Qn: "A test-tube filled with some water is placed in a beaker containing some ice. If salt is added to the ice, the water in the test-tube will freeze. But I thought salt will lower the freezing point of water and hence the ice in the beaker will melt? Isn’t that the reason why we sprinkle salt on snow on a roadway in winter to melt the ice? If the ice in the beaker will melt, how can the water in the test-tube became frozen?"
Short Ans: salt lowers melting point of ice, so the ice melts. Ice melting absorbs thermal energy and so the temperature drops. Therefore the water in the test-tube gets frozen.
Longer answer: It is a common error to confuse the word "melting" with increasing temperature. But melting of ice actually absorbs thermal energy from its surroundings. This is analogous to water evaporating. Water evaporating absorbs heat, and it is the mechanism of how we cool down when we perspire and dogs pant.
So there are actually 2 steps as to why salt reduces the temperature of ice. First, salt reduces the melting point of ice. This causes the ice to melt. Then, the melting of the ice reduces the temperature of the ice due to it absorbing the thermal energy.