# Word problems

## Filling a Cistern: Three Pipes, No Numbers

(A new question of the week) Today we’ll look at a classic algebra word problem: Finding how long it takes to fill a cistern through two pipes, with a drain open. But usually these problems are given with specific numbers, as a simple exercise in algebra. What if it’s all variables? the discussion provides some …

## Maximum Volume of a Box: Two Interpretations

(A new question of the week) Often the hardest part of solving a problem is interpreting what it means. Math is precise; human language can be ambiguous, and assumptions can be hidden. Today, we look at a multi-variable calculus problem that looked enough like a classic single-variable maximization problem to fool the reader into not …

## A Rate Problem: Two Speeds, Two Ways

(A new question of the week) A question we got at the end of March asked about a standard kind of algebra word problem that can be solved in a couple very different ways. It illustrates several choices that can be made (both about the meaning of the problem and how to solve it), as …

## An Introduction to Trigonometry

(An archive question of the week) While I’m showing some recent explanations of basic trigonometry techniques, this is a good time to look at an even more basic explanation of the essentials of the subject for a beginner. Right triangle trigonometry Here is the question, from 2001: Trigonometry in a Nutshell I’m in 8th grade …

## Negating Logic Statements: How to Say “Not”

Last time, I started a series exploring aspects of the translation of English statements to or from formal logical terms and symbols, which will lead to discussions of converse and contrapositive, and eventually of logical arguments. We’ve looked at how to translate concepts of “or” (disjunction) and “if” (conditional); but our goals will also require …

## Translating Logic Statements

The next few posts will examine aspects of logic, both symbolic logic, and how we talk about theorems in general. We’ll start here with issues in interpreting the wording of logic, and some of the semantic difficulties we face. English isn’t logical. (Well, I suppose humans in general aren’t logical.) Which kind of OR? We’ll …

## Conditional Probability and Multiple Choice

(A new question of the week) A recent question about probability has ties to Venn diagrams, tables, and Bayes’ Theorem. Questions about answering multiple-choice questions are common; this one offers a twist that provided opportunity to discuss several important concepts. Here is the initial question, from August: On a multiple choice question, only one answer is …

## Venn Diagrams: Language Issues

(A new question of the week) I mentioned that we have had a number of questions related to Venn diagrams recently. Here I would like to show a couple of these, from a Philippine student. Even fluent English speakers can get confused in these problems; observing how a student new to the language misinterprets details …

## Venn Diagrams: Over the Top

(An archive question of the week) Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. A mixture of tools will make the work easier. Here it …

## Venn Diagram Puzzles: Logic and Ambiguity

Puzzles involving two or three properties of people or objects, commonly solved by Venn diagrams, are popular with teachers, and perhaps not so popular with students! Several have asked about them recently, leading me to catalog our past answers about them, to find the most useful examples to point to. Some of these are very …