Simplify The Following Expression:$\[ \frac{5}{6} \times \frac{11}{14} = \\]$\[ \frac{5 \times 11}{6 \times 14} \\]

by ADMIN 117 views

Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently. In this article, we will focus on simplifying a given expression involving fractions. We will break down the process into manageable steps, making it easy to understand and follow.

The Given Expression

The given expression is:

56×1114=5×116×14\frac{5}{6} \times \frac{11}{14} = \frac{5 \times 11}{6 \times 14}

Step 1: Multiply the Numerators

To simplify the expression, we start by multiplying the numerators (the numbers on top) of the two fractions.

5×11=555 \times 11 = 55

Step 2: Multiply the Denominators

Next, we multiply the denominators (the numbers on the bottom) of the two fractions.

6×14=846 \times 14 = 84

Step 3: Write the Product of the Numerators over the Product of the Denominators

Now, we write the product of the numerators over the product of the denominators.

5584\frac{55}{84}

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

Finding the Greatest Common Divisor (GCD)

To find the GCD of 55 and 84, we can use the Euclidean algorithm or list the factors of each number.

Factors of 55: 1, 5, 11, 55 Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The greatest common divisor of 55 and 84 is 1.

Conclusion

Since the GCD of 55 and 84 is 1, the fraction 5584\frac{55}{84} is already in its simplest form.

Final Answer

The simplified expression is:

5584\frac{55}{84}

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in finance, simplifying expressions can help us calculate interest rates, investment returns, and other financial metrics. In science, simplifying expressions can help us model complex systems, make predictions, and understand the behavior of physical phenomena.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

  • Always start by multiplying the numerators and denominators separately.
  • Use the Euclidean algorithm or list the factors of each number to find the GCD.
  • Simplify the fraction by dividing both the numerator and the denominator by the GCD.
  • Check your answer by plugging it back into the original expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

  • Not multiplying the numerators and denominators separately.
  • Not finding the GCD of the numerator and the denominator.
  • Not simplifying the fraction by dividing both the numerator and the denominator by the GCD.
  • Not checking the answer by plugging it back into the original expression.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify expressions efficiently and accurately. Remember to always multiply the numerators and denominators separately, find the GCD of the numerator and the denominator, and simplify the fraction by dividing both the numerator and the denominator by the GCD. With practice and patience, you can become proficient in simplifying expressions and tackle complex mathematical problems with confidence.