Practice Ambiguity in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A situation where a mathematical expression, statement, or notation can be interpreted in more than one valid way, leading to different results.

Ambiguity is a fork in the road with no sign — different readers take different paths and arrive at different answers, each thinking they are right.

Showing a random 20 of 50 problems.

Example 1

medium

Evaluate

2 + 3 \cdot 4^2

using the standard order of operations.

Example 2

challenge

A poll says '40% of people prefer A to B; 30% prefer B to A.' What does the remaining 30% mean ambiguously? Under the assumption 'no preference', give the percentage who do not prefer A.

Example 3

challenge

The integral

\int \dfrac{1}{x} \, dx

is often written as

\ln |x| + C

. Why is the absolute value needed? Evaluate

\int_{-2}^{-1} \dfrac{1}{x} \, dx

Example 4

easy

Read '1/2x' as

\dfrac{1}{2x}

x = 5

. Give the value.

Example 5

medium

The instruction 'simplify

\frac{6}{8}

' is ambiguous about target form. Give the reduced fraction's numerator.

Example 6

hard

Some define

0^0 = 1

(combinatorial convention); others say undefined. Using the combinatorial convention, evaluate

\binom{5}{0} \cdot 0^0

Example 7

hard

In '$5 off coupon, then 10% off' vs '10% off, then $5 off' on a $50 item, give the final price under the first order.

Example 8

medium

A 'random number between 1 and 10' inclusive of both endpoints — how many integer choices?

Example 9

easy

The statement '

x

is close to 0' is ambiguous in mathematics. Suggest an unambiguous mathematical version.

Example 10

medium

In '

\log 100

' with no base shown, take base

10

. What is the value?

Example 11

easy

How many distinct values can

2^{3^2}

take depending on grouping? State the standard one.

Example 12

easy

Evaluate

10 - 4 - 3

left-to-right.

Example 13

medium

A set is 'closed under addition.' Does '

\{0\}

' qualify? Answer

1

for yes.

Example 14

medium

'A number times its successor is 12.' Set up and solve for the positive integer.

Example 15

medium

\mathbb{N}

the set of positive integers or non-negative integers? Give the smallest element under the 'positive integers' convention.

Example 16

medium

Two students compute

8 - 3 - 2

. One gets

3

, one gets

7

. Which is correct and why?

Example 17

challenge

For which real

x

does the ambiguous expression

x^{x^x}

have

2^{(2^2)}

as its value? Give that

x

, then the value.

Example 18

medium

In a probability problem 'red or blue' could mean inclusive or exclusive. If the problem allows both (inclusive), and

P(\text{red}) = 0.3

P(\text{blue}) = 0.4

P(\text{both}) = 0.1

, find

P(\text{red or blue})

Example 19

easy

How many real solutions to

x^2 = 9

? Give the sum of the solutions.

Example 20

easy

What is

\log 1000

under the common (base 10) convention?

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Related Concepts

Notation Overload