Algebra, Inequalities

Inequalities are statements comparing two expressions using symbols like <, >, \le, \ge. This topic involves solving inequalities (linear, quadratic, absolute value), proving algebraic or geometric inequalities (e.g., AM-GM), and understanding their properties.

• 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

• AVERAGE SPEED

  In a recent motor ride it was found that we had gone at the rate of ten miles an hour, but we did the return journey over the same route, owing to the roads being more clear of traffic, at fifteen miles an hour. What was our average speed? Do not be too hasty in your answer to this simple little question, or it is pretty certain that you will be wrong.
  Topics:

  Algebra -> Inequalities -> Averages / Means Algebra -> Word Problems -> Motion Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 67

• THE TRUSSES OF HAY

  Farmer Tompkins had five trusses of hay, which he told his man Hodge to weigh before delivering them to a customer. The stupid fellow weighed them two at a time in all possible ways, and informed his master that the weights in pounds were `110, 112, 113, 114, 115, 116, 117, 118, 120`, and `121`. Now, how was Farmer Tompkins to find out from these figures how much every one of the five trusses weighed singly? The reader may at first think that he ought to be told "which pair is which pair," or something of that sort, but it is quite unnecessary. Can you give the five correct weights?
  Topics:

  Algebra -> Equations Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 101

• THE MINERS' HOLIDAY

  Seven coal-miners took a holiday at the seaside during a big strike. Six of the party spent exactly half a sovereign each, but Bill Harris was more extravagant. Bill spent three shillings more than the average of the party. What was the actual amount of Bill's expenditure?
  Topics:

  Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 125

• SLOW CRICKET

  In the recent county match between Wessex and Nincomshire the former team were at the wickets all day, the last man being put out a few minutes before the time for drawing stumps. The play was so slow that most of the spectators were fast asleep, and, on being awakened by one of the officials clearing the ground, we learnt that two men had been put out leg-before-wicket for a combined score of `19` runs; four men were caught for a combined score or `17` runs; one man was run out for a duck's egg; and the others were all bowled for `3` runs each. There were no extras. We were not told which of the men was the captain, but he made exactly `15` more than the average of his team. What was the captain's score?
  Topics:

  Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 388