How do you solve #(7k/8) - (3/4) - (5k/16) = (3/8)#?

Question

Avery Alexander · Accepted Answer

#color(purple)( k = -14# #(7k/8) - (3/4) - (5k/16) = (3/8)# Solving the mixed fraction first: #(56 + k)/8 - (3/4) - ((80 +k))/16 = (3/8)# Here, the L.C.M is #16# #((56 + k). 2)/cancel16 - (12/cancel16) - ((80 +k))/cancel16 = (6/cancel16)# # 112 +2k - 12 - 80 - k = 6# #20 + k = 6# #color(purple)( k = -14#

Avery Alexander · Answer

undefined To solve the equation $\frac{7k}{8} - \frac{3}{4} - \frac{5k}{16} = \frac{3}{8}$, follow these steps:

1. **Find a common denominator for the fractions involved:** The denominators here are 8, 4, 16, and 8. The least common denominator (LCD) is 16.

2. **Express all fractions with the common denominator (16):**
   - $\frac{7k}{8} = \frac{14k}{16}$ (by multiplying both numerator and denominator by 2),
   - $\frac{3}{4} = \frac{12}{16}$ (by multiplying both numerator and denominator by 4),
   - $\frac{5k}{16}$ remains unchanged since its denominator is already 16,
   - $\frac{3}{8} = \frac{6}{16}$ (by multiplying both numerator and denominator by 2).

3. **Rewrite the equation with these equivalents:**
   $$\frac{14k}{16} - \frac{12}{16} - \frac{5k}{16} = \frac{6}{16}$$

4. **Combine like terms:**
   - Combine the k terms: $\frac{14k}{16} - \frac{5k}{16} = \frac{9k}{16}$.
   - The equation now is $\frac{9k}{16} - \frac{12}{16} = \frac{6}{16}$.

5. **Solve for k:**
   - First, isolate the k term by adding $\frac{12}{16}$ to both sides of the equation:
     $$\frac{9k}{16} = \frac{6}{16} + \frac{12}{16}$$
     $$\frac{9k}{16} = \frac{18}{16}$$
   - Then, solve for k by multiplying both sides by the reciprocal of $\frac{9}{16}$ (which is $\frac{16}{9}$):
     $$k = \frac{18}{16} \times \frac{16}{9}$$
     $$k = \frac{18}{9}$$
     $$k = 2$$

So, $k = 2$.