When cold water at 0 ˚C is added to the warm water, some energy is passed from the warm to the cold water. As a result, the temperature of both amounts of water starts to change immediately and eventually becomes the same.
On the contrary, in case of adding the same amount of ice at 0 ˚C, some energy must be spend to melt the ice first. During the melting, the temperature of the ice remains the same – 0 ˚C, while the temperature of the hot water decreases. When all ice is melted away, the same amount of cold water at 00C remains as in the initial situation of the previous case. But now the temperature of the surrounding warm water becomes less, for some heat energy has been already consumed by the ice.
Thus, is better to use ice because it can consume more heat energy then the equal mass of water at 0 ˚C.
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