Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers.
We need to simplify the given expression. Let's start by simplifying the inside of the parenthesis on the left part of the expression. \begin{align*}
((5p+1)&-2p\cdot4)(3)+(4-1\div3)(6p-9)\\
&=(5p+1-8p)(3)+(4-1\div3)(6p-9)\\
&=(-3p+1)(3)+(4-1\div3)(6p-9)
\end{align*} Next we can distribute the 3 to get $$3\cdot(-3p)+3\cdot1+(4-1\div3)(6p-9),$$ which equals $-9p+3+(4-1\div3)(6p-9)$ . The left part looks simplified so now we can focus on the right part. Let's subtract what's in the left parentheses then distribute. \begin{align*}
-9p+3+(4-1\div3)(6p-9)&=-9p+3+(\frac{4\cdot3}{3}-\frac{1}{3})(6p-9)\\
&=-9p+3+\frac{11}{3}(6p-9)\\
&=-9p+3+\frac{11}{3}\cdot6p-\frac{11}{3}\cdot9\\
&=-9p+3+\frac{11\cdot6p}{3}-\frac{11\cdot9}{3}\\
&=-9p+3+\frac{66p}{3}-\frac{99}{3}\\
&=-9p+3+\frac{3\cdot22p}{3}-\frac{3\cdot33}{3}\\
&=-9p+3+22p-33\\
&=22p-9p+3-33\\
&=\boxed{13p-30}\\
\end{align*}
13p-30
Level 5
Prealgebra
Problem: Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To simplify the expression, first distribute the terms in each part of the expression. Start with the first part: 

1. Expand $((5p + 1) - 2p \cdot 4)(3)$.
2. Calculate $2p \cdot 4$ to get $8p$.
3. Substitute this back into the expression to get $(5p + 1 - 8p)(3)$, which simplifies to $(-3p + 1)(3)$.
4. Distribute the 3: $-9p + 3$.

Next, simplify the second part: 

1. Expand $(4 - 1 \div 3)(6p - 9)$.
2. Calculate $1 \div 3$ to get $\frac{1}{3}$.
3. Substitute this back into the expression to get $(4 - \frac{1}{3})(6p - 9)$.
4. Convert 4 to a fraction: $4 = \frac{12}{3}$, so $\frac{12}{3} - \frac{1}{3} = \frac{11}{3}$.
5. Now, distribute $\frac{11}{3}$: $\frac{11}{3}(6p - 9)$.
6. Calculate $\frac{11}{3} \cdot 6p = \frac{66p}{3} = 22p$ and $\frac{11}{3} \cdot -9 = -\frac{99}{3} = -33$.

Now combine the results from both parts:

1. Combine $-9p + 3$ and $22p - 33$.
2. For the $p$ terms: $-9p + 22p = 13p$.
3. For the constant terms: $3 - 33 = -30$.

Thus, the simplified expression is $13p - 30$. The final answer is $13p - 30$.<|END_OF_LATENT|> The final answer is $13p-30$.