The initial temperature was 19.27 °C.
The guiding principle is the Law of Conservation of Energy: the sum of all the energy transfers must add up to zero.
The formula for the heat q gained or lost by a substance is
q = mCΔT
where
m = the mass of the substance.
C = its specific heat capacity.
ΔT = T_f - T_i = the change in temperature.
In this problem, there are two heat transfers.
Heat lost by gold + heat gained by water = 0
m _1C_1ΔT_1 + m_2c_2ΔT_2= 0
m_1 = 13.5 g; C_1 = 0.130 J·°C^(-1)g^(-1); ΔT_1 = T_f – T_i = 20.00 °C – 125.0 °C = -105.0 °C
m_2 = 60.0 g; C_2 = 4.184 J·°C^(-1)g^(-1); ΔT_2 = ?
q_1 = m_1C_1ΔT_1 = 13.5 g × 0.130 J·°C^(-1)g^(-1) × -105.0 °C = -184.3 J
q_2 = m_2C_2ΔT_2 = 60.0 g × 4.184 J·°C^(-1)g^(-1) × ΔT_2
= 251.0 ΔT_2 J·°C^(-1)
q_1+ q_2 = -184.3 J + 251.0 ΔT_2 J·°C^(-1) = 0
251.0 ΔT_2 °C^(-1) = 184.3
ΔT_2 = 184.3/251.0 °C^(-1) = 0.734°C
ΔT_2 = T_f - T_i = 20.00 °C -T_i = 0.734 °C
T_i = 20.00 °C – 0.734 °C = 19.27 °C