Algebra, Equations

An equation is a statement that two mathematical expressions are equal. Solving an equation involves finding the values of variables that make the statement true. Questions cover various types: linear, quadratic, polynomial, rational, radical, and systems of equations.

Diophantine Equations

  • THE JUNIOR CLERK'S PUZZLE

    Two youths, bearing the pleasant names of Moggs and Snoggs, were employed as junior clerks by a merchant in Mincing Lane. They were both engaged at the same salary—that is, commencing at the rate of £`50` a year, payable half-yearly. Moggs had a yearly rise of £`10`, and Snoggs was offered the same, only he asked, for reasons that do not concern our puzzle, that he might take his rise at £`2, 10`s. half-yearly, to which his employer (not, perhaps, unnaturally!) had no objection.

    Now we come to the real point of the puzzle. Moggs put regularly into the Post Office Savings Bank a certain proportion of his salary, while Snoggs saved twice as great a proportion of his, and at the end of five years they had together saved £`268, 15`s. How much had each saved? The question of interest can be ignored.

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  • DOMESTIC ECONOMY

    Young Mrs. Perkins, of Putney, writes to me as follows: "I should be very glad if you could give me the answer to a little sum that has been worrying me a good deal lately. Here it is: We have only been married a short time, and now, at the end of two years from the time when we set up housekeeping, my husband tells me that he finds we have spent a third of his yearly income in rent, rates, and taxes, one-half in domestic expenses, and one-ninth in other ways. He has a balance of £`190` remaining in the bank. I know this last, because he accidentally left out his pass-book the other day, and I peeped into it. Don't you think that a husband ought to give his wife his entire confidence in his money matters? Well, I do; and—will you believe it?—he has never told me what his income really is, and I want, very naturally, to find out. Can you tell me what it is from the figures I have given you?"

    Yes; the answer can certainly be given from the figures contained in Mrs. Perkins's letter. And my readers, if not warned, will be practically unanimous in declaring the income to be—something absurdly in excess of the correct answer!

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  • THE COSTERMONGER'S PUZZLE

    "How much did yer pay for them oranges, Bill?"

    "I ain't a-goin' to tell yer, Jim. But I beat the old cove down fourpence a hundred."

    "What good did that do yer?"

    "Well, it meant five more oranges on every ten shillin's-worth."

    Now, what price did Bill actually pay for the oranges? There is only one rate that will fit in with his statements.

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  • MAMMA'S AGE

    Tommy: "How old are you, mamma?"

    Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years."

    Tommy: "That's a lot, isn't it? And how old are you, papa?"

    Papa: "Just six times as old as you, my son."

    Tommy: "Shall I ever be half as old as you, papa?"

    Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day."

    Tommy: "And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing——"

    Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache."

    Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma?

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  • THEIR AGES

    "My husband's age," remarked a lady the other day, "is represented by the figures of my own age reversed. He is my senior, and the difference between our ages is one-eleventh of their sum." Sources:

  • THE FAMILY AGES

    When the Smileys recently received a visit from the favourite uncle, the fond parents had all the five children brought into his presence. First came Billie and little Gertrude, and the uncle was informed that the boy was exactly twice as old as the girl. Then Henrietta arrived, and it was pointed out that the combined ages of herself and Gertrude equalled twice the age of Billie. Then Charlie came running in, and somebody remarked that now the combined ages of the two boys were exactly twice the combined ages of the two girls. The uncle was expressing his astonishment at these coincidences when Janet came in. "Ah! uncle," she exclaimed, "you have actually arrived on my twenty-first birthday!" To this Mr. Smiley added the final staggerer: "Yes, and now the combined ages of the three girls are exactly equal to twice the combined ages of the two boys." Can you give the age of each child? Sources:

  • MRS. TIMPKINS'S AGE

    Edwin: "Do you know, when the Timpkinses married eighteen years ago Timpkins was three times as old as his wife, and to-day he is just twice as old as she?"

    Angelina: "Then how old was Mrs. Timpkins on the wedding day?"

    Can you answer Angelina's question?

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  • A CENSUS PUZZLE

    Mr. and Mrs. Jorkins have fifteen children, all born at intervals of one year and a half. Miss Ada Jorkins, the eldest, had an objection to state her age to the census man, but she admitted that she was just seven times older than little Johnnie, the youngest of all. What was Ada's age? Do not too hastily assume that you have solved this little poser. You may find that you have made a bad blunder! Sources:

  • 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?

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  • 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?

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