Finding the Value of x in the Equation

The problem asks us to find the value of the variable x in the given equation involving mixed numbers.

The equation is:

\(5 \frac{3}{4} + x + 2 \frac{1}{2} = 10 \frac{1}{8}\)

Step-by-Step Solution

1. Convert Mixed Numbers to Improper Fractions

It's often easier to work with improper fractions when solving equations. Let's convert each mixed number:

• \(5 \frac{3}{4} = \frac{(5 \times 4) + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}\)
• \(2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}\)
• \(10 \frac{1}{8} = \frac{(10 \times 8) + 1}{8} = \frac{80 + 1}{8} = \frac{81}{8}\)

Substituting these into the equation, we get:

\(\frac{23}{4} + x + \frac{5}{2} = \frac{81}{8}\)

2. Combine Known Fractions on the Left Side

Let's add the fractions \(\frac{23}{4}\) and \(\frac{5}{2}\). To add fractions, they must have a common denominator. The least common multiple of 4 and 2 is 4.

Convert \(\frac{5}{2}\) to a fraction with a denominator of 4:

\(\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}\)

Now add the fractions:

\(\frac{23}{4} + \frac{10}{4} = \frac{23 + 10}{4} = \frac{33}{4}\)

The equation now becomes:

\(\frac{33}{4} + x = \frac{81}{8}\)

3. Isolate x

To find x, we need to subtract \(\frac{33}{4}\) from both sides of the equation:

\(x = \frac{81}{8} - \frac{33}{4}\)

4. Subtract the Fractions

To subtract the fractions, they need a common denominator. The least common multiple of 8 and 4 is 8.

Convert \(\frac{33}{4}\) to a fraction with a denominator of 8:

\(\frac{33}{4} = \frac{33 \times 2}{4 \times 2} = \frac{66}{8}\)

Now subtract:

\(x = \frac{81}{8} - \frac{66}{8} = \frac{81 - 66}{8} = \frac{15}{8}\)

5. Convert the Improper Fraction Back to a Mixed Number

The value of x is \(\frac{15}{8}\). To express this as a mixed number, divide the numerator (15) by the denominator (8):

\(15 \div 8 = 1\) with a remainder of \(15 - (1 \times 8) = 15 - 8 = 7\).

So, \(\frac{15}{8}\) as a mixed number is \(1 \frac{7}{8}\).

Thus, the value of x is \(1 \frac{7}{8}\).

Verification (Optional)

We can check our answer by substituting \(x = 1 \frac{7}{8}\) back into the original equation.

\(5 \frac{3}{4} + 1 \frac{7}{8} + 2 \frac{1}{2}\)

Using improper fractions: \(\frac{23}{4} + \frac{15}{8} + \frac{5}{2}\)

Find a common denominator, which is 8:

\(\frac{23 \times 2}{4 \times 2} = \frac{46}{8}\)

\(\frac{5 \times 4}{2 \times 4} = \frac{20}{8}\)

Summing the fractions: \(\frac{46}{8} + \frac{15}{8} + \frac{20}{8} = \frac{46 + 15 + 20}{8} = \frac{81}{8}\)

Convert back to a mixed number: \(\frac{81}{8} = 10 \frac{1}{8}\).

This matches the right side of the original equation, confirming our value for x is correct.

The value of x is \(1 \frac{7}{8}\).