If `(13)/(16)x-(3)/(8)=(2)/(3)+(3)/(10)`, what is the value of x?

Options

A. `(17)/(8)`
B. `(9)/(4)`
C. `(8)/(5)`
D. `(17)/(10)`

Answer

To solve the equation \(\frac{13}{16}x - \frac{3}{8} = \frac{2}{3} + \frac{3}{10}\), we will follow these steps:

### Step 1: Simplify the Right Side
First, we need to find a common denominator for the fractions on the right side, which are \(\frac{2}{3}\) and \(\frac{3}{10}\). The least common multiple (LCM) of 3 and 10 is 30.

\[ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \]
\[ \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \]

Now, we can add these two fractions:

\[ \frac{20}{30} + \frac{9}{30} = \frac{29}{30} \]

### Step 2: Rewrite the Equation
Now we can rewrite the original equation with the simplified right side:

\[ \frac{13}{16}x - \frac{3}{8} = \frac{29}{30} \]

### Step 3: Eliminate the Fraction on the Left Side
To eliminate the fraction on the left side, we can find a common denominator for \(\frac{13}{16}x\) and \(-\frac{3}{8}\). The LCM of 16 and 8 is 16.

Rewriting \(-\frac{3}{8}\) with a denominator of 16:

\[ -\frac{3}{8} = -\frac{3 \times 2}{8 \times 2} = -\frac{6}{16} \]

Now the equation looks like this:

\[ \frac{13}{16}x - \frac{6}{16} = \frac{29}{30} \]

### Step 4: Combine the Left Side
Now we can combine the fractions on the left side:

\[ \frac{13}{16}x - \frac{6}{16} = \frac{13x - 6}{16} \]

So the equation becomes:

\[ \frac{13x - 6}{16} = \frac{29}{30} \]

### Step 5: Cross-Multiply
To eliminate the fractions, we can cross-multiply:

\[ 30(13x - 6) = 29 \times 16 \]

Calculating \(29 \times 16\):

\[ 29 \times 16 = 464 \]

So we have:

\[ 30(13x - 6) = 464 \]

### Step 6: Distribute and Simplify
Now distribute 30 on the left side:

\[ 390x - 180 = 464 \]

### Step 7: Isolate \(x\)
Add 180 to both sides:

\[ 390x = 464 + 180 \]
\[ 390x = 644 \]

Now divide both sides by 390:

\[ x = \frac{644}{390} \]

### Step 8: Simplify the Fraction
We can simplify \(\frac{644}{390}\) by finding the greatest common divisor (GCD) of 644 and 390, which is 2:

\[ x = \frac{644 \div 2}{390 \div 2} = \frac{322}{195} \]

### Final Answer
Thus, the value of \(x\) is:

\[ x = \frac{322}{195} \]

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