What would you like to understand?

The calculation assumes a population of 210 people: 10 librarians and 200 farmers. With 40% of librarians fitting the description (yielding 4 matching librarians) and 10% of farmers fitting it (yielding 20 matching farmers), there are 24 total matches.

Conditions: Population consists only of librarians and farmers with a 1:20 ratio; Likelihoods are stipulated as 40% for librarians and 10% for farmers; Conditioning requires $P(E) > 0$

To compute $\gcd(5295, 4321)$, apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with $5295 = 1 \cdot 4321 + 974$.

Conditions: The inputs are 5295 and 4321.; The Euclidean algorithm is used.; The division algorithm is applied at each step.

The Euclidean algorithm computes the greatest common divisor by repeatedly applying the division algorithm. Starting with two natural numbers $a$ and $b$, you divide the larger by the smaller to get a quotient and a remainder.

Conditions: The inputs $a$ and $b$ are natural numbers.; The division algorithm is applied repeatedly.; The process stops when a remainder equals 0.

For a twice differentiable function on an interval, a positive second derivative throughout that interval indicates upward concavity (concave up), while a negative second derivative indicates downward concavity (concave down). This corresponds to whether the tangent slope is increasing or decreasing.

Conditions: The function is twice differentiable on the interval under discussion.; The strict sign statement applies where the instantaneous slope-change rate has that sign.

Matrix-vector multiplication is reframed as scaling the transformed basis vectors (which are the columns of the matrix) by the input vector's components and summing them. This constructs the final position within the skewed coordinate system defined by the matrix, rather than just following a tedious arithmetic recipe.

Conditions: Input and output coordinates use the fixed standard basis.; The matrix represents an active linear map moving vectors.

To find the left-hand limit as x approaches a value c (denoted $\lim_{x \to c^-} f(x)$), observe the behavior of the function's curve strictly for input values less than c, moving towards c. If the y-values grow without bound in either the positive or negative direction, the finite real limit does not exist.

Conditions: The function has a vertical asymptote or discontinuity at $x=c$.; You are evaluating the limit from the left side (inputs strictly less than c).

To find the x-intercepts of the curve $y = x^2 - 4$, set the function equal to zero ($x^2 - 4 = 0$) and factor the expression as a difference of squares into $(x+2)(x-2) = 0$. Applying the zero-product property yields the solutions $x = -2$ and $x = 2$, which are the points where the graph crosses the x-axis.

Conditions: The curve is defined by the quadratic equation $y = x^2 - 4$.; The goal is to find where the graph crosses the x-axis.; The algebraic method used is factoring the difference of perfect squares.

To calculate the directional derivative for a specific vector, multiply each partial derivative by the corresponding component of the direction vector and sum them. For $v=[-1,2]$, this means taking -1 times the partial derivative with respect to x, plus 2 times the partial derivative with respect to y.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $v=[-1,2]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

To find the point on the curve $y = x^2 - 4$ at $x = -1$, substitute $x = -1$ into the equation. Evaluating $(-1)^2 - 4$ yields $1 - 4 = -3$, so the coordinates of the point are $(-1, -3)$.

Conditions: The curve is defined by the equation $y = x^2 - 4$.; The x-coordinate of the desired point is given as $-1$.; The point must lie on the curve.

To find the magnitude of a vector from its components, you calculate the square root of the sum of the squares of those components. This process effectively measures the Euclidean length of the vector in the coordinate plane.

Conditions: The vector is defined by its components in a 2D Cartesian coordinate system.; The coordinate system is orthonormal.

In a 2D plane, the first component dictates horizontal movement (right positive, left negative) and the second dictates vertical movement (up positive, down negative) from the origin to the tip.

Conditions: Standard Cartesian coordinate system; Vector anchored at the origin

To compute the greatest common divisor of 1785 and 546, apply the Euclidean algorithm by repeatedly dividing the previous divisor by the previous remainder. Start with 1785 divided by 546.

Conditions: The inputs are 1785 and 546.; The Euclidean algorithm is used.; The division algorithm is applied at each step.