Simplify : `2/3 (2a-3b) -3/4 (2a +5/3 b)`

Answer

To simplify the expression \( \frac{2}{3} (2a - 3b) - \frac{3}{4} (2a + \frac{5}{3} b) \), we will follow these steps:

### Step 1: Distribute the coefficients
Distributing \( \frac{2}{3} \) and \( -\frac{3}{4} \) to the terms inside the parentheses:

\[ \frac{2}{3} \cdot 2a - \frac{2}{3} \cdot 3b - \frac{3}{4} \cdot 2a - \frac{3}{4} \cdot \frac{5}{3} b \]

### Step 2: Calculate each term
Calculating each term:

1. \( \frac{2}{3} \cdot 2a = \frac{4}{3} a \)
2. \( -\frac{2}{3} \cdot 3b = -2b \)
3. \( -\frac{3}{4} \cdot 2a = -\frac{3}{2} a \)
4. \( -\frac{3}{4} \cdot \frac{5}{3} b = -\frac{5}{4} b \)

Now we can rewrite the expression:

\[ \frac{4}{3} a - 2b - \frac{3}{2} a - \frac{5}{4} b \]

### Step 3: Combine like terms
Now, we will combine the terms involving \( a \) and the terms involving \( b \).

For \( a \):
\[ \frac{4}{3} a - \frac{3}{2} a \]

To combine these, we need a common denominator. The least common multiple of 3 and 2 is 6.

Rewriting the fractions:
\[ \frac{4}{3} = \frac{8}{6} \quad \text{and} \quad \frac{3}{2} = \frac{9}{6} \]

Now combine:
\[ \frac{8}{6} a - \frac{9}{6} a = -\frac{1}{6} a \]

For \( b \):
\[ -2b - \frac{5}{4} b \]

Again, we need a common denominator. The least common multiple of 1 and 4 is 4.

Rewriting \( -2b \):
\[ -2b = -\frac{8}{4} b \]

Now combine:
\[ -\frac{8}{4} b - \frac{5}{4} b = -\frac{13}{4} b \]

### Step 4: Write the final simplified expression
Putting it all together, we have:
\[ -\frac{1}{6} a - \frac{13}{4} b \]

### Final Answer:
\[ -\frac{1}{6} a - \frac{13}{4} b \]