Practice Error Analysis in Math

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

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Quick Recap

The systematic study of how errors arise in calculations or models, how large they are, and how they propagate through subsequent steps.

Error analysis asks "how wrong could my answer be?" โ€” not just "what is my answer?" โ€” because every measurement and approximation carries uncertainty.

Showing a random 20 of 50 problems.

Example 1

medium
A student factors x2+5x+6x^2 + 5x + 6 as (x+2)(x+4)(x+2)(x+4). Check by expansion and correct.

Example 2

hard
A student 'proves' 1=21 = 2 by writing a=ba = b, multiplying both sides by aa, subtracting b2b^2, and dividing by aโˆ’ba-b. Identify the error.

Example 3

easy
A student repeatedly makes sign errors when distributing -(x - 3). What is the correct expansion?

Example 4

medium
A student cancels incorrectly: x2+3xx=x2+3\dfrac{x^2+3x}{x} = x^2+3. Identify and correct the error.

Example 5

medium
A student computes the mean of 10, 20, 30 as (10+20+30)/2 = 30. Diagnose the error and give the correct mean.

Example 6

easy
A student writes ab+c=ab+ac\frac{a}{b+c} = \frac{a}{b} + \frac{a}{c}. Is this valid?

Example 7

medium
A student says: 'If a>ba > b then a2>b2a^2 > b^2.' Find a counterexample.

Example 8

easy
A speedometer reads 60 with a possible error of +/-2. Express the reading as a range.

Example 9

easy
A student writes logโก(a+b)=logโกa+logโกb\log(a+b) = \log a + \log b. Diagnose the error.

Example 10

medium
A rectangle has length 10 +/- 0.1 and width 5 +/- 0.1. Estimate the relative error in the area using relative-error addition.

Example 11

medium
An approximation of pi as 3.14 is used to compute a circle's circumference C = 2*pi*r with r = 10. Find the percent error in C from using 3.14 instead of 3.14159.

Example 12

medium
Two independent measurements have relative errors 3% and 4%. For their product, estimate the combined relative error using error addition.

Example 13

easy
Two added measurements each have absolute error 0.2. What is the maximum total absolute error in the sum?

Example 14

medium
A student computes 5โˆ’3ร—2=45 - 3 \times 2 = 4. Identify the error and give the correct result.

Example 15

challenge
Two quantities a = 100 +/- 1 and b = 99 +/- 1 are subtracted. Compute the result, its absolute error, and the relative error, and explain why subtraction of near-equal numbers is dangerous.

Example 16

hard
For f=x3f = x^3 with ฮ”xx=1%\frac{\Delta x}{x} = 1\%, estimate the percent error in ff.

Example 17

hard
A pendulum period is T=2ฯ€L/gT = 2\pi\sqrt{L/g}. If LL is known to 2%2\% and gg to 1%1\%, estimate the percent error in TT.

Example 18

easy
A measurement is 48 but the true value is 50. What is the percent error?

Example 19

easy
A student writes (x + 2)^2 = x^2 + 4. Identify the missing term in this common error.

Example 20

hard
Estimate the relative error in f=x2yf = \frac{x^2}{y} if ฮ”xx=2%\frac{\Delta x}{x} = 2\% and ฮ”yy=5%\frac{\Delta y}{y} = 5\%.
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