Practice Assumptions in Math

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

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Quick Recap

Statements accepted as true without proof that form the starting conditions for a mathematical argument or model.

What are we assuming to be true? Everything follows from these starting points.

Showing a random 20 of 50 problems.

Example 1

easy
A shopkeeper assumes 'each customer buys exactly one item.' Under this assumption, if 50 customers visit, how many items are sold? State one way the assumption could fail.

Example 2

easy
A formula uses log⁡x\log x. What is assumed about xx?

Example 3

easy
A probability calculation multiplies P(A)⋅P(B)P(A) \cdot P(B) for two events. What is assumed?

Example 4

easy
A geometry claim 'parallel lines never meet' relies on which assumed framework?

Example 5

medium
Find the error: 'Let x=yx=y. Then x2=xyx^2 = xy. So x2−y2=xy−y2x^2 - y^2 = xy - y^2, (x−y)(x+y)=y(x−y)(x-y)(x+y)=y(x-y), hence x+y=yx+y=y, so 2y=y2y=y, giving 1=21=2.'

Example 6

medium
A problem assumes 'the rope is inextensible.' If the rope actually stretches, which conclusions of the analysis become invalid?

Example 7

medium
A model of compound interest uses annual compounding. Restating it with the SAME nominal rate but monthly compounding changes the result. Which assumption caused the change?

Example 8

medium
The slope formula y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1} assumes what?

Example 9

medium
A modeller treats a population's growth with P(t)=P0ertP(t)=P_0 e^{rt}. What assumption is implicit?

Example 10

challenge
A combinatorial argument counts arrangements assuming all objects are distinguishable, then is applied to identical balls. Explain the error and give the corrected count for placing 55 identical balls into 33 boxes.

Example 11

medium
A budget problem divides total cost evenly among friends. What two assumptions does 'evenly' hide?

Example 12

easy
True or false: assumptions that are never stated explicitly cannot cause errors.

Example 13

medium
A proof says 'WLOG, assume a≤ba \le b.' What does 'WLOG' mean?

Example 14

medium
True or false: a model that perfectly fits data has no assumptions.

Example 15

easy
What is an axiom?

Example 16

medium
In '1x+1y=1\frac{1}{x} + \frac{1}{y} = 1, find integer solutions', what implicit constraint/assumption must hold for the fractions to be defined?

Example 17

challenge
An economic model assumes 'rational actors with perfect information.' Construct a real scenario where this assumption fails and explain the consequence for the model's predictions.

Example 18

medium
What assumption is needed to write (x)2=x(\sqrt{x})^2 = x?

Example 19

medium
In a 'fair coin' problem, what is the embedded assumption?

Example 20

medium
'A fair die is rolled.' How does the word 'fair' change the assumptions, and what probability does each face get?
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