For a given maturity, given three equally spaced option strikes $K_1,K_2,K_3$ a "butterfly" combination consists of shorting 2 of the middle strike calls and buying one each of the "wing" or lateral calls. This position has a positive cost i.e. $c_1+c_3-2 c_2 >=0$ (why? because it has a positive payoff for $S_T\approx K_2$ and zero payoff elsewhere).
In the example given we have $10+5-2*7=1>=0$
It can be shown (Gordon has already shown it above) that the same payoff can be obtained with puts: You short two of the middle strike puts and buy one each of the wing puts. By no arbitrage the cost $p_1+p_3-2 p_2$ will be the same as the cost with calls we found above.
However in general $p_1\ne c_1,p_2\ne c_2,p_3\ne c_3$. If we assume zero interest rates (as Zumba showed above) we will have $p_i=c_i-S_0+K_i$ instead (by Put Call Parity).
If the example given we have $p_1=10-61+55=4$, $p_2=7-61+60=6$, $p_3=5-61+65=9$. Notice that the call prices $10,7,5$ are decreasing with strike while the put prices $4,6,9$ are increasing in strike. Nevertheless $p_1+p_3-2 p_2=4+9-2*6=1$ is the same as the cost we found with calls. All as expected.
Hope this clarifies a few things. (Note that there is no need to reverse the sign of the positions, if we buy $c_i$ we also buy (not sell) $p_i$).
What if interest rates are non-zero? Then we have $p_i=c_i-S_0+PV(K_i)$. So
Because $K_2=(K_1+K_3)/2$ we have $2 PV(K_2)= PV(K_1)+PV(K_3)$. Simplifying the above we have that$$p_1+p_3-2 p_2=c_1+c_3-2 c_2$$
So the equal cost of the put butterfly and the call butterfly is true in general, for any level of interest rate.