Algebra, Word Problems

Word problems present mathematical challenges in a narrative or real-world context. Solving them requires translating the text into mathematical equations or expressions and then applying appropriate mathematical techniques. These can span arithmetic, algebra, geometry, etc.

Motion Problems Solving Word Problems "From the End" / Working Backwards

  • PHEASANT-SHOOTING

    A Cockney friend, who is very apt to draw the long bow, and is evidently less of a sportsman than he pretends to be, relates to me the following not very credible yarn:—

    "I've just been pheasant-shooting with my friend the duke. We had splendid sport, and I made some wonderful shots. What do you think of this, for instance? Perhaps you can twist it into a puzzle. The duke and I were crossing a field when suddenly twenty-four pheasants rose on the wing right in front of us. I fired, and two-thirds of them dropped dead at my feet. Then the duke had a shot at what were left, and brought down three-twenty-fourths of them, wounded in the wing. Now, out of those twenty-four birds, how many still remained?"

    It seems a simple enough question, but can the reader give a correct answer?

    Sources:

  • THE GARDENER AND THE COOK

    A correspondent, signing himself "Simple Simon," suggested that I should give a special catch puzzle in the issue of The Weekly Dispatch for All Fools' Day, `1900`. So I gave the following, and it caused considerable amusement; for out of a very large body of competitors, many quite expert, not a single person solved it, though it ran for nearly a month.

     

    "The illustration is a fancy sketch of my correspondent, 'Simple Simon,' in the act of trying to solve the following innocent little arithmetical puzzle. A race between a man and a woman that I happened to witness one All Fools' Day has fixed itself indelibly on my memory. It happened at a country-house, where the gardener and the cook decided to run a race to a point `100` feet straight away and return. I found that the gardener ran `3` feet at every bound and the cook only `2` feet, but then she made three bounds to his two. Now, what was the result of the race?"

    A fortnight after publication I added the following note: "It has been suggested that perhaps there is a catch in the 'return,' but there is not. The race is to a point `100` feet away and home again—that is, a distance of `200` feet. One correspondent asks whether they take exactly the same time in turning, to which I reply that they do. Another seems to suspect that it is really a conundrum, and that the answer is that 'the result of the race was a (matrimonial) tie.' But I had no such intention. The puzzle is an arithmetical one, as it purports to be."

    Sources:

  • Question

    The journey from home to school takes Yossi `20` minutes. One day, when Yossi was already on his way to school, he remembered that he had forgotten his pen at home. If he continues to school now, he will arrive `3` minutes before the bell. And if he goes back home to get the pen, he will be `7 ` minutes late for class. What fraction of the way to school had Yossi traveled when he remembered that he didn't have a pen?

    Note: Yossi walks at a constant speed the entire time.

    Sources:

  • The Restless Fly

    Cities A and B are 300 kilometers apart. Two cyclists start simultaneously from A and B, heading towards each other. Their speeds are constant and equal to `30` km/h and `20` km/h, respectively. At the same moment, a fly departs from city A, flying at a speed of `100` km/h. The fly overtakes the first cyclist and flies until it meets the cyclist who left from city B. The moment the fly meets the cyclist, it immediately turns around and flies back until it meets the first cyclist again, and then it turns around again, and so on, until the cyclists meet. How many kilometers did the fly travel?

  • Question

    At `12:00`, the two hands of a clock coincide. In exactly how much time will they coincide again? How many times within a 24-hour period will they coincide?

  • Question

    How long will it take a train traveling at a speed of `60` km/h and with a length of `50` meters to pass through a tunnel with a length of `50` meters?

  • Question

    Nina and Meir left their home at the same time and went to visit their grandmother. The route from their home to their grandmother's house is 3 km long, and benches are located along it.

    Nina sits on every bench along the way to eat one cookie. She eats each cookie for the same whole number of minutes. Meir also stops and sits on every bench to eat one cookie.

    It takes Meir twice as long as Nina to eat a cookie. It is also known that Nina walks at a speed of 3 km/h, and Meir at a speed of 4 km/h.

    It turned out that Meir and Nina arrived at their grandmother's house at the same time.

    How many benches were along the way? Find all the possibilities and justify your answer.

    Sources:

  • Horse, Camel, and Donkey in a Circle

    On a circular track of length 92, there is a horse, a donkey, and a camel that start from the same point and begin walking along the circle.
    The horse and the camel walk counterclockwise, and the donkey walks clockwise. The camel's speed is 1 meter per second, the donkey's is 3, and the horse's is 5.

    In how many seconds will all three meet again?

    Note: The meeting does not necessarily have to be at the starting point

    Sources:

  • Clock Angle

    How many minutes after 7:00 will the angle between the hour and minute hands be exactly one degree for the first time?
    Note: The clock hands move continuously at a constant speed.

    Sources:

  • A PUZZLING WATCH

    A friend pulled out his watch and said, "This watch of mine does not keep perfect time; I must have it seen to. I have noticed that the minute hand and the hour hand are exactly together every sixty-five minutes." Does that watch gain or lose, and how much per hour? Sources: