Understanding the Question: Calculating the Value of X

The question asks us to find the value of the variable \(X\) based on a given mathematical expression involving fractions. The expression is:

\( X=\frac{7}{5}+\left\{\frac{1}{5}-\frac{1}{3}\right\} \times \frac{5}{2}~\)

To solve this, we need to follow the correct order of operations. A commonly used rule for the order of operations is BODMAS or PEMDAS.

• Brackets (or Parentheses)
• Orders (or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In our expression for the value of X, we have addition, subtraction within braces, and multiplication. We will tackle the operations in the following order:

1. The subtraction inside the braces \(\left\{\frac{1}{5}-\frac{1}{3}\right\}\).
2. The multiplication of the result from step 1 by \(\frac{5}{2}\).
3. The addition of \(\frac{7}{5}\) to the result from step 2.

Step-by-Step Calculation to Find the Value of X

Let's calculate the value of X following the order of operations:

Step 1: Calculate the expression inside the braces

We need to calculate \(\frac{1}{5}-\frac{1}{3}\). To subtract fractions, they must have a common denominator. The least common multiple (LCM) of 5 and 3 is 15.

Convert each fraction to have a denominator of 15:

• \(\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}\)
• \(\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\)

Now, subtract the fractions:

\(\frac{3}{15} - \frac{5}{15} = \frac{3-5}{15} = \frac{-2}{15}\)

So, the expression inside the braces is equal to \(\frac{-2}{15}\).

Step 2: Perform the multiplication

Now the expression for \(X\) becomes:

\( X=\frac{7}{5}+\left\{\frac{-2}{15}\right\} \times \frac{5}{2}~\)

Next, we perform the multiplication:

\(\frac{-2}{15} \times \frac{5}{2}\)

Multiply the numerators together and the denominators together:

\(\frac{-2 \times 5}{15 \times 2} = \frac{-10}{30}\)

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10:

\(\frac{-10 \div 10}{30 \div 10} = \frac{-1}{3}\)

So, the multiplication part of the expression is equal to \(\frac{-1}{3}\).

Step 3: Perform the final addition

Now the expression for \(X\) is:

\( X=\frac{7}{5} + \left(\frac{-1}{3}\right) \)

This is equivalent to:

\( X=\frac{7}{5} - \frac{1}{3} \)

Again, to add or subtract fractions, they need a common denominator. The LCM of 5 and 3 is 15.

Convert each fraction to have a denominator of 15:

• \(\frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15}\)
• \(\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\)

Now, perform the subtraction:

\(\frac{21}{15} - \frac{5}{15} = \frac{21-5}{15} = \frac{16}{15}\)

Therefore, the value of X is \(\frac{16}{15}\).

Summary of Finding the Value of X

We followed the order of operations to evaluate the expression for \(X\). Here is a summary:

• Calculate the subtraction inside the braces: \(\frac{1}{5}-\frac{1}{3} = \frac{3}{15}-\frac{5}{15} = \frac{-2}{15}\)
• Substitute this back: \( X=\frac{7}{5}+\left\{\frac{-2}{15}\right\} \times \frac{5}{2}~\)
• Perform the multiplication: \(\frac{-2}{15} \times \frac{5}{2} = \frac{-10}{30} = \frac{-1}{3}\)
• Substitute this back: \( X=\frac{7}{5} + \left(\frac{-1}{3}\right) = \frac{7}{5} - \frac{1}{3} \)
• Perform the final subtraction: \(\frac{7}{5} - \frac{1}{3} = \frac{21}{15} - \frac{5}{15} = \frac{16}{15}\)

The final value of X is \(\frac{16}{15}\).

Revision Table: Key Concepts for Finding the Value of X

Additional Information: Working with Fraction Expressions

When dealing with fraction expressions like the one to find the value of X, it's crucial to be careful with signs, especially when subtracting or dealing with negative results from operations within brackets. Always simplify fractions at each step if possible to keep the numbers manageable, although it is not strictly necessary until the final answer.

Understanding common denominators is fundamental for adding or subtracting fractions. The least common multiple (LCM) is often preferred as it results in smaller numbers, but any common multiple will work, although you might need to simplify the final answer more extensively.

Multiplication of fractions is straightforward: multiply straight across (numerator times numerator, denominator times denominator). Division of fractions involves multiplying the first fraction by the reciprocal of the second fraction.

Practicing various problems involving mixed operations with fractions helps build confidence and accuracy in finding the value of X or similar variables in future questions.