To solve the equation [tex]\(\frac{1}{2} a + \frac{1}{4} a + \frac{1}{8} a + 25 = a\)[/tex], let's proceed step-by-step.
1. Combine the terms involving [tex]\(a\)[/tex] on the left side of the equation:
[tex]\[
\frac{1}{2}a + \frac{1}{4}a + \frac{1}{8}a + 25 = a
\][/tex]
2. Find a common denominator for the fractions [tex]\(\frac{1}{2}\)[/tex], [tex]\(\frac{1}{4}\)[/tex], and [tex]\(\frac{1}{8}\)[/tex]. The common denominator is 8.
[tex]\[
\frac{1}{2}a = \frac{4}{8}a
\][/tex]
[tex]\[
\frac{1}{4}a = \frac{2}{8}a
\][/tex]
[tex]\[
\frac{1}{8}a = \frac{1}{8}a
\][/tex]
3. Now add these fractions:
[tex]\[
\frac{4}{8}a + \frac{2}{8}a + \frac{1}{8}a = \frac{7}{8}a
\][/tex]
So the equation becomes:
[tex]\[
\frac{7}{8}a + 25 = a
\][/tex]
4. Isolate the term involving [tex]\(a\)[/tex] on the left side by subtracting [tex]\(\frac{7}{8}a\)[/tex] from both sides of the equation:
[tex]\[
\frac{7}{8}a + 25 - \frac{7}{8}a = a - \frac{7}{8}a
\][/tex]
Simplifying, we get:
[tex]\[
25 = a - \frac{7}{8}a
\][/tex]
5. Combine the [tex]\(a\)[/tex] terms on the right side:
[tex]\[
25 = \frac{8}{8}a - \frac{7}{8}a
\][/tex]
Simplifying:
[tex]\[
25 = \frac{1}{8}a
\][/tex]
6. Solve for [tex]\(a\)[/tex] by multiplying both sides of the equation by 8:
[tex]\[
25 \times 8 = a
\][/tex]
[tex]\[
200 = a
\][/tex]
Thus, the solution to the equation [tex]\(\frac{1}{2}a + \frac{1}{4}a + \frac{1}{8}a + 25 = a\)[/tex] is:
[tex]\[
a = 200
\][/tex]
