Math questions, answered
59 short answers from 12 guides. Each set links to the full guide for the worked examples and the reasoning behind the answer.
Area and Volume Formulas, and Why They Work
What is Heron's formula for the area of a triangle?
Heron's formula finds a triangle's area from its three sides when the height cannot be measured. First take s = (a + b + c) / 2, then A = √(s(s − a)(s − b)(s − c)). A 13, 14, 15 triangle has s = 21 and an area of √7056 = 84. Irregular plots can be split into triangles and the Heron areas added.
Why is a square metre 10.764 square feet?
A metre is 3.2808 feet, but area converts both dimensions, so the length factor is squared: a square metre is 10.764 square feet. Volume converts three dimensions, so a cubic metre is 35.315 cubic feet. Forgetting to square or cube the factor is the single largest error in practical area and volume work.
What is the difference between a US gallon and an imperial gallon?
A US gallon is 231 cubic inches, exactly 3.785411784 litres. An imperial (UK) gallon is exactly 4.54609 litres, about 20% larger. Both definitions are exact, and a litre is 1,000 cm³, so converting a tank's cubic volume to gallons depends on which gallon is meant.
How do you calculate the volume of a frustum?
A frustum is a cone with the top cut off, like a bucket, and its volume is the big cone minus the small one: ⅓πh(R² + Rr + r²). Set r to zero and it is a cone; set r = R and it is a cylinder. A pot with radii of 15 cm and 10 cm and a 30 cm height holds 14.92 litres.
What is the most common mistake in area calculations?
The most common error is using the slanted side instead of the perpendicular height, the straight-line distance at a right angle to the base, which always overestimates the area. Another is entering a diameter as a radius, which quadruples a circle's area and multiplies a sphere's volume by eight.
Exponent Rules: Negative, Zero and Fractional Powers Explained
Why is anything to the power of zero equal to 1?
Any non-zero number to the power of zero is 1 because it is the only value that keeps the exponent rules consistent. Dividing a power by itself, b³ / b³, must give b⁰ by the subtraction rule, and it must also equal 1. On a ladder of powers of 2, each step down divides by 2, so 2¹ = 2 steps down to 2⁰ = 1.
What are the rules of exponents?
The core exponent rules are bᵐ × bⁿ = bᵐ⁺ⁿ, bᵐ / bⁿ = bᵐ⁻ⁿ, (bᵐ)ⁿ = bᵐⁿ and (ab)ⁿ = aⁿbⁿ. From those follow b⁰ = 1, b⁻ⁿ = 1 / bⁿ, b^(1/n) = ⁿ√b and b^(m/n) = (ⁿ√b)ᵐ. Every one of them comes from insisting that multiplying powers of the same base adds their exponents.
Why is 8 to the power 0.667 not exactly 4?
Because 0.667 is not 2/3. It is 667/1000, so 8^0.667 asks for the 1000th root of 8⁶⁶⁷, which is about 4.0028, while 8^(2/3) is exactly 4. Rounding an exponent changes the question, so enter repeating exponents such as 2/3 as fractions rather than decimals.
Do exponents distribute over addition?
No. Exponents do not distribute over addition, so (a + b)² is a² + 2ab + b², not a² + b². They do distribute over multiplication: (ab)ⁿ = aⁿbⁿ, so (2 × 5)³ = 8 × 125 = 1000, the same as 10³.
What is the largest known prime number?
The largest known prime is M136279841, which is one less than 2^136279841. The Great Internet Mersenne Prime Search found it in October 2024, and it has 41,024,320 digits. That digit count comes from floor(n × log₁₀ 2) + 1, so it can be found without writing the number out.
How to Calculate a Weighted Average: Grades, GPA, Prices and Pitfalls
How do I calculate average cost per share?
Divide the total spent by the number of shares owned. Buying 100 shares at $42.10, 250 at $38.75 and 50 at $45.00 costs $16,147.50 for 400 shares, an average cost of $40.36875 per share. Averaging the three prices without weights gives $41.95, which overstates the cost because most shares were bought at the lowest price.
How do I average percentages from groups of different sizes?
Weight each group's average or percentage by the size of the group instead of averaging them directly. A class of 10 students averaging 90 and a class of 30 averaging 70 have an overall average of (10 × 90 + 30 × 70) / 40 = 75, not 80. The same rule applies to combining any rates, weighting by group size or the rate's denominator.
What is Simpson's paradox?
Simpson's paradox is when weighting reverses a comparison. In a well-known 1986 study of kidney stone treatments, treatment A had a higher success rate than treatment B for small stones and for large stones separately, yet a lower success rate overall, because treatment A was used mostly on large stones, which are harder to treat, so the two overall rates used very different weights.
How do you calculate an average star rating?
Weight each star level by its number of reviews and divide by the total count. A product with 120 five-star, 80 four-star, 30 three-star, 10 two-star and 15 one-star reviews averages (5 × 120 + 4 × 80 + 3 × 30 + 2 × 10 + 1 × 15) / 255 = 1045 / 255 = 4.10 stars.
How to Calculate Percent Error (and What It Does Not Tell You)
What is the difference between accuracy and precision?
Accuracy is how close the average of repeated measurements is to the true value, measured with the percent error of the mean. Precision is how closely the trials agree with each other, whatever the true value, measured with the standard deviation or the relative standard deviation. A set of readings can be very precise yet inaccurate, which is the signature of a systematic error.
How do you calculate relative standard deviation?
Relative standard deviation (RSD) is the sample standard deviation divided by the mean, times 100. Five readings of g, 9.79, 9.82, 9.85, 9.76 and 9.80 m/s², have a mean of 9.804 and a sample standard deviation of about 0.034 m/s², so the RSD is about 0.34%. RSD measures precision, not accuracy.
Can you calculate percent error when the true value is zero?
No. Percent error is undefined when the true value is zero, because the formula divides by it, so report the absolute error instead. Temperatures in Celsius or Fahrenheit are also scale-dependent: an error of 1 degree is 10% at 10 °C but about 0.35% at 283 K, so compute on an absolute scale or report the error in degrees.
How many significant figures should a percent error have?
Two or three significant figures is plenty for a percent error, because the error itself is rarely known better than that. Keep full precision through the calculation and round once, at the end. Rounding the measured value first distorts the answer: rounding 9.6 to 10 before comparing with 9.81 turns a 2.1% error into a 1.9% overestimate.
What is mean absolute percent error?
Mean absolute percent error (MAPE) averages each trial's own percent error. It is never smaller than the percent error of the mean, because averaging the trials first lets highs and lows cancel. Forecasters use MAPE to judge how far individual predictions miss, while scientists more often report the percent error of the mean together with the relative standard deviation.
Logarithms Explained: Bases, ln, Change of Base and Real Uses
What are the product, quotient and power rules for logarithms?
The product rule is log_b(xy) = log_b x + log_b y, the quotient rule is log_b(x/y) = log_b x − log_b y, and the power rule is log_b(xⁿ) = n · log_b x. Only products, quotients and powers inside the log simplify: log(x + y) is not log x + log y, and (log x)² is not 2 log x.
Who invented logarithms?
John Napier published the first logarithm tables in 1614, and Henry Briggs produced base-10 tables soon after. They were invented because of the product rule: adding two numbers from a table is far easier than multiplying two eight-digit numbers by hand. The slide rule, used by engineers until the 1970s, is that rule made physical.
How many bits does it take to store a number?
A number up to N needs floor(log₂ N) + 1 bits. log₂ 1000 is 9.97, so 1000 needs 10 bits. The same base-2 logarithm explains binary search: finding one item among a million takes at most ceil(log₂ 1,000,000) = 20 comparisons, because each step halves the remaining range.
What are examples of logarithmic scales?
Common logarithmic scales include pH, where each unit is a tenfold change in hydrogen-ion activity; decibels, where +10 dB is ten times the power; earthquake magnitude, where each step is ten times the amplitude and about 32 times the energy; bits, where each one doubles the number of values; and musical octaves, which double the frequency.
How do you find the base of a logarithm?
Rewrite the logarithm as a power and take a root. For log_b 81 = 4, rewrite it as b⁴ = 81, so b = 81^(1/4) = 3. The base must be positive and not 1, so although (−3)⁴ is also 81, −3 is not a valid base. Similarly, log_b 2 = 0.5 gives b = 4.
Matrix Operations by Hand: Determinant, Inverse and Row Reduction
What are the three elementary row operations?
The three elementary row operations are swapping two rows, scaling a row by a non-zero number, and replacing a row by itself plus a multiple of another row. None of them changes the solution set of the system the matrix represents. Gaussian and Gauss-Jordan elimination use them to compute the determinant, inverse, rank and solutions.
How do you find the determinant of a 3x3 matrix?
Row-reduce the matrix to triangular form using only swaps and row replacements, multiply the diagonal entries, and flip the sign once for each swap. Adding a multiple of one row to another leaves the determinant unchanged, while scaling a row by k multiplies it by k. Cofactor expansion also works but needs n! terms, 720 for a 6×6.
How do I solve a system of linear equations with RREF?
Write the coefficients and right-hand sides as an augmented matrix and reduce it to reduced row echelon form, then read the solution from the last column. For 2x + y − z = 8, −3x − y + 2z = −11 and −2x + y + 2z = −3, that gives x = 2, y = 3, z = −1. A row reducing to [0 0 0 | 5] means no solution.
What is an ill-conditioned matrix?
An ill-conditioned matrix is one where tiny input changes cause huge output changes. The classic example is the Hilbert matrix, with entries 1/(i + j − 1): the 3×3 case has determinant 1/2160 and an all-integer inverse. Floating point gets close at 3×3, but by around 12×12 a standard double-precision inverse is essentially meaningless, which is why exact fractions help.
How do you transpose a matrix?
Transposing a matrix swaps its rows and columns, so row i of A becomes column i of Aᵀ and a 2×3 matrix becomes a 3×2. By contrast, addition needs both matrices to have the same shape, and A×B needs A to have as many columns as B has rows.
Percentages, Fractions and Ratios: How the Arithmetic Fits Together
How do you add fractions with different denominators?
To add fractions with different denominators, rewrite them over a common denominator, ideally the least common multiple, which keeps the numbers small. For 1/4 + 1/6, the LCM of 4 and 6 is 12, so the sum is 3/12 + 2/12 = 5/12. Any common multiple works, but a larger one needs a bigger simplification at the end.
What is the Euclidean algorithm for the GCF?
The Euclidean algorithm finds the greatest common factor by dividing the larger number by the smaller, keeping the remainder and repeating until the remainder is 0. For 1071 and 462 the remainders are 147, 21 and 0, so the GCF is 21. Euclid wrote it down around 300 BC, and it needs only a handful of steps even for huge numbers.
Why do some fractions give repeating decimals?
A fraction repeats because long division can only produce remainders smaller than the divisor, so a remainder eventually comes back and the digits cycle. A fraction in lowest terms ends only when its denominator has no prime factors except 2 and 5, the factors of 10. So 3/8 = 0.375 ends, while 1/6 and 3/13 repeat.
How do you simplify a ratio?
To simplify a ratio, divide every term by their greatest common factor, exactly as with a fraction, so 24 : 36 becomes 2 : 3. If the terms are decimals, first multiply all of them by a power of ten to make whole numbers: 1.5 : 2.25 becomes 150 : 225 and then 2 : 3.
How do you check a long division answer with a remainder?
Check a long division answer by multiplying the quotient by the divisor and adding the remainder, which must give the original number. For 7452 ÷ 13 the answer is 573 remainder 3, and 573 × 13 + 3 = 7452. The same result can be written as the mixed number 573 3/13.
Quadratics, Scientific Notation and Roman Numerals: Writing Numbers Well
What is the numerically stable quadratic formula?
The numerically stable quadratic formula computes q = −(b + sign(b) × √D) / 2, then x₁ = q / a and x₂ = c / q. It avoids subtracting two nearly equal numbers, which in double precision can wreck the small root: for x² + 10⁸x + 1 = 0 the textbook formula returns −7.45 × 10⁻⁹ instead of about −1 × 10⁻⁸.
How do I check the roots of a quadratic?
Use Vieta's formulas: the roots of ax² + bx + c = 0 should add up to −b/a and multiply to c/a. For 2x² − 4x − 6, the roots −1 and 3 add to 2, which is 4/2, and multiply to −3, which is −6/2. A mismatch means a mistake somewhere.
How many significant figures should a calculated result have?
For multiplication and division, keep the smallest number of significant figures among the inputs, so 3.0 × 10⁸ times 2.50 × 10⁻³ is 7.5 × 10⁵. For addition and subtraction, match the fewest decimal places, so 12.11 + 18.0 is 30.1. Exact counts and defined constants never limit the result.
Why does a spreadsheet show a long number as 1.23457E+15?
A spreadsheet switches long numbers such as IDs or card numbers to E notation, and when it shows 1.23457E+15 the digits beyond the 15th have usually already been lost. Format the column as text before importing, because no notation can bring those digits back once they are gone.
What are the six subtractive pairs in Roman numerals?
Standard Roman numerals have only six subtractive pairs: IV (4), IX (9), XL (40), XC (90), CD (400) and CM (900). Only I, X and C are ever subtracted, and only from the next two symbols up. Each decimal place is written separately, so 49 is XLIX, not IL, and 1999 is MCMXCIX, not MIM.
Rounding Rules: Half Up, Half Even and Why Software Disagrees
What is the difference between Excel ROUND, ROUNDUP and TRUNC?
Excel's ROUND rounds to the nearest value with ties away from zero, so 2.5 becomes 3 and −2.5 becomes −3. ROUNDUP always moves away from zero, so 2.4 becomes 3, while ROUNDDOWN and TRUNC always move toward zero, so −2.6 becomes −2. Despite their names, ROUNDUP and ROUNDDOWN have nothing to do with halves.
Why does rounding make totals drift upward?
Rounding every tie in the same direction adds a small bias. The values 0.5, 1.5, 2.5 and 3.5 add up to 8, but rounded half away from zero they become 1, 2, 3 and 4, totalling 10. Rounded half to even they become 0, 2, 2 and 4, which still total 8, because ties go up and down equally often.
Why does Python round(2.5) return 2?
Python 3's round() uses round half to even, the default rounding mode of the IEEE 754 standard, so a tie goes to the nearest even digit: round(2.5) is 2 and round(3.5) is 4. That surprises people who expect half away from zero, which is what Excel's ROUND and school rounding use.
Should I round in stages or only once?
Round once, at the end. Rounding in stages can change the answer: 2.445 rounded to two places is 2.45, and that rounded to one place is 2.5, but 2.445 rounded straight to one place is 2.4, because it is below the midpoint 2.45. ASTM E29 also says to round in a single step from the full-precision value.
Does round half up mean the same thing in Java and JavaScript?
No. In JavaScript, Math.round rounds ties toward positive infinity, so −2.5 becomes −2. In Java's BigDecimal, RoundingMode.HALF_UP rounds ties away from zero, so −2.5 becomes −3, and HALF_DOWN rounds ties toward zero. When documenting a rounding choice, describe the behaviour, such as "ties away from zero", rather than relying on a name.
Solving Systems of Linear Equations by Elimination, Step by Step
What row operations are allowed in elimination?
Three row operations never change the solutions of a linear system: swapping two rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. In equation form, that means swapping equations, scaling an equation, or adding a multiple of one equation to another. Elimination uses these moves to remove one variable at a time.
What is reduced row echelon form?
Reduced row echelon form (RREF) is the matrix produced by Gauss-Jordan elimination, where every pivot is scaled to 1 and the entries both above and below each pivot are cleared. For a system with one solution, the left part becomes the identity matrix and the answers sit in the last column of the augmented matrix with nothing left to do.
How does the rank of a matrix tell you the number of solutions?
Compare the rank of the coefficient matrix A, the rank of the augmented matrix [A | b] and the number of unknowns n. If rank A is less than rank [A | b], there is no solution. If both ranks equal n, there is exactly one solution. If both are equal but less than n, there are infinitely many, with n minus rank free variables. This is the Rouché–Capelli theorem.
What is a parametric solution of a linear system?
A parametric solution describes infinitely many solutions by setting each free variable, an unknown without a pivot, to a parameter such as t. For example, x = 11/2 − (3/2)t, y = 1/2 + (1/2)t, z = t describes a whole line of solutions in three-dimensional space, and choosing t = 1 gives the particular solution (4, 1, 1).
How old is the elimination method for solving equations?
Elimination is very old. The Chinese text The Nine Chapters on the Mathematical Art, compiled around two thousand years ago, solves systems of three equations with counting rods arranged in the same tabular form as a modern augmented matrix. The same method still scales from two equations to thousands and is what computer libraries run under the hood.
Solving Triangles and Slopes: Sines, Cosines and Rise Over Run
When do you use the law of cosines?
Use the law of cosines, c² = a² + b² − 2ab·cos C, when you know three sides (SSS) and need the angles, or two sides and the angle between them (SAS) and need the third side. When C is 90°, cos C is 0 and the law collapses into the Pythagorean theorem. For sides 5, 6 and 7 it gives angle A as 44.42°.
When do you use the law of sines?
Use the law of sines, a / sin A = b / sin B = c / sin C, once you know one side and the angle opposite it. That is what the ASA and AAS cases become after subtracting the two known angles from 180° to find the third. The common ratio equals 2R, where R is the radius of the circle through all three corners.
How do I convert percent grade to degrees?
Take the arctangent of the slope, where the slope is the percent grade divided by 100. Grade and angle are not proportional: a 100% grade is 45°, not 90°, and a 6% road is only about 3.4°. At small slopes the angle in degrees is roughly 57.3 times the slope, but that shortcut drifts quickly above about 20%.
What is the maximum slope for a wheelchair ramp?
The 2010 ADA Standards for Accessible Design limit a ramp's running slope to 1:12, which is a grade of 8.33% or an angle of 4.76°. A ramp rising 30 cm over 3 m has a slope of 0.1, a 10% grade, so it is too steep. At 30 cm of rise it needs at least 3.6 m of run.
How do I find the equation of a line from two points?
Find the slope m = (y₂ − y₁) / (x₂ − x₁), then the intercept b = y₁ − m·x₁, giving y = mx + b. For (1, 2) and (4, 7) the slope is 5/3 and the intercept 1/3, so y = (5/3)x + 1/3, or 5x − 3y = −1 in standard form. A vertical line has an undefined slope and the equation x = constant.
Square Roots, Cube Roots and nth Roots: Simplifying Radicals Properly
What is simplest radical form?
A radical is in simplest radical form when three conditions hold: the radicand has no factor that is a perfect nth power other than 1, there is no fraction under the radical, and there is no radical in a denominator. So √72 becomes 6√2, ∛54 becomes 3∛2, and √(1/2) is written as √2 / 2.
What is the principal square root?
The principal square root is the non-negative one of a number's two square roots, and it is the one the radical sign picks. Both 6 and −6 square to 36, but √36 is 6. Solving x² = 36 gives x = ±√36 = ±6 because the ± is written separately. The same convention holds for every even index, so ⁴√81 is 3.
How do you estimate a square root without a calculator?
First bracket it between perfect squares: √50 lies between 7 and 8, much closer to 7. Then refine a guess g with one Newton step, (g + x/g) / 2. Starting from 7 for √50 gives (7 + 50/7) / 2 = 7.0714, against a true value of 7.0710678, and each further step roughly doubles the number of correct digits.
Is the square root of a sum equal to the sum of the square roots?
No. √(a + b) is not √a + √b: √(9 + 16) = √25 = 5, while √9 + √16 = 7. Radicals have a product rule, ⁿ√(ab) = ⁿ√a × ⁿ√b, and a quotient rule for positive numbers, but there is no sum rule. Mixing them up is one of the most common radical mistakes.
Why does (−8)^(1/3) return NaN in programming languages?
Many programming languages compute x^(1/3) as exp(ln(x)/3), and because the natural logarithm of a negative number is not real, they return NaN for (−8)^(1/3), even though the real cube root is −2. Dedicated cube-root functions avoid the problem: JavaScript's Math.cbrt(−8) correctly returns −2.