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  1. 090Two trains start 100 km apart on the same track and head toward each other at 50 km/h each. A bird starts at the front of one train and flies back and forth between them at 75 km/h until the trains meet. How far does the bird fly?Logic and brainteasersCoreConsulting style brainteasersSales and trading

    Try it first

    Answer before you start drawing zig-zags.

    Show the worked solution

    75 km. The trains close the 100 km gap at a combined 100 km/h, so they meet after exactly one hour. The bird flies without stopping for that hour at 75 km/h, so it covers 75 km. You never need to track the individual legs, though they do add up: 60 km, then 12, then 2.4, each a fifth of the one before, summing to 75.

    What question should you ask first?

    If a friend walks a dog in the park for an hour and the dog runs at 10 km/h the whole time, the dog covers 10 km however it zig-zags between trees. Nobody measures each dash. Distance is speed times time, so find how long the bird flies and the shape of its path stops mattering. The trains settle the time: they approach each other at 50 plus 50, which is 100 km/h, and 100 km apart means one hour.

    Find when the trains meet, and the bird's zig-zag adds itself up0 km50 km100 km030 min1 hourTimeTrain A, 50 km/hTrain B, 50 km/hbird turnsbird, 75 km/hThe hard way: legsLeg 160 kmLeg 212 kmLeg 32.40 kmLeg 40.48 kmeach leg 1/5 of the last60 / (1 - 1/5) = 75 kmThe easy way75 km/h x 1 hour = 75 km
    The trains meet after one hour, so the bird flying at 75 km/h covers 75 km, and summing its legs of 60, 12 and 2.4 km and onward, each a fifth of the last, reaches the same 75 km by the long road.

    If the interviewer insists on the legs, how do you sum them?

    On the first leg the bird and the oncoming train close at 75 plus 50, which is 125 km/h, so they meet after 100 / 125 = 0.8 hours; the bird has flown 60 km. In that time the trains have closed 80 km, leaving 20 km between them. The next leg is the same problem a fifth the size: 12 km, then 2.4, and so on. Each leg is a fixed fraction of the last, which makes the total a geometric series that sums to 60 / (1 - 1/5) = 75 km.

    The relationship
    60+12+2.4+⋯=601−15=75 km60 + 12 + 2.4 + \dots = \frac{60}{1 - \tfrac{1}{5}} = 75 \text{ km}
    60the first leg in km
    1/5the ratio of each leg to the one before
    What it says in wordsA series where each term is a fixed fraction of the last sums to the first term over one minus the fraction.

    Why ask this in a banking interview? It is a test of whether you look for the simpler quantity before diving into the mechanics, which is what a good analyst does with a messy model: find the number that settles the question and compute that. Answer the short way, then offer the series as a check if there is time.

    Where candidates lose it

    Candidates who start computing the first leg often keep going, and by the third leg they are lost in fractions with the clock running. The interviewer is watching whether you pause to ask what actually determines the answer.

    The opposite loss is saying 75 instantly with no reason. Give the one line, combined speed 100 and so one hour, so the answer is audibly reasoned.

    What the interviewer asks next

    • How many times does the bird turn around?
    • If train A runs at 60 km/h and train B at 40, does the answer change?
    • Where in a model have you found a shortcut quantity that made a messy calculation unnecessary?