Question

Solve for a,a. Express your answer as a proper or improper fraction in simplest terms.
minus, start fraction, 3, divided by, 4, end fraction, equals, minus, one half, a, minus, one fifth
−
4
3
​
=
−
2
1
​
a−
5
1
​

Answer 1

To solve the equation

\[
-\frac{3}{4} = -\frac{1}{2} a - \frac{1}{5},
\]

we first isolate \(-\frac{1}{2} a\). We can do this by adding \(\frac{1}{5}\) to both sides of the equation:

\[
-\frac{3}{4} + \frac{1}{5} = -\frac{1}{2} a.
\]

Now we need to find a common denominator to combine the fractions on the left side. The least common multiple of 4 and 5 is 20. Now we'll convert each fraction:

\[
-\frac{3}{4} = -\frac{15}{20}, \quad \frac{1}{5} = \frac{4}{20}.
\]

Now we can combine the fractions:

\[
-\frac{15}{20} + \frac{4}{20} = -\frac{15 - 4}{20} = -\frac{11}{20}.
\]

So we have:

\[
-\frac{11}{20} = -\frac{1}{2} a.
\]

Next, we want to get rid of the negative signs. We'll multiply both sides by -1:

\[
\frac{11}{20} = \frac{1}{2} a.
\]

Now to solve for \(a\), we can multiply both sides by 2:

\[
a = 2 \cdot \frac{11}{20} = \frac{22}{20}.
\]

Now we simplify \(\frac{22}{20}\):

\[
a = \frac{22 \div 2}{20 \div 2} = \frac{11}{10}.
\]

So the solution for \(a\) is

\[
\boxed{\frac{11}{10}}.
\]