Solve `3/4n+2/5n=46` for n

Options

A. 2
B. 13.5
C. 16.7
D. 40

Answer

To solve the equation \( \frac{3}{4}n + \frac{2}{5}n = 46 \) for \( n \), we can follow these steps:

### Step 1: Combine like terms
First, we can factor out \( n \) from the left side of the equation:
\[ n \left( \frac{3}{4} + \frac{2}{5} \right) = 46 \]

### Step 2: Find a common denominator
To combine the fractions \( \frac{3}{4} \) and \( \frac{2}{5} \), we need to find a common denominator. The least common multiple (LCM) of 4 and 5 is 20.

### Step 3: Rewrite the fractions
Now, we can rewrite each fraction with a denominator of 20:
\[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \]
\[ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \]

### Step 4: Combine the fractions
Now we can add the two fractions:
\[ \frac{15}{20} + \frac{8}{20} = \frac{15 + 8}{20} = \frac{23}{20} \]

### Step 5: Substitute back into the equation
Now we substitute back into the equation:
\[ n \left( \frac{23}{20} \right) = 46 \]

### Step 6: Isolate \( n \)
To isolate \( n \), we can multiply both sides by the reciprocal of \( \frac{23}{20} \):
\[ n = 46 \times \frac{20}{23} \]

### Step 7: Simplify the right side
Now we can simplify:
\[ n = \frac{46 \times 20}{23} \]
Calculating \( 46 \div 23 = 2 \):
\[ n = 2 \times 20 = 40 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{40} \]

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