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⋅422 + 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 ∫1x dx\int \dfrac{1}{x} \, dx is often written as ln⁡∣x∣+C\ln |x| + C. Why is the absolute value needed? Evaluate ∫−2−11x dx\int_{-2}^{-1} \dfrac{1}{x} \, dx.

Example 4

easy
Read '1/2x' as 12x\dfrac{1}{2x} at x=5x = 5. Give the value.

Example 5

medium
The instruction 'simplify 68\frac{6}{8}' is ambiguous about target form. Give the reduced fraction's numerator.

Example 6

hard
Some define 00=10^0 = 1 (combinatorial convention); others say undefined. Using the combinatorial convention, evaluate (50)⋅00\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 'xx is close to 0' is ambiguous in mathematics. Suggest an unambiguous mathematical version.

Example 10

medium
In 'log⁡100\log 100' with no base shown, take base 1010. What is the value?

Example 11

easy
How many distinct values can 2322^{3^2} take depending on grouping? State the standard one.

Example 12

easy
Evaluate 10−4−310 - 4 - 3 left-to-right.

Example 13

medium
A set is 'closed under addition.' Does '{0}\{0\}' qualify? Answer 11 for yes.

Example 14

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

Example 15

medium
Is N\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 8−3−28 - 3 - 2. One gets 33, one gets 77. Which is correct and why?

Example 17

challenge
For which real xx does the ambiguous expression xxxx^{x^x} have 2(22)2^{(2^2)} as its value? Give that xx, 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(red)=0.3P(\text{red}) = 0.3, P(blue)=0.4P(\text{blue}) = 0.4, P(both)=0.1P(\text{both}) = 0.1, find P(red or blue)P(\text{red or blue}).

Example 19

easy
How many real solutions to x2=9x^2 = 9? Give the sum of the solutions.

Example 20

easy
What is log⁡1000\log 1000 under the common (base 10) convention?
← Back to Ambiguity Worked Examples

Related Concepts

Notation Overload