Calculating 10 1/2 X 1 3/4 X 4 3/5 A Step By Step Solution

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In this comprehensive guide, we will delve into the step-by-step process of calculating the product of mixed numbers: 10 1/2, 1 3/4, and 4 3/5. This mathematical exploration will not only provide you with the solution but also enhance your understanding of fraction manipulation and multiplication. Whether you're a student grappling with mixed numbers or simply seeking to refresh your math skills, this detailed explanation will equip you with the knowledge and confidence to tackle similar calculations with ease.

Understanding Mixed Numbers and Improper Fractions

To effectively multiply mixed numbers, it's crucial to first understand their composition. A mixed number combines a whole number and a proper fraction, such as 10 1/2. To convert a mixed number into an improper fraction, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, while the denominator remains the same.

Let's apply this to our first mixed number, 10 1/2:

  1. Multiply the whole number (10) by the denominator (2): 10 * 2 = 20
  2. Add the numerator (1): 20 + 1 = 21
  3. Keep the same denominator (2)

Therefore, 10 1/2 is equivalent to the improper fraction 21/2. We can apply the same process to the other mixed numbers in our equation. For 1 3/4:

  1. Multiply the whole number (1) by the denominator (4): 1 * 4 = 4
  2. Add the numerator (3): 4 + 3 = 7
  3. Keep the same denominator (4)

So, 1 3/4 transforms into the improper fraction 7/4. Lastly, let's convert 4 3/5:

  1. Multiply the whole number (4) by the denominator (5): 4 * 5 = 20
  2. Add the numerator (3): 20 + 3 = 23
  3. Keep the same denominator (5)

Thus, 4 3/5 becomes the improper fraction 23/5. Now that we've successfully converted all mixed numbers into improper fractions, our equation looks like this: 21/2 x 7/4 x 23/5. This transformation is a critical step towards simplifying the multiplication process.

Multiplying Improper Fractions: A Step-by-Step Approach

Now that we have converted the mixed numbers into improper fractions, the multiplication process becomes more straightforward. Multiplying fractions involves multiplying the numerators together and then multiplying the denominators together. This process is consistent regardless of the number of fractions being multiplied.

Let's apply this to our equation: 21/2 x 7/4 x 23/5

First, we multiply the numerators:

21 * 7 * 23 = 3381

Next, we multiply the denominators:

2 * 4 * 5 = 40

This gives us the improper fraction 3381/40. While this is a correct answer, it's often more useful to express the result as a mixed number. Converting an improper fraction back to a mixed number allows for easier comprehension of the magnitude of the value. The improper fraction 3381/40 represents the result of our multiplication, but it's not in its simplest or most easily understood form.

Converting Improper Fractions to Mixed Numbers

To convert the improper fraction 3381/40 back into a mixed number, we perform long division. We divide the numerator (3381) by the denominator (40). The quotient becomes the whole number part of the mixed number, the remainder becomes the numerator of the fractional part, and the denominator remains the same.

Performing the long division, we find that 3381 divided by 40 is 84 with a remainder of 21. This means that:

  • The whole number part is 84.
  • The numerator of the fractional part is 21.
  • The denominator remains 40.

Therefore, the mixed number equivalent of 3381/40 is 84 21/40. This final conversion provides a clear and concise answer to our original problem. We have successfully multiplied the mixed numbers and expressed the result in a way that is easy to interpret.

Simplifying Fractions: Ensuring the Most Concise Answer

Before declaring our final answer, it's essential to check if the fractional part of the mixed number can be simplified. Simplifying fractions involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. This process reduces the fraction to its lowest terms, providing the most concise representation of the value.

In our mixed number 84 21/40, we need to determine if 21/40 can be simplified. The factors of 21 are 1, 3, 7, and 21. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The greatest common factor (GCF) of 21 and 40 is 1. Since the GCF is 1, the fraction 21/40 is already in its simplest form. This means that our mixed number, 84 21/40, is the most simplified and accurate answer to our original equation.

Final Answer and Conclusion

After meticulously converting mixed numbers to improper fractions, multiplying them, converting the result back to a mixed number, and simplifying the fractional part, we arrive at our final answer. The product of 10 1/2, 1 3/4, and 4 3/5 is 84 21/40. This comprehensive process demonstrates the importance of understanding fraction manipulation and the step-by-step approach required to solve such problems accurately.

Therefore, 10 rac{1}{2} imes 1 rac{3}{4} imes 4 rac{3}{5} = N, N = 84 21/40. This detailed explanation not only provides the solution but also reinforces the underlying mathematical principles involved. By understanding these principles, you can confidently tackle similar calculations and expand your mathematical proficiency. Remember, practice and a methodical approach are key to mastering fraction operations and achieving accurate results.