Add the expressions 4 minus two-thirds b + one-fourth a and One-half a + one-sixth b minus 7. What is the simplified sum?

Question

Negative one-sixth a + StartFraction 5 over 12 EndFraction b minus 3
Three-fourths a minus one-half b minus 3
Three-fourths a + StartFraction 5 over 6 EndFraction b minus 3
StartFraction 5 over 12 EndFraction a minus StartFraction 1 over 6 EndFraction b minus 3

Answer 1

To find the sum of the expressions \( 4 - \frac{2}{3}b + \frac{1}{4}a \) and \( \frac{1}{2}a + \frac{1}{6}b - 7 \), we will combine like terms.

Let's first write both expressions clearly:

1. \( 4 - \frac{2}{3}b + \frac{1}{4}a \)
2. \( \frac{1}{2}a + \frac{1}{6}b - 7 \)

Now, we can combine these expressions:

\[ (4 - 7) + \left(\frac{1}{4}a + \frac{1}{2}a\right) + \left(-\frac{2}{3}b + \frac{1}{6}b\right) \]

Let’s simplify each component:

1. Constant terms: \[ 4 - 7 = -3 \]

2. For the \(a\) terms: \[ \frac{1}{4}a + \frac{1}{2}a = \frac{1}{4}a + \frac{2}{4}a = \frac{3}{4}a \]

3. For the \(b\) terms: To combine \(-\frac{2}{3}b\) and \(\frac{1}{6}b\), we need a common denominator. The least common denominator of 3 and 6 is 6.

   • Convert \(-\frac{2}{3}b\) to sixths: \[ -\frac{2}{3}b = -\frac{4}{6}b \]
   • Now combine: \[ -\frac{4}{6}b + \frac{1}{6}b = -\frac{4 - 1}{6}b = -\frac{3}{6}b = -\frac{1}{2}b \]

Now, combine all simplified parts together:

\[ -\frac{3}{4} + \frac{3}{4}a - \frac{1}{2}b \]

Thus, the final combined expression is:

\[ \frac{3}{4}a - \frac{1}{2}b - 3 \]

So the simplified sum of the expressions is:

\[ \frac{3}{4}a - \frac{1}{2}b - 3 \]

None of the outlined answer choices match the final answer. Let’s double-check in questions or make corrections.