Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \: (1) \: \: Gain \: \% = 8 \: \% \qquad \: \\ \\& \qquad \:\sf \: (2) \: \: SP \: = \: \$ \: 559 \\ \\& \qquad \:\sf \: (3) \: \: CP \: = \: \$ \: 16\end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given that,
[tex]\qquad\sf \:CP = \$ \: 345 \\ \\ [/tex]
[tex]\qquad\sf \:SP = \$ \: 372.60 \\ \\ [/tex]
Now,
[tex]\qquad\sf\implies \sf \:SP > CP \\ \\ [/tex]
So, there is gain in this transaction.
So,
[tex]\qquad\sf \:Gain\% = \dfrac{SP - CP}{CP} \times 100\% \\ \\ [/tex]
So, on substituting the values, we get
[tex]\qquad\sf \:Gain\% = \dfrac{372.60 - 345}{345} \times 100\% \\ \\ [/tex]
[tex]\qquad\sf \:Gain\% = \dfrac{27.60}{345} \times 100\% \\ \\ [/tex]
[tex]\qquad\sf \:Gain\% = \dfrac{2760}{345} \% \\ \\ [/tex]
[tex]\qquad\sf\implies \bf \:Gain \: \% = 8 \: \% \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Given that,
[tex]\qquad\sf \:CP = \$ \: 645 \\ \\ [/tex]
[tex]\qquad\sf \:Loss = 13\dfrac{1}{3}\% = \dfrac{40}{3}\% \\ \\ [/tex]
So,
[tex]\qquad\sf \:SP = \dfrac{(100 - Loss\%) \times CP}{100} \\ \\ [/tex]
[tex]\qquad\sf \:SP = \dfrac{\left(100 - \dfrac{40}{3} \right) \times 645}{100} \\ \\ [/tex]
[tex]\qquad\sf \:SP = \dfrac{\left( \dfrac{300 - 40}{3} \right) \times 645}{100} \\ \\ [/tex]
[tex]\qquad\sf \:SP = \dfrac{260 \times 645}{300} \\ \\ [/tex]
[tex]\qquad\sf \:SP = \dfrac{26 \times 645}{30} \\ \\ [/tex]
[tex]\qquad\sf \:SP = \dfrac{13 \times 645}{15} \\ \\ [/tex]
[tex]\qquad\sf \:SP = 13 \times 43\\ \\ [/tex]
[tex]\qquad\sf\implies \bf \:SP \: = \$ \: 559\\ \\ [/tex]
[tex]\large\underline{\sf{Solution-3}}[/tex]
Given that,
[tex]\qquad\sf \:SP = \$ \: 34.40 \\ \\ [/tex]
[tex]\qquad\sf \:Gain\% = 7\dfrac{1}{2}\% = \dfrac{15}{2} \% \\ \\ [/tex]
So,
[tex]\qquad\sf \:CP = \dfrac{100 \times SP}{100 + Gain\%} \\ \\ [/tex]
[tex]\qquad\sf \:CP = \dfrac{100 \times 34.40}{100 + \dfrac{15}{2} } \\ \\ [/tex]
[tex]\qquad\sf \:CP = \dfrac{ 3440}{ \dfrac{200 + 15}{2} } \\ \\ [/tex]
[tex]\qquad\sf \:CP = \dfrac{3440 \times 2}{215} \\ \\ [/tex]
[tex]\qquad\sf\implies \bf \:CP \: = \: \$ \: 16 \\ \\ [/tex]