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.

• The Sophisticated Task

  Hannah has a basket with `13` apples. Hannah wants to know the total weight of all these apples. Rachel has a digital scale, and she is willing to help Hannah, but only under the following conditions: In each weighing, Hannah can weigh exactly `2` apples, and the number of weighings cannot exceed `8`.

  Explain how, under these conditions, Hannah can know the total weight of the apples.

  Topics:

  Combinatorics -> Double Counting Algebra -> Equations Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Algorithm Theory -> Weighing
  Sources:
  • problems.ru web site

• Clock Tower

  A bell in a clock tower rings `3` times in `12` seconds. How long does it take to ring `6` times?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  There are three sacks of nuts. The first sack has `6` kilograms of nuts less than the other two sacks combined. The second sack has `10` kilograms of nuts less than the other two sacks combined.

  How many kilograms of nuts are in the third sack?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • problems.ru web site

• What is more worthwhile?

  What is more worthwhile – to receive a discount of `20%`, or to buy a gift voucher worth `120%` of its price?

  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages

• 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

  There are chairs with `4` legs and with `3` legs in a room. When people sat on all the chairs, there were `39` legs in the room (no one remained standing). How many chairs of each type are there in the room?

  Topics:

  Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Algebra -> Equations -> Diophantine Equations

• Coins and Cakes

  Miriam has coins of two shekels and of five shekels.

  If she pays only with two-shekel coins, she will be 60 shekels short of buying 4 cakes.

  If she pays only with five-shekel coins, she will be 60 shekels short of buying 5 cakes.

  In total, she is 60 shekels short of buying 6 cakes. How much does a cake cost?

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  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 2

• Question

  A box contains pencils in three colors: red, green, and blue, totaling `20` pencils. There are `6` times more blue pencils than green pencils. There are fewer red pencils than green pencils. How many red pencils are in the box?

  Topics:

  Arithmetic Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  A grasshopper can jump `80` centimeters forward or `50` centimeters backward. Can the grasshopper move away from its starting point in fewer than `7` jumps to a distance of exactly one meter and `70` cm?

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  Topics:

  Arithmetic Number Theory -> Chinese Remainder Theorem Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations

• Birds and Seeds

  Nine identical birds eat less than `1001` seeds for lunch, and ten such birds eat more than `1100` seeds for lunch. How many seeds does one bird eat for lunch?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division