A train running at a uniform speed passes a bridge $275$ m long in $15$ seconds and another bridge $425$ m long in $21$ seconds. Find the length of the train.

Let the length of the train be L and its speed be v. Using the formula distance equals speed multiplied by time, the first condition gives L plus 275 equals 15 times v, and the second condition gives L plus 425 equals 21 times v. Subtract the first equation from the second equation to eliminate L, resulting in 150 equals 6 times v, so the speed v is 25 m/s. Substitute v back into the first equation to get L plus 275 equals 15 times 25, which gives L equals 375 minus 275, resulting in a train length of 100 m.

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