Algebra

Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.

Algebraic Techniques Equations Inequalities Word Problems Sequences

• MOTHER AND DAUGHTER

  "Mother, I wish you would give me a bicycle," said a girl of twelve the other day.

  "I do not think you are old enough yet, my dear," was the reply. "When I am only three times as old as you are you shall have one."

  Now, the mother's age is forty-five years. When may the young lady expect to receive her present?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 45

• MARY AND MARMADUKE

  Marmaduke: "Do you know, dear, that in seven years' time our combined ages will be sixty-three years?"

  Mary: "Is that really so? And yet it is a fact that when you were my present age you were twice as old as I was then. I worked it out last night."

  Now, what are the ages of Mary and Marmaduke?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 46

• ROVER'S AGE

  "Now, then, Tommy, how old is Rover?" Mildred's young man asked her brother.

  "Well, five years ago," was the youngster's reply, "sister was four times older than the dog, but now she is only three times as old."

  Can you tell Rover's age?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 47

• CONCERNING TOMMY'S AGE

  Tommy Smart was recently sent to a new school. On the first day of his arrival the teacher asked him his age, and this was his curious reply: "Well, you see, it is like this. At the time I was born—I forget the year—my only sister, Ann, happened to be just one-quarter the age of mother, and she is now one-third the age of father." "That's all very well," said the teacher, "but what I want is not the age of your sister Ann, but your own age." "I was just coming to that," Tommy answered; "I am just a quarter of mother's present age, and in four years' time I shall be a quarter the age of father. Isn't that funny?"

  This was all the information that the teacher could get out of Tommy Smart. Could you have told, from these facts, what was his precise age? It is certainly a little puzzling.

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 48

• NEXT-DOOR NEIGHBOURS

  There were two families living next door to one another at Tooting Bec—the Jupps and the Simkins. The united ages of the four Jupps amounted to one hundred years, and the united ages of the Simkins also amounted to the same. It was found in the case of each family that the sum obtained by adding the squares of each of the children's ages to the square of the mother's age equalled the square of the father's age. In the case of the Jupps, however, Julia was one year older than her brother Joe, whereas Sophy Simkin was two years older than her brother Sammy. What was the age of each of the eight individuals?
  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 49

• THE BAG OF NUTS

  Three boys were given a bag of nuts as a Christmas present, and it was agreed that they should be divided in proportion to their ages, which together amounted to `17 1/2` years. Now the bag contained `770` nuts, and as often as Herbert took four Robert took three, and as often as Herbert took six Christopher took seven. The puzzle is to find out how many nuts each had, and what were the boys' respective ages.

  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Number Theory -> Division
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 50

• HOW OLD WAS MARY?

  Here is a funny little age problem, by the late Sam Loyd, which has been very popular in the United States. Can you unravel the mystery?The combined ages of Mary and Ann are forty-four years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old is Mary? That is all, but can you work it out? If not, ask your friends to help you, and watch the shadow of bewilderment creep over their faces as they attempt to grip the intricacies of the question.
  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 51

• WHAT WAS THE TIME?

  "I say, Rackbrane, what is the time?" an acquaintance asked our friend the professor the other day. The answer was certainly curious.

  "If you add one quarter of the time from noon till now to half the time from now till noon to-morrow, you will get the time exactly."

  What was the time of day when the professor spoke?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 57

• A TIME PUZZLE

  How many minutes is it until six o'clock if fifty minutes ago it was four times as many minutes past three o'clock?
  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 58

• 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?
  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Motion Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 59