GED Mathematical Reasoning Test flashcards

Exponential growth occurs when a quantity increases by a constant factor over equal time intervals. When dealing with very large or small numbers, scientific notation is used to express them concisely.

• Initial amount multiplied by the growth factor raised to the power of time periods.
• Scientific notation expresses numbers as a coefficient times a power of 10.
• When multiplying numbers in scientific notation, multiply coefficients and add exponents.
• Adjust coefficient to be between 1 and 10, adjusting exponent of 10 accordingly.

Memory trick: Start with the base, multiply by the factor, for each time you pass.

To multiply fractions, multiply the numerators together and the denominators together. Convert mixed numbers to improper fractions first.

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

Memory trick: Many cookies need many fractions multiplied just right.

Sequential fractional changes involve applying a fraction-based increase or decrease to a quantity, and then applying another fraction-based change to the *new* resulting quantity.

• Calculate the first change based on the initial value.
• Update the value after the first change.
• Calculate the second change based on the *updated* value.
• Apply the second change to find the final value.

Memory trick: First, increase the base; then, from that new place, decrease and embrace.

To find a percentage of a number, convert the percentage to a decimal (divide by 100) and then multiply it by the given number.

• Percentage means 'per hundred'.
• To convert a percentage to a decimal, divide by 100 or move the decimal point two places to the left.
• Multiply the decimal equivalent by the total number.

Memory trick: Percent to decimal, then multiply, for the part you want to find.

When a value undergoes multiple percentage changes, each change is applied to the *new* value resulting from the previous change, not the original value.

• For a decrease of X%, multiply by (1 - X/100).
• For an increase of Y%, multiply by (1 + Y/100).
• Changes are multiplicative, not additive, to the original percentage.

Memory trick: First change, new base, second change, new pace.

To add or subtract fractions, they must have a common denominator. Find the least common multiple (LCM) of the denominators, convert the fractions, then add or subtract the numerators.

• Find a common denominator (LCM).
• Convert fractions to equivalent fractions with the common denominator.
• Add or subtract the numerators.
• Keep the common denominator.

Memory trick: Common ground for adding, difference for subtracting, fractions unite, then simplify to make it right.

The sum of all percentages representing parts of a whole must equal 100%. To find an unknown part, subtract known parts (as percentages) from 100%, then calculate that percentage of the total.

• Total percentage = 100%.
• Convert percentages to decimals for calculations (divide by 100).
• Multiply decimal percentage by the total number to find the count for that part.

Memory trick: All parts add to 100, subtract what you know, the rest is found, then multiply slow.

Converting a percentage to a fraction involves writing the percentage as a fraction with a denominator of 100 and then simplifying the fraction to its lowest terms.

• A percentage represents 'parts per hundred'.
• Write the percentage number as the numerator over 100.
• Simplify the fraction by dividing both numerator and denominator by their greatest common divisor.
• The total number of items (120) is extraneous information for finding the *fraction* with no defects.

Memory trick: Percent over a hundred, then simplify till it's done.

Dividing a number by a decimal involves moving the decimal point in the divisor and dividend to make the divisor a whole number, then performing standard division.

• Move decimal in divisor to make it a whole number.
• Move decimal in dividend the same number of places.
• Perform long division.
• Round down for 'full pieces' scenarios.

Memory trick: Divide and conquer pieces, dropping any remainder.

To calculate a discount, first find the discount amount by multiplying the original price by the discount percentage (as a decimal). Then, subtract the discount amount from the original price to find the final price.

• Discount Amount = Original Price × Discount Rate.
• Final Price = Original Price - Discount Amount.
• Discount rates are usually given as percentages.

Memory trick: Percent off, then subtract, for the price you'll pay, don't look back.

To find the equation of a line given two points, first calculate the slope using the slope formula, then use either the point-slope form or substitute a point and the slope into the slope-intercept form to find the y-intercept.

• Slope formula: m = (y2 - y1) / (x2 - x1)
• Point-slope form: y - y1 = m(x - x1)
• Slope-intercept form: y = mx + b
• Any point (x, y) on the line satisfies the equation.

Memory trick: Slope First, Then Point-Slope, or Intercept!

Slope is a measure of the steepness of a line, calculated as the vertical change (rise) divided by the horizontal change (run) between two points on the line.

• Formula: m = rise / run
• Positive slope: line goes up from left to right
• Negative slope: line goes down from left to right
• Zero slope: horizontal line

Memory trick: Rise up, Run across, that's how slope is made!

An exponential growth function, typically y = a(b)^x or y = a(1+r)^x, models situations where a quantity increases by a fixed percentage over equal time intervals.

• 'a' is the initial amount or starting value.
• 'b' (or 1+r) is the growth factor, where 'r' is the growth rate as a decimal.
• 'x' (or t) is the number of time periods.
• The growth factor is always greater than 1.

Memory trick: Growth Factor is '1 + Rate' – 'R' for 'Rise' in value!

The y-intercept of a graph is the point where the graph crosses the y-axis. In a function, it represents the output value when the input value is zero, often signifying an initial amount, starting point, or fixed cost.

• In y = mx + b, 'b' is the y-intercept.
• To find it, set x=0 and solve for y.
• Its meaning is highly dependent on the context of the problem.

Memory trick: Y-intercept is the 'Y' at the 'start' (x=0)!

Evaluating a function means substituting a given value for the input variable (often x or t) into the function's equation and then calculating the resulting output value.

• Input values are typically given as a number or a variable expression.
• Substitute the input value wherever the variable appears in the function.
• Follow the order of operations to simplify the expression and find the output.

Memory trick: Input into the machine, output the result!

A break-even point occurs when the cost of production equals the revenue generated from sales. Mathematically, it's the point where two functions (cost and revenue) intersect, found by setting their equations equal to each other.

• Cost function typically includes a fixed cost (y-intercept) and a variable cost (slope).
• Revenue function usually starts at zero and has a positive slope (price per item).
• The solution (x, y) represents the quantity (x) and the amount of money (y) at which break-even occurs.

Memory trick: When money in equals money out!

The equation (x^2/a^2) + (y^2/b^2) = 1 describes an ellipse centered at the origin, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes.

• a is semi-major axis (half longest diameter)
• b is semi-minor axis (half shortest diameter)
• If a = b, it's a circle (radius a)
• Foci are on the major axis

Memory trick: Ellipse is Squished Circle: A and B Define!

A linear equation can be determined from two given points by first calculating the slope, and then using one of the points and the slope to find the y-intercept.

• Slope (m) = (y2 - y1) / (x2 - x1)
• Equation form: y = mx + b
• Initial value often corresponds to the y-intercept (b).

Memory trick: Slope First, Then Intercept - that's the linear concept!

The practical range of a function is the set of all output values (y or W(t)) that are realistic and meaningful within the context of a real-world problem, considering both the function's behavior and external constraints.

• Dependent on the practical domain.
• Restricted by physical constraints (e.g., non-negative quantities).
• Limited by explicit problem conditions (e.g., 'cannot weigh less than X').
• Often involves evaluating the function at the boundaries of the practical domain.

Memory trick: Output Limits, Real-World Fits!

Slope quantifies how much the dependent variable (y) changes for every unit change in the independent variable (x), representing a rate.

• Slope = Δy / Δx (rise over run)
• Positive slope means y increases as x increases
• Negative slope means y decreases as x increases

Memory trick: Rise Up, Run Across – Slope's the Rate!

For a quadratic function in standard form f(x) = ax^2 + bx + c, the vertex is the point where the parabola reaches its maximum (if a < 0) or minimum (if a > 0) value. The x-coordinate of the vertex is given by x = -b / (2a).

• If 'a' is negative, the parabola opens downwards, and the vertex is a maximum.
• If 'a' is positive, the parabola opens upwards, and the vertex is a minimum.
• The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.

Memory trick: Vertex 'X' is '-B over 2A'!

A linear equation in the form y = mx + b represents a straight line. 'm' is the slope, indicating the rate of change, and 'b' is the y-intercept, indicating the initial value or starting point.

• Slope (m) describes how much y changes for each unit change in x.
• Y-intercept (b) is the value of y when x is 0.
• Contextual meaning is crucial: slope is often a rate, y-intercept is an initial amount.

Memory trick: Y = mx + b tells all!

The set of all possible output values of a function that are reasonable or make sense in a real-world context, given the practical domain.

• Depends on the practical domain of the function.
• Represents the 'output' values (e.g., height, cost, population).
• Often found by evaluating the function at the extremes of the practical domain.

Memory trick: Range 'R'eaches from the 'R'elevant 'R'esults.

The slope of a line is a measure of its steepness and direction, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

• Formula: m = (y2 - y1) / (x2 - x1)
• Positive slope means the line rises from left to right.
• Negative slope means the line falls from left to right.
• Zero slope is a horizontal line; undefined slope is a vertical line.

Memory trick: Slope is 'Rise over Run' – remember the vertical first!

The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function. For a parabola y = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).

• For a < 0, the parabola opens downwards, and the vertex is a maximum.
• For a > 0, the parabola opens upwards, and the vertex is a minimum.
• The x-coordinate of the vertex gives the input value where the max/min occurs.

Memory trick: Vertex 'V'alue is 'V'ery 'V'ital for 'V'aluable 'V'alues.

Solving a linear inequality involves finding the range of values for a variable that make the inequality true. The process is similar to solving linear equations, with one key difference: multiplying or dividing by a negative number reverses the inequality sign.

• Isolate the variable using inverse operations.
• If you multiply or divide both sides by a negative number, flip the inequality sign.
• The solution is a range of values, often represented on a number line or in interval notation.

Memory trick: Balance the scale, flip if you negate!

For a function of the form y = A sin(Bx + C) + D, the amplitude is |A|, and the vertical shift is D. The range is [D - |A|, D + |A|], meaning the maximum value is D + |A| and the minimum is D - |A|.

• Standard sine/cosine range: [-1, 1]
• Amplitude (A) scales the range: [-A, A]
• Vertical shift (D) moves the range: [D-A, D+A]
• Maximum value = D + A

Memory trick: Amplitude stretches, Midline shifts – Max is up, Min is down!

The practical domain of a function is the set of all input values (x or t) for which the function is defined and makes sense in the context of a real-world problem.

• Restricted by physical constraints (e.g., time cannot be negative).
• Restricted by problem statements (e.g., 'during the first 10 hours').
• Often a subset of the mathematical domain.
• Must result in meaningful output values.

Memory trick: Real-world Rules, not just Math's Tools!

The half-life of a substance is the time required for half of the initial amount of the substance to undergo decay. On a decay graph, it's the time taken for the amount to drop to 50% of its initial value.

• Find initial amount (y-intercept).
• Calculate half of the initial amount.
• Locate this half-amount on the y-axis.
• Trace horizontally to the graph, then vertically down to the x-axis to find the time.

Memory trick: Start Amount, Halve It, Find the Time!

An exponential growth function models situations where a quantity increases by a fixed percentage or factor over equal time intervals. It's typically represented as y = a(b)^(t/k), where 'a' is the initial value, 'b' is the growth factor, 't' is time, and 'k' is the time period for the growth factor.

• Base 'b' is usually > 1 (e.g., 2 for doubling, 3 for tripling).
• Initial amount 'a' is the y-intercept.
• Exponent (t/k) adjusts for growth occurring over specific intervals (k).
• Used for populations, investments, and compound interest.

Memory trick: Exponential Growth is 'A' times 'B' to the 'T over K'!

For a quadratic function f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex. The x-coordinate of the vertex is -b/(2a). The y-coordinate (the max/min value) is found by plugging this x-coordinate back into the function.

• If 'a' is positive, the parabola opens upward, and the vertex is a minimum.
• If 'a' is negative, the parabola opens downward, and the vertex is a maximum.
• The vertex provides the extreme value (highest or lowest point) of the function.

Memory trick: Find the peak's location, then its height!

A function where the independent variable appears as an exponent, leading to growth or decay by a constant multiplicative factor.

• Form: y = a * b^x (a is initial value, b is growth/decay factor)
• Characterized by rapid increase or decrease
• Common in population growth, compound interest, radioactive decay

Memory trick: Linear is a line, Quad is a curve, Expo is a boom!

This involves setting up a quadratic equation to represent the area of a geometric shape, typically a rectangle, when one dimension is expressed in terms of another, and the total area is known.

• Area of a rectangle = length × width.
• Express one dimension in terms of the other (e.g., L = W + x).
• Substitute into the area formula, resulting in a quadratic equation.
• Solve the quadratic equation and discard non-physical solutions (e.g., negative length/width).

Memory trick: Area of a 'Rect'angle is 'L' times 'W', often leading to 'Quad'ratics!

In A(t) = A_0 * b^(t/k), A_0 is initial amount, b is decay factor (e.g., 0.5 for half-life), and k is the decay period (e.g., half-life).

• A_0 = initial amount
• b = decay factor (fraction remaining after one period)
• k = length of one decay period
• t = time elapsed

Memory trick: Start with A_0, Decay by B, Time by K!

An exponential decay function, typically y = a(b)^x or y = a(1-r)^x, models situations where a quantity decreases by a fixed percentage over equal time intervals.

• 'a' is the initial amount.
• 'b' (or 1-r) is the decay factor, where 'r' is the decay rate as a decimal.
• 'x' (or t) is the number of time periods.
• The decay factor (b) is always between 0 and 1 (0 < b < 1).

Memory trick: Decay is 'E'xponential when it's 'E'very time by a 'P'ercentage!

The y-intercept of a function (the value of y when x=0) represents the initial amount, starting value, or fixed cost in a real-world scenario.

• Always occurs when the independent variable (x) is zero.
• Represents the 'beginning' state or fixed value.
• Found by setting x=0 and solving for y.
• Crucial for understanding initial conditions in models.

Memory trick: Y-Intercept: X is Zero, It's the Start!

Exponential decay models describe quantities that decrease at a rate proportional to their current value. The general form is A(t) = a(b)^t, where 'a' is the initial amount, 'b' is the decay factor (0 < b < 1), and 't' is time.

• The decay factor 'b' is found by dividing consecutive terms or by solving for 'b' using two points.
• The initial amount 'a' is the value of A(t) when t=0.
• Exponential decay curves approach the x-axis but never touch it (asymptote).

Memory trick: Start with 'a', multiply by 'b' every 't'!

Parallel lines are lines in a plane that are always the same distance apart and never intersect. They have the same slope.

• Same slope (m1 = m2)
• Different y-intercepts (b1 ≠ b2)
• Never intersect
• Can be vertical (undefined slope)

Memory trick: Same Slope, Different Stop (y-intercept)!

The slope of a line represents the average rate of change of the dependent variable with respect to the independent variable over an interval. It's calculated as the ratio of the change in y to the change in x.

• Formula: Average Rate of Change = (Change in Y) / (Change in X).
• Similar to the slope formula (y2 - y1) / (x2 - x1).
• Units are typically 'units of Y per unit of X'.
• For non-linear functions, it's the slope of the secant line between two points.

Memory trick: Rate of Change is 'Rise over Run' between two points, like a slope!

The graph of a function visually depicts the relationship between input and output values. Different types of functions (linear, quadratic, exponential) have distinct graphical shapes.

• Linear: Straight line (constant rate of change).
• Quadratic: Parabola (rate of change is linear).
• Exponential: Curve with increasing (growth) or decreasing (decay) steepness.
• Constant: Horizontal line (zero rate of change).

Memory trick: Linear is 'Line', Quadratic is 'Curve', Exponential 'Explodes'!

To solve for a variable in an exponent, logarithms are typically used. The property log(b^x) = x * log(b) is key to bringing the exponent down.

• If b^x = y, then x = log_b(y).
• Can use common log (log₁₀) or natural log (ln).
• Property: log(A^B) = B * log(A).
• When dividing by a negative number in an inequality, flip the inequality sign.

Memory trick: Log It Down, Isolate and Solve!

The vertex of a parabola is the highest or lowest point on its graph. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by the formula x = -b/(2a).

• If a > 0, the parabola opens upward, and the vertex is a minimum point.
• If a < 0, the parabola opens downward, and the vertex is a maximum point.
• The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.

Memory trick: Vertex is the peak or valley!

Linear functions have a constant rate of change (slope) and are of the form y = mx + b. Exponential functions have a constant multiplication factor (growth/decay rate) and are of the form y = ab^x, where the variable is in the exponent.

• Linear graphs are straight lines; exponential graphs are curves.
• Linear change involves adding/subtracting the same amount; exponential change involves multiplying/dividing by the same factor.
• The independent variable in an exponential function is an exponent.

Memory trick: Linear is a line, Exponential is an exponent!

An exponential growth function models situations where a quantity increases by a constant percentage over equal time intervals. It is defined by an initial amount and a growth factor.

• General form: y = a * b^x or y = P₀(1 + r)^t
• P₀ (or 'a') is the initial amount.
• r is the growth rate (as a decimal).
• b (or 1+r) is the growth factor.

Memory trick: Linear adds, Exponential multiplies – know the difference!

For a parabola, the function decreases until it reaches its vertex (if it opens upwards) or increases until it reaches its vertex (if it opens downwards). The axis of symmetry (-b/2a) marks the turning point.

• Parabola opens up (a>0): Decreases then increases.
• Parabola opens down (a<0): Increases then decreases.
• Vertex x-coordinate: -b/(2a).
• Intervals are defined relative to the vertex's x-coordinate.

Memory trick: Vertex is the Turn, 'a' tells the Trend!

Interpreting a graph involves extracting specific information by locating points, reading values from axes, and understanding the relationship between variables represented by the curve or line.

• Horizontal axis usually represents the independent variable (input).
• Vertical axis usually represents the dependent variable (output).
• To find output for a given input, locate input on horizontal axis, move to graph, then to vertical axis.
• To find input for a given output, locate output on vertical axis, move to graph, then to horizontal axis.

Memory trick: Axis, Line, Point – Information you've got!

An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant (common difference). An arithmetic series is the sum of the terms of an arithmetic sequence.

• General term: an = a1 + (n-1)d, where a1 is the first term and d is the common difference.
• Sum of n terms: Sn = n/2 * (a1 + an) or Sn = n/2 * (2a1 + (n-1)d).
• To find the next term, add the common difference.

Memory trick: Add the first and last, multiply by half the count!

For a quadratic function f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex. If 'a' is negative, it's a maximum; if 'a' is positive, it's a minimum. The x-coordinate of the vertex is -b/(2a).

• Vertex x-coordinate: x = -b / (2a)
• Vertex y-coordinate (max/min value): f(-b / (2a))
• Parabola opens down (a<0) means vertex is a maximum.
• Parabola opens up (a>0) means vertex is a minimum.

Memory trick: Vertex's Point, Peak or Valley, that's the extreme!

Simple interest is a quick method of calculating the interest charge on a loan or investment. It is determined by multiplying the principal amount by the interest rate and the number of periods.

• Formula: I = P × R × T
• Rate (R) must be in decimal form.
• Time (T) must be in years.

Memory trick: Principal times Rate times Time, that's how simple interest climbs!

Compound percentage growth occurs when an initial quantity increases by a certain percentage, and then the new total increases by that same percentage in subsequent periods, leading to exponential growth.

• Growth is applied to the new total each period.
• Can be calculated using the formula P * (1 + r)^n, where P is initial, r is rate, n is periods.
• Results in a larger total increase than simple interest over time.

Memory trick: Grow, then grow on the new 'glow'.

Division with decimals in a practical context involves dividing a total quantity by a per-unit quantity to determine the number of units that can be made or obtained. The result often requires considering only whole units.

• Divide total by unit amount.
• The quotient indicates the number of units.
• For 'how many full units', disregard any fractional remainder.

Memory trick: Total cloth, cut by shirt, gives whole shirts.

Area calculation determines the total surface within a 2D shape. Unit conversion adjusts quantities between different units of measurement.

• Area of a rectangle = length × width.
• Real-world problems often require rounding up to ensure sufficient quantity.
• Ensure units are consistent before performing calculations.

Memory trick: Always paint the wall, then divide by coverage, and always round up!

The median is the middle value in a dataset when the numbers are arranged in numerical order. If there's an even number of data points, the median is the average of the two middle numbers.

• Requires data to be ordered (ascending or descending).
• Less affected by outliers than the mean.
• For odd N: middle value; for even N: average of two middle values.

Memory trick: Mean is average, Median is middle, Mode is most.

Dividing mixed numbers involves converting them into improper fractions, then multiplying by the reciprocal of the divisor.

• Convert mixed numbers to improper fractions first.
• Dividing by a number is the same as multiplying by its reciprocal.
• Simplify the resulting fraction if possible, and convert back to a mixed number for the final answer.

Memory trick: Mixed to improper, then flip and multiply, the answer will fly!

To subtract fractions with different denominators, first find a common denominator, convert both fractions to equivalent fractions with that common denominator, and then subtract the numerators while keeping the common denominator.

• Find the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCM as denominator.
• Subtract the numerators and keep the common denominator.
• Simplify the resulting fraction if possible.

Memory trick: Start minus end, common ground, difference found.

A proportion states that two ratios are equal. When one of the ratios involves a fraction, set up the proportion and use cross-multiplication to solve for the unknown variable.

• Set up as (a/b) = (c/d).
• Cross-multiply: a*d = b*c.
• Solve for the unknown variable.
• Convert improper fractions to mixed numbers if needed.

Memory trick: Equal ratios, cross-multiply, find the missing piece.

To express a fraction (part/whole) as a percentage, divide the part by the whole to get a decimal, then multiply the decimal by 100.

• Percentage means 'out of one hundred'.
• Formula: (Part / Whole) * 100%.
• Useful for comparing different ratios.

Memory trick: Part over whole, then times a hundred for the goal.

To find a remaining fractional part after some parts have been removed, first sum the parts removed, and then subtract this sum from the whole, represented as 1.

• A whole is represented as 1 (or n/n).
• Fractions must have a common denominator for addition/subtraction.
• The remaining fraction is 1 - (sum of removed fractions).

Memory trick: The whole pie minus eaten slices equals what's left.

Sequential discounts are applied one after another, where each subsequent discount is calculated on the *already discounted* price, not the original price. Fractions and percentages can be mixed in these calculations.

• Apply discounts one at a time, in order.
• Each discount is based on the current price, not the original.
• Convert percentages to decimals and fractions to decimals for easier calculation.

Memory trick: Discount, then discount the new price again.

To calculate a percentage of a number expressed in scientific notation, convert the percentage to a decimal, then multiply the decimal by the coefficient of the scientific notation number, keeping the power of 10 consistent, and finally adjust to proper scientific notation if needed.

• Convert percentage to decimal (e.g., 15% = 0.15).
• Multiply the decimal by the coefficient of the scientific notation.
• Adjust the coefficient and exponent to maintain proper scientific notation (coefficient between 1 and 10).

Memory trick: Percent becomes decimal, multiply by number, then adjust the power.