GED Mathematical Reasoning Test flashcards

To solve problems involving unit rates where quantities are fractional, first determine how many 'units' are present in the total, then multiply that number by the fractional amount required per unit.

• Find how many times the base unit (e.g., 15 sq ft) fits into the total (e.g., 60 sq ft).
• Multiply this count by the fractional rate (e.g., 2/3 bag per base unit).
• Convert improper fractions to mixed numbers for practical answers.

Memory trick: Sections first, then multiply by fractional need.

The break-even point for a quadratic profit function is found by setting the profit (P) to zero and solving the resulting quadratic equation for the variable (e.g., selling price or quantity).

• Represents the points where total revenue equals total cost.
• Often yields two solutions for a quadratic equation.
• Solutions can be found by factoring, completing the square, or using the quadratic formula.

Memory trick: Zero profit means solving the quadratic puzzle.

This involves converting a real-world scenario described in words into a mathematical equation that uses variables to represent unknown quantities and operations to describe relationships.

• Identify the unknown quantities and assign variables (e.g., x, y).
• Look for keywords that indicate mathematical operations (e.g., 'sum' for addition, 'product' for multiplication).
• Set up the equation based on the given relationships and total values.

Memory trick: Read carefully, assign variables, build the math bridge.

The process of rewriting an algebraic expression in a more compact and understandable form by performing operations like distribution, combining like terms, and expanding products.

• Follow the order of operations (PEMDAS/BODMAS).
• Distribute multiplication over addition/subtraction.
• Combine terms with the same variable and exponent.

Memory trick: Expand, Distribute, Combine Clearly.

The process of finding the value of a variable in the exponent of an equation by using the natural logarithm (ln), especially when the base is 'e'.

• Isolate the exponential term first.
• Take the natural logarithm (ln) of both sides.
• Use the logarithm property ln(b^x) = x * ln(b) to bring the exponent down.
• Solve for the variable algebraically.

Memory trick: Exponential 'e': Isolate, then Natural Log it!

Substituting a specific value for the input variable into a function's rule to find the corresponding output value.

• Replace variable with given value.
• Follow order of operations (PEMDAS).
• Output is the function's value at that input.

Memory trick: Input In, Output Out: Plug it in, then compute!

For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, x = -b / (2a), represents the optimal input value (e.g., dosage, price) that leads to the maximum or minimum output (e.g., growth, profit).

• Used when a problem asks for the input that maximizes or minimizes an output.
• The sign of 'a' determines if it's a maximum (a<0) or minimum (a>0).
• The vertex formula is directly used to find this optimal input.

Memory trick: Vertex Finds the 'X' Factor.

A method to find the values of variables that satisfy two or more linear equations simultaneously by solving one equation for a variable and substituting that expression into the other equation.

• Isolate one variable in one of the equations.
• Substitute the expression for that variable into the other equation.
• Solve the resulting single-variable equation.
• Substitute the found value back into one of the original equations to find the other variable.

Memory trick: Substitution: Swap it out, then solve it!

Problems involving multiple entities working together or individually to complete a task, where their rates of work are a fraction of the task completed per unit of time.

• Rate = 1 / Time to complete.
• Work = Rate × Time.
• Combined work rates are added.

Memory trick: Rate is One Over Time: How much done per hour!

For a quadratic function in the form f(x) = ax^2 + bx + c, if a < 0, the parabola opens downwards and has a maximum value at its vertex.

• The x-coordinate of the vertex is given by x = -b / (2a).
• The maximum value is the y-coordinate of the vertex, f(-b / (2a)).
• Used to find peak values in real-world scenarios.

Memory trick: Vertex Vigorously Varies Value.

A mathematical statement involving an exponential expression and an inequality symbol (>, <, ≥, ≤). Solving it means finding the range of values for the variable that makes the inequality true.

• Often requires taking logarithms of both sides.
• Remember to reverse the inequality sign if multiplying or dividing by a negative number.
• Can be solved graphically by finding intersection points.

Memory trick: Logarithms Level the Exponential Playing Field.

A problem type where two or more quantities with different concentrations are mixed to form a new quantity with a desired concentration. Solved by setting up equations for total quantity and total amount of a specific component.

• Typically involves two variables and two equations.
• One equation for total volume/weight, another for total amount of substance.
• Can be solved using substitution, elimination, or graphing methods.

Memory trick: Mix Volumes, Match Concentrations.

To multiply a mixed number by a fraction, first convert the mixed number into an improper fraction. Then, multiply the numerators and multiply the denominators.

• Convert mixed numbers to improper fractions before multiplying.
• Multiply numerators together.
• Multiply denominators together.
• Simplify the resulting fraction if possible.

Memory trick: Mixed fractions must be 'improper' before you multiply them properly!

The point at which total cost and total revenue are equal, meaning there is no net loss or gain. It is found by setting the cost function equal to the revenue function.

• Cost = Revenue.
• Profit = 0.
• Found by solving a linear equation or system.

Memory trick: Cost Meets Revenue: Find where expenses just equal income!

Converting a real-world scenario involving time, rates, and quantities into a linear equation to solve for an unknown variable.

• Identify the unknown quantity and assign a variable.
• Translate relationships between quantities into mathematical expressions.
• Form an equation using the given total or equality condition.

Memory trick: Variable Values Vividly Vex.

Translating a real-world problem into a quadratic equation by identifying the unknown, known values, and the relationship that forms a second-degree polynomial.

• Identify the variable.
• Formulate expressions for quantities.
• Set up equation based on given conditions.
• Resulting equation will have x^2 term.

Memory trick: Identify, Relate, Equate: Find what's unknown, how it connects, then set it equal!

Calculating the value of an exponential function at a specific point, particularly when dealing with phenomena that double over a fixed period.

• General form: P(t) = P0 * b^(t/D), where b is the growth factor (2 for doubling).
• P0 is the initial amount, t is elapsed time, D is the doubling period.
• The exponent (t/D) represents the number of doubling periods.

Memory trick: Doubling: Start, then multiply by two for each period!

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

• Each x-value (input) must have only one y-value (output).
• It can be represented as a set of ordered pairs, a graph, or an equation.
• The vertical line test can be used to determine if a graph represents a function.

Memory trick: One Input, One Output; No Cheating!

Word problems involving the area of geometric shapes (especially rectangles) where one dimension is expressed in terms of another often lead to quadratic equations.

• Area of a rectangle = length × width.
• Define variables carefully based on the relationships given.
• The resulting equation can usually be solved by factoring, using the quadratic formula, or completing the square.

Memory trick: Dimensions Dictate the Equation.

Finding the value of a variable in an exponent by applying logarithms to both sides of the equation, utilizing logarithm properties.

• Isolate the exponential term first.
• Take log (natural or common) of both sides.
• Use log property: log(a^b) = b*log(a).

Memory trick: Isolate, Log, Divide: Get the base alone, then use logs to find the power!

For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, given by x = -b / (2a), represents the input value (often time) at which the function reaches its maximum or minimum output.

• If 'a' is negative, the vertex is a maximum.
• If 'a' is positive, the vertex is a minimum.
• The formula x = -b / (2a) is crucial for finding the time/input of the extremum.

Memory trick: Vertex Time Varies, Value Varies.

Equations where the variable appears in the exponent. Solved by isolating the exponential term and then applying logarithms to both sides to bring the exponent down.

• Isolate the exponential term first.
• Take the logarithm (natural log or common log) of both sides.
• Use the logarithm property log(b^x) = x log(b) to bring down the exponent.
• Solve the resulting linear equation for the variable.

Memory trick: Logarithms Level Exponents.

The slope (m) in a linear equation y = mx + b represents the rate of change of the dependent variable (y) with respect to the independent variable (x).

• Indicates how much 'y' changes for each unit change in 'x'.
• Can be positive (increasing), negative (decreasing), zero (constant), or undefined (vertical line).
• Often interpreted as a 'rate' in real-world scenarios.

Memory trick: Y-intercept is your START, Slope is your RATE.

The average rate of change of a function over an interval is the slope of the secant line connecting the two endpoints of the interval.

• Calculated as (change in y) / (change in x)
• Represents how much one quantity changes, on average, per unit change in another quantity
• For linear functions, it is constant and equal to the slope

Memory trick: Rate is Rise over Run, always remember the slope!

Finding the range(s) of values for the variable that satisfy an inequality involving a quadratic expression, often by finding roots and testing intervals.

• Set inequality to zero.
• Find roots of the quadratic equation.
• Test intervals or use parabola's shape.

Memory trick: Roots, Test, Shade: Find where the parabola crosses, then check the sections!

The practical domain of a function represents the set of all realistic input values that make sense within the context of a real-world problem.

• Considers physical or practical limitations.
• Often restricted to non-negative numbers or specific ranges.
• Different from the mathematical domain, which includes all values for which the function is defined.

Memory trick: Real-world limits define the input road.