathematics and explore the process of calculating a simple expression involving fractions. The expression we will be working with is 37×815\frac{3}{7} \times \frac{8}{15}. This type of problem is commonly encountered in algebra and arithmetic, and understanding how to solve it is essential for success in these subjects.

Understanding the Expression

Before we begin calculating the expression, it's essential to understand what it represents. The expression 37×815\frac{3}{7} \times \frac{8}{15} is a product of two fractions. To calculate this expression, we need to multiply the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately.

Multiplying Fractions

When multiplying fractions, we follow a simple rule: we multiply the numerators together and the denominators together. This can be represented as:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

In our expression, the numerators are 3 and 8, and the denominators are 7 and 15. Applying the rule, we get:

37×815=3×87×15\frac{3}{7} \times \frac{8}{15} = \frac{3 \times 8}{7 \times 15}

Calculating the Numerator and Denominator

Now that we have the expression in the form a×cb×d\frac{a \times c}{b \times d}, we can calculate the numerator and denominator separately.

The numerator is the product of 3 and 8, which is:

3×8=243 \times 8 = 24

The denominator is the product of 7 and 15, which is:

7×15=1057 \times 15 = 105

Simplifying the Fraction

Now that we have the numerator and denominator, we can simplify the fraction by dividing both numbers by their greatest common divisor (GCD). In this case, the GCD of 24 and 105 is 3.

Dividing both numbers by 3, we get:

24105=835\frac{24}{105} = \frac{8}{35}

Conclusion

In this article, we have explored the process of calculating a simple expression involving fractions. We have seen how to multiply fractions, calculate the numerator and denominator, and simplify the fraction by dividing both numbers by their greatest common divisor. By following these steps, we can confidently calculate expressions like 37×815\frac{3}{7} \times \frac{8}{15}.

Real-World Applications

Calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15} has numerous real-world applications. For example, in finance, we may need to calculate the interest on a loan or investment. In science, we may need to calculate the probability of an event occurring. In engineering, we may need to calculate the stress on a material or the force required to move an object.

Tips and Tricks

When calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15}, it's essential to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps, we can ensure that our calculations are accurate and reliable.

Common Mistakes

When calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15}, there are several common mistakes to watch out for:

  • Forgetting to multiply the numerators and denominators separately: This can lead to incorrect results.
  • Not simplifying the fraction: Failing to simplify the fraction can make it difficult to work with and may lead to errors.
  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.

Conclusion

In conclusion, calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15} is a straightforward process that involves multiplying the numerators and denominators separately and simplifying the fraction by dividing both numbers by their greatest common divisor. By following the steps outlined in this article and avoiding common mistakes, we can confidently calculate expressions like this and apply them to real-world problems.

Final Answer

The final answer to the expression 37×815\frac{3}{7} \times \frac{8}{15} is:

Q&A: Frequently Asked Questions

Q: What is the formula for multiplying fractions? A: The formula for multiplying fractions is ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.

Q: How do I calculate the numerator and denominator when multiplying fractions? A: To calculate the numerator and denominator, you simply multiply the numbers together. For example, in the expression 37×815\frac{3}{7} \times \frac{8}{15}, the numerator is 3×8=243 \times 8 = 24 and the denominator is 7×15=1057 \times 15 = 105.

Q: What is the greatest common divisor (GCD) and how do I use it to simplify a fraction? A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To simplify a fraction, you divide both the numerator and denominator by their GCD. For example, in the expression 24105\frac{24}{105}, the GCD of 24 and 105 is 3. Dividing both numbers by 3, we get 835\frac{8}{35}.

Q: What are some common mistakes to watch out for when calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15}? A: Some common mistakes to watch out for include:

  • Forgetting to multiply the numerators and denominators separately
  • Not simplifying the fraction
  • Not following the order of operations (PEMDAS)

Q: How do I apply the order of operations (PEMDAS) when calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15}? A: To apply the order of operations (PEMDAS), you follow these steps:

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What are some real-world applications of calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15}? A: Some real-world applications of calculating expressions like 37×815\frac{3}{7} \times \frac{8}{15} include:

  • Finance: Calculating interest on a loan or investment
  • Science: Calculating the probability of an event occurring
  • Engineering: Calculating the stress on a material or the force required to move an object

Q: How do I simplify a fraction with a variable in the numerator or denominator? A: To simplify a fraction with a variable in the numerator or denominator, you can use the same steps as before. For example, if you have the expression 3x7y\frac{3x}{7y}, you can simplify it by dividing both the numerator and denominator by their GCD.

Q: What is the difference between a fraction and a decimal? A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. A decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part. For example, the fraction 37\frac{3}{7} is equivalent to the decimal 0.428571.

Q: How do I convert a fraction to a decimal? A: To convert a fraction to a decimal, you can divide the numerator by the denominator. For example, to convert the fraction 37\frac{3}{7} to a decimal, you can divide 3 by 7, which gives you 0.428571.

Q: What is the difference between a mixed number and an improper fraction? A: A mixed number is a way of expressing a fraction as a combination of a whole number and a fraction. An improper fraction is a way of expressing a fraction as a ratio of two numbers, where the numerator is greater than the denominator. For example, the mixed number 217\frac{1}{7} is equivalent to the improper fraction 157\frac{15}{7}.

Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, you can multiply the whole number part by the denominator and add the numerator. For example, to convert the mixed number 217\frac{1}{7} to an improper fraction, you can multiply 2 by 7 and add 1, which gives you 157\frac{15}{7}.

Q: What are some common mistakes to watch out for when converting fractions to decimals or mixed numbers? A: Some common mistakes to watch out for include:

  • Forgetting to divide the numerator by the denominator when converting a fraction to a decimal
  • Not simplifying the fraction before converting it to a decimal or mixed number
  • Not following the order of operations (PEMDAS) when converting a fraction to a decimal or mixed number