Reasoning with Equations & Inequalities — Challenge

Before you solve anything, translate. Most of the difficulty in a word problem is turning English into algebra.

What these problems cover

These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

Mia, Tom and Ana play goalkeeper, striker and defender, one each. Mia is not the goalkeeper. The defender lives next door to Mia. The goalkeeper and Tom attend the same school. Who is the goalkeeper?

  1. Mia
  2. Tom
  3. Ana
  4. Cannot be determined
  5. Nobody
Worked answer

Ana

The goalkeeper and Tom attend the same school, so they are different people and Tom is not the goalkeeper.

Mia is not the goalkeeper either, which leaves only Ana.

As a check, the defender lives next door to Mia, so Mia is not the defender; Mia is therefore the striker and Tom the defender.

Problem 2

Ben, Cara and Dev are questioned about a broken window. Ben says "Dev did it." Cara says "I did not do it." Dev says "Ben is lying." Exactly one of the three is telling the truth. Who broke the window?

  1. Ben
  2. Cara
  3. Dev
  4. Cannot be determined
  5. Nobody
Worked answer

Cara

Test each possibility and count the true statements.

If Ben did it, then Cara's statement is true and Dev's is also true, which is two.

If Dev did it, then Ben's statement is true and Cara's is true, which is again two.

If Cara did it, Ben's statement is false, Cara's is false, and only Dev's is true. That is exactly one, so Cara broke the window.

Problem 3

All cats are mammals, and all mammals are animals. Which conclusion must follow?

  1. All animals are cats
  2. All cats are animals
  3. Some mammals are not animals
  4. No cats are animals
  5. All mammals are cats
Worked answer

All cats are animals

Every cat is a mammal, and every mammal is an animal, so following the chain gives that every cat is an animal.

The reverse does not follow: knowing all cats are animals says nothing about animals that are not cats.

The remaining options contradict the two statements given.

Problem 4

In a race with no ties, Alex finished before Bo, and Cal finished after Bo. Who finished last?

  1. Alex
  2. Bo
  3. Cal
  4. Cannot be determined
  5. They tied
Worked answer

Cal

Alex before Bo puts Alex ahead of Bo.

Cal after Bo puts Cal behind Bo.

So the order is Alex, then Bo, then Cal, and Cal finished last.

Problem 5

Five people meet and each shakes hands once with every other person. How many handshakes take place?

  1. 5
  2. 8
  3. 10
  4. 20
  5. 25
Worked answer

10

Each of the five people shakes hands with the other four, which suggests 5· 4 = 20.

But that counts each handshake twice, once from each side.

So the number of handshakes is 202 = 10.

Problem 6

A box holds only red and blue balls. It is false that all the balls are red. What must be true?

  1. All the balls are blue
  2. At least one ball is blue
  3. No balls are red
  4. Exactly half are blue
  5. Cannot be determined
Worked answer

At least one ball is blue

Saying it is false that all the balls are red means the claim fails for at least one ball.

That ball is not red, and the only other colour available is blue, so at least one ball is blue.

It does not follow that every ball is blue: there may still be red ones.

Problem 7

Four teams each play every other team exactly once. How many matches are played?

  1. 4
  2. 6
  3. 8
  4. 12
  5. 16
Worked answer

6

Each of the four teams plays the other three, giving 4· 3 = 12.

Each match involves two teams, so every match has been counted twice.

The number of matches is 122 = 6.

Problem 8

If it rains, the match is cancelled. The match was not cancelled. What follows?

  1. It rained
  2. It did not rain
  3. The match was cancelled
  4. Nothing follows
  5. It will rain tomorrow
Worked answer

It did not rain

The rule says rain guarantees cancellation. So if it had rained, the match would have been cancelled.

The match was not cancelled, so it cannot have rained.

This is valid reasoning. The invalid version runs the other way: knowing the match was cancelled would not prove it rained, since something else might have caused it.

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