$$\frac{2p+1}{3}+\frac{p-1}{2}=4\frac{1}{2}$$
$p=4$
$$2\left(2p+1\right)+3\left(p-1\right)=3\left(4\times 2+1\right)$$
$$4p+2+3\left(p-1\right)=3\left(4\times 2+1\right)$$
$$4p+2+3p-3=3\left(4\times 2+1\right)$$
$$7p+2-3=3\left(4\times 2+1\right)$$
$$7p-1=3\left(4\times 2+1\right)$$
$$7p-1=3\left(8+1\right)$$
$$7p-1=3\times 9$$
$$7p-1=27$$
$$7p=27+1$$
$$7p=28$$
$$p=\frac{28}{7}$$
$$p=4$$
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