quotients

(noun)

A result obtained by dividing one quantity by another.

Related Terms

Examples of quotients in the following topics:

  • Reaction Quotients

    • The reaction quotient is a measure of the relative amounts of reactants and products during a chemical reaction at a given point in time.
    • The reaction quotient, Q, is a measure of the relative amounts of reactants and products during a chemical reaction at a given point in time.
    • Just as for the equilibrium constant, the reaction quotient can be a function of activities or concentrations.
    • Three properties can be derived from this definition of the reaction quotient:
    • Calculate the reaction quotient, Q, and use it to predict whether a reaction will proceed in the forward or reverse direction

  • Difference Quotients

    • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
    • It is also known as Newton's quotient:
    • The difference quotient is the average slope of a function between two points.
    • In this case, the difference quotient is know as a derivative, a useful tool in calculus.
    • Relate the difference quotient in algebra to the derivative in calculus

  • Dividing Polynomials

    • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
    • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
    • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
    • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
    • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

  • Logarithms of Quotients

    • By applying the product, power, and quotient rules, you could write this expression as:
    • Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients

  • Predicting the Direction of a Reaction

    • Equilibrium constants and reaction quotients can be used to predict whether a reaction will favor the products or the reactants.
    • If a reaction is not at equilibrium, you can use the reaction quotient, Q, to see where the reaction is in the pathway:

  • Strategy for General Problem Solving

    • If the units are ignored, the quotients do not numerically equal 1, but 1/12 or 12.
    • Since the two quotients are equal to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1.
    • You can also use these quotients to convert from inches to feet or from feet to inches.
    • If there is confusion regarding which quotient to use in the conversion, just make sure the units cancel out correctly.
    • The units behave just like numbers in products and quotients—they can be multiplied and divided.

  • The Derivative and Tangent Line Problem

    • The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
    • As the point $q$ approaches $p$, which corresponds to making $h$ smaller and smaller, the difference quotient should approach a certain limiting value $k$, which is the slope of the tangent line at the point $p$.
    • Then there is a unique value of $k$ such that, as $h$approaches $0$, the difference quotient gets closer and closer to $k$, and the distance between them becomes negligible compared with the size of $h$, if $h$ is small enough.
    • This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function $f$.

  • The Remainder Theorem and Synthetic Division

    • Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder.
    • This gives the quotient $x^2-9x-27$ and the remainder $-123$.
    • So the quotient must be the second degree polynomial $x^2 - 9x - 27$.

  • History of Intelligence Testing

    • The abbreviation "IQ" comes from the term intelligence quotient, first coined by the German psychologist William Stern in the early 1900s (from the German Intelligenz-Quotient).
    • He proposed that an individual's intelligence level be measured as a quotient (hence the term "intelligence quotient") of their estimated mental age divided by their chronological age.
    • The original formula for the quotient was Mental Age/Chronological Age x 100.

  • Division and Factors

    • So given two polynomials $D(x)$ (the dividend) and $d(x)$ (the divisor), we are looking for two polynomials $q(x)$ (the quotient) and $r(x)$ (the $remainder)$ such that $D(x) = d(x)q(x) + r(x)$ and the degree of $r(x)$ is strictly smaller than the degree of $d(x).$
    • Conceptually, we want to see how many copies of $d(x)$ are contained in $D(x)$ (this is the quotient) and then how far $D(x)$ is away from being a multiple of $d(x)$ (this is the remainder).
    • Again looking at the highest degree terms, we see that $4x^2 = 2x\cdot2x$, so we write down $2x$ as the second term in the quotient and proceed as before:
    • We see that the quotient $q(x)$ $3x^2+2x+6$ and the remainder $r(x)$ is $22$, so
    • (Of course, the quotient will also be a factor.)