Arithmetic

Arithmetic is the fundamental branch of mathematics dealing with numbers and the basic operations: addition, subtraction, multiplication, and division. Questions involve performing these operations, understanding number properties (like integers, fractions, decimals), and solving related word problems.

• Question

  Over the course of a year, the price of apples increased twice by `50%` each time, and before Rosh Hashanah, they were discounted by `50%`. What is the new price of a kilogram of apples if, at the beginning of the year, they cost `8` shekels per kilogram?

  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages

• Question

  Every student in the gymnasium knows at least one of the following two ancient languages: Latin or Ancient Greek. It is known that `85%` of all students know Latin, and `75%` know Greek. What is the percentage of students who know both languages?

  Topics:

  Set Theory Arithmetic -> Percentages

• Question

  The company "The Diligent Builder" is engaged in stockpiling trees in a magical forest in Canada. A nature protection association called "The Green Avenger" wants to protect the forest and opposes the company's activity. As a result, the company's CEO said the following sentence:

  "`99%` of the trees in the forest are maple trees. In the coming year, we are going to cut down only maple trees, and as a result, the percentage of maple trees in the forest in a year will become `98%`."

  What percentage of the trees in the forest do the Diligent Builders intend to cut down?

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  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages Logic -> Reasoning / Logic -> Paradoxes

• The Orange Path

  In the illustration, there is an orange path surrounding a blue square. The area of the path is 44% of the area of the square.
  What is the width of the orange path as a percentage relative to the side length of the blue square?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems Arithmetic -> Percentages
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 3

• THE TWO AEROPLANES

  A man recently bought two aeroplanes, but afterwards found that they would not answer the purpose for which he wanted them. So he sold them for £`600` each, making a loss of `20` per cent, on one machine and a profit of `20` per cent, on the other. Did he make a profit on the whole transaction, or a loss? And how much?
  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 9

• 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.

  Topics:

  Algebra -> Equations Algebra -> Word Problems Arithmetic -> Percentages
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 26

• Question

  Avi lives in apartment number `45`. On which floor is the apartment located if there are `2` apartments on the first floor of his building, and `4` apartments on every other floor? (Note: The building does not have a floor `0`)

  Topics:

  Algebra -> Word Problems Arithmetic -> Division with Remainder
  Sources:
  • problems.ru web site

• Question

  Find all natural numbers with the following property: when divided by 7, their remainder is equal to their quotient.

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Arithmetic -> Division with Remainder

• Question

  Prove that among five integers, it is possible to choose two whose difference is divisible by `4`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Combinatorics -> Pigeonhole Principle Number Theory -> Division -> Parity (Even/Odd) Arithmetic -> Division with Remainder

• Question

  In Ella's building, there are three entrances and `13` floors (from `0` to `12`, and the apartments are on every floor except the ground floor). In each entrance, on each floor, there are five apartments. Ella lives in apartment number `73`. In which entrance and on which floor does she live?

  Topics:

  Algebra -> Word Problems Number Theory -> Division Arithmetic -> Division with Remainder