GED Mathematical Reasoning Test flashcards

The surface area of a sphere is the total area of its outer surface, calculated by 4 times pi (π) times the square of its radius.

• Formula: A = 4πr²
• Measured in square units.
• Represents the 2D surface of a 3D object.

Memory trick: Four Pi R Squared, for the sphere's full area declared!

The distance around the edge of a circle.

• Formula: C = 2πr or C = πd
• Units are linear (e.g., feet, meters)
• Used for measuring perimeters of circular objects

Memory trick: Round the Radius, Get the Circumference

The volume of a rectangular prism (a box) is the amount of three-dimensional space it occupies. It is found by multiplying its length, width, and height.

• Formula: V = Length × Width × Height
• Units are always cubic units (e.g., in³, cm³)
• Represents the capacity of the container

Memory trick: Length, Width, Height, for Volume's full might!

The area of a uniform strip surrounding a shape, calculated by multiplying the perimeter of the inner shape by the path's width.

• Assumes path is on the outside or inside of the perimeter
• Formula: Perimeter × Width (for a uniform path)
• Requires careful attention to whether the path is added to or subtracted from dimensions

Memory trick: Perimeter Path Paves Perfectly

The amount of two-dimensional space enclosed within a trapezoid.

• Formula: A = 0.5 × (b1 + b2) × h
• b1 and b2 are the lengths of the parallel sides
• h is the perpendicular height between the parallel sides

Memory trick: Half Sum of Bases Times Height

Substituting a given value for the independent variable (often time) into an exponential function to find the corresponding dependent variable value.

• Requires knowledge of base 'e' and calculator use.
• Follow order of operations (exponents first).
• Pay attention to rounding instructions.

Memory trick: Input a number, get an output number.

The sum of the areas of all surfaces of a cylinder, including the two circular bases and the curved lateral surface.

• Composed of two circular bases (2 * πr²) and one rectangular lateral surface (2πrh).
• Formula: A = 2πr² + 2πrh or A = 2πr(r + h).
• Units are square units.

Memory trick: Total surface area is like wrapping paper for the whole can.

The area of a rectangle is the amount of space enclosed within its sides. It is calculated by multiplying its length by its width.

• Formula: Area = Length × Width
• Units are always square units (e.g., ft², m²)
• Used for measuring flat surfaces like floors, walls, or land plots

Memory trick: Length times Width, that's the Area's stride!

Finding the value(s) of the variable(s) that make a linear equation true. This often involves isolating the variable on one side of the equation.

• Linear equations have variables raised to the power of 1.
• Operations (addition, subtraction, multiplication, division) must be applied equally to both sides.
• The goal is to isolate the variable.

Memory trick: Balance the scale to hit the target variable.

The amount of three-dimensional space a cube occupies.

• Calculated by multiplying length × width × height.
• For a cube, all sides are equal, so V = side³.
• Units are cubic units (e.g., cubic feet, cubic meters).

Memory trick: Volume is how much stuff fits inside the box.

The area of a circle is the total space enclosed within its circumference, calculated by pi (π) times the square of its radius.

• Formula: A = πr²
• Measured in square units.
• π is approximately 3.14159.

Memory trick: Pi R Squared, for the circle's full space to be shared!

The amount of three-dimensional space a cylinder occupies, or the capacity it can hold.

• Calculated by multiplying the area of the base (πr²) by the height (h).
• Formula: V = πr²h.
• Units are cubic units (e.g., cubic meters, liters).

Memory trick: Volume is like filling a can with liquid.

The amount of two-dimensional space enclosed by the three sides of a triangle.

• Calculated by half the product of its base and height.
• Formula: A = ½ × base × height.
• The height must be perpendicular to the base.
• Units are square units.

Memory trick: A triangle is half a rectangle, so its area is half too.

In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).

• Applies only to right-angled triangles.
• Formula: a² + b² = c².
• Used to find an unknown side length when two sides are known.

Memory trick: Pythagoras helps find the longest side of a right triangle.

The area of a square can be calculated directly if only its diagonal length is known.

• Formula: Area = d²/2
• Derived from Pythagorean theorem: s² + s² = d² => 2s² = d² => s² = d²/2
• Where 'd' is the length of the diagonal

Memory trick: Diagonal Squared, Then Halved, for Square Area

The sum of the area of the square base and the areas of all four triangular lateral faces.

• Formula: Base Area + Lateral Area
• Base Area = side²
• Lateral Area = 4 × (0.5 × base side × slant height)

Memory trick: Base Plus Four Triangles for Pyramid's Skin

Sets of three positive integers a, b, and c, such that a² + b² = c².

• Common triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
• Knowing these can quickly solve Pythagorean theorem problems.
• Any multiple of a triple is also a Pythagorean triple (e.g., 6, 8, 10).

Memory trick: Pythagorean triples are like special codes for right triangles.

The distance formula calculates the straight-line distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane.

• Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Essentially an application of the Pythagorean theorem.
• Results in a positive value.

Memory trick: Subtract, Square, Add, then Square Root, to find the path's true loot!

The area of the curved side of a cylinder, excluding the top and bottom circular bases.

• Formula: 2πrh
• Used for painting, wrapping, or covering the side of cylindrical objects
• Units are typically square units (e.g., ft², m²)

Memory trick: Radius Height Paint

For a parabola, the vertex is the highest or lowest point. For h(x) = ax^2 + bx + c, the x-coordinate of the vertex is -b/(2a). For h(x) = a(x-r1)(x-r2), the x-coordinate is (r1+r2)/2.

• If 'a' is negative, the parabola opens downward, and the vertex is a maximum.
• If 'a' is positive, the parabola opens upward, and the vertex is a minimum.
• The y-coordinate of the vertex is the maximum/minimum value of the function.

Memory trick: Vertex is where the parabola turns – the peak or the pit.

A scale factor is the ratio by which all dimensions of an object are multiplied to create a scaled version (either larger or smaller) while maintaining proportionality.

• Scale factor = Model dimension / Actual dimension
• All corresponding lengths are scaled by the same factor.
• Units must be consistent when calculating the scale factor (e.g., all cm or all m).

Memory trick: Units match, ratio found, then apply to the new ground!

The amount of three-dimensional space occupied by a sphere, or its capacity.

• Formula: V = (4/3)πr³
• r is the radius of the sphere
• Units are cubic units (e.g., ft³, m³)

Memory trick: Four-thirds Pi-R-Cubed for Sphere's Fluid

Calculating the surface area of a cylinder often involves summing the areas of its circular bases (top and bottom) and its lateral (curved) surface, adapting for specific painting/covering needs.

• Lateral Surface Area: 2πrh
• Area of one circular base: πr²
• Total Surface Area: 2πr² + 2πrh
• For top and lateral only: πr² + 2πrh

Memory trick: Top circle, curved side, sum them up, then divide!

The lateral surface area of a cone is the area of its curved side, excluding the base, calculated using its radius and slant height.

• Formula: A = πrl, where r is radius and l is slant height.
• Slant height (l) is the distance from the apex to a point on the circumference of the base.
• Measured in square units.

Memory trick: Pi R L, for the cone's side spell!

The ratio by which all dimensions of an object are multiplied to create a scaled copy.

• Calculated as (new dimension) / (original dimension).
• A scale factor less than 1 indicates a reduction; greater than 1 indicates an enlargement.
• Used to find corresponding lengths in similar figures or models.

Memory trick: Scale factor is the 'magic number' that makes things bigger or smaller proportionally.

The perimeter of a rectangle is the total distance around its outer edge, calculated by adding twice its length and twice its width.

• Formula: P = 2(L + W) or P = 2L + 2W
• Measured in linear units (e.g., feet, meters).
• Represents the boundary of a 2D shape.

Memory trick: Length plus width, then double that, multiply by cost, and that's that!

Used to find the length of a side in a right-angled triangle when the lengths of the other two sides are known, often applied in real-world scenarios involving heights, distances, and diagonals.

• Formula: a² + b² = c²
• a and b are the lengths of the legs, c is the hypotenuse
• Useful for finding lengths in construction, navigation, etc.

Memory trick: Pole to Ground, Hypotenuse Found

A comparison of two quantities, often used to describe a relationship between different dimensions, such as rise over run.

• Expressed as a:b or a/b
• Can be used to scale measurements proportionally
• Important for understanding slopes and gradients

Memory trick: Rise and Run, Proportional Fun

Using geometric formulas (like perimeter or area) and given relationships to set up and solve linear equations for unknown dimensions.

• Write down relevant geometric formulas.
• Express unknown dimensions in terms of a single variable.
• Substitute expressions into the formula and solve the resulting linear equation.

Memory trick: Geometry facts, linear equation acts!

Solving a quadratic equation in a real-world context involves setting the function equal to a given value and then using methods like factoring, completing the square, or the quadratic formula to find the unknown variable.

• Always set the equation to zero (ax² + bx + c = 0) before solving.
• The quadratic formula is d = [-b ± sqrt(b² - 4ac)] / (2a).
• In real-world problems, only positive and realistic solutions are typically valid.

Memory trick: Set the output, then solve for the input using the quadratic formula.

Converting a real-world scenario that involves a range or limit into a mathematical inequality using variables and inequality symbols.

• Identify the unknown quantities and assign variables.
• Look for keywords that indicate inequality symbols (e.g., 'at least', 'no more than', 'less than', 'greater than').
• Formulate the inequality based on the relationships described.

Memory trick: Words to limits, inequality fits!

The point where total cost equals total revenue, resulting in zero profit. For linear functions, it's found by setting the cost function equal to the revenue function and solving for the quantity.

• Break-even occurs when Cost = Revenue.
• Set C(x) = R(x) and solve for x (the quantity).
• The fixed costs are covered, and there is no profit or loss.

Memory trick: Cost equals Revenue, then you're even!

The vertex of a parabola represents the maximum or minimum value of a quadratic function. For a function in the form ax² + bx + c, the x-coordinate of the vertex is -b/(2a).

• If 'a' > 0, the parabola opens upwards, and the vertex is a minimum.
• If 'a' < 0, the parabola opens downwards, and the vertex is a maximum.
• Substitute the x-coordinate of the vertex back into the function to find the maximum/minimum value.

Memory trick: Vertex finds the highest or lowest peak!

To solve exponential equations where the base is 'e' or a general base, use logarithms. The natural logarithm (ln) is particularly useful for base 'e' because ln(e^x) = x.

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

Memory trick: Isolate 'e', then 'ln' to free the 't'.

Solving equations where the variable is in the exponent by using natural logarithms (ln) to bring the exponent down.

• Isolate the exponential term.
• Take the natural logarithm of both sides (ln).
• Use the logarithm property ln(e^x) = x to simplify.
• Solve for the variable.

Memory trick: Exponential out, Logarithm in!

To solve an exponential equation where both sides can be expressed with the same base, rewrite both sides using that common base, then equate the exponents and solve the resulting equation.

• Identify a common base for all terms.
• Rewrite numbers as powers of the common base.
• If b^x = b^y, then x = y.
• Solve the resulting linear or quadratic equation.

Memory trick: Make bases match, then drop them and solve the powers.

For a parabolic path or shape modeled by a quadratic function y = ax² + bx + c, the maximum height (or depth) is the y-coordinate of the vertex. The x-coordinate of the vertex, -b/(2a), gives the horizontal position where this maximum occurs.

• Parabola opens down if 'a' is negative (max height).
• Vertex x-coordinate = -b/(2a).
• Vertex y-coordinate (max height) = f(-b/(2a)).

Memory trick: The vertex is the arch's highest point, both left-right and up-down.

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.

• Vertex x-coordinate: -b/(2a).
• Substitute x-coordinate into the function to find the max/min value.
• The sign of 'a' determines if the parabola opens up (min) or down (max).

Memory trick: Vertex is the peak or valley of your quadratic journey.

For a quadratic function f(x) = ax² + bx + c, if the leading coefficient 'a' is positive, the parabola opens upwards, and the vertex represents the minimum value of the function.

• Minimum occurs at the vertex x-coordinate: x = -b/(2a).
• Substitute this x-value into the function to find the minimum y-value.
• A positive 'a' indicates a minimum, a negative 'a' indicates a maximum.

Memory trick: Vertex is the bottom of the bowl, find its time coordinate.

For a linear function in the form y = mx + b, 'b' represents the y-intercept, which is the initial value or starting point when x=0.

• The 'b' term in y = mx + b is the initial value.
• The initial value occurs when the independent variable (x or t) is zero.
• It represents the starting amount or condition before any change occurs.

Memory trick: The 'b' in mx+b is where you 'Begin'!

A linear cost function models the total cost (C) as a sum of fixed costs and variable costs, where variable costs are directly proportional to the number of units produced (x).

• C = mx + b, where m is variable cost per unit and b is fixed cost.
• Fixed costs are incurred regardless of production volume.
• Variable costs change with the production volume.

Memory trick: Fixed costs are your party's base fee, variable costs are per guest.

The vertex of a parabola represents the maximum or minimum value of a quadratic function. For a function in the form ax² + bx + c, the x-coordinate of the vertex is -b/(2a).

• If 'a' > 0, the parabola opens upwards, and the vertex is a minimum.
• If 'a' < 0, the parabola opens downwards, and the vertex is a maximum.
• Substitute the x-coordinate of the vertex back into the function to find the maximum/minimum value.

Memory trick: Vertex finds the peak or pit!

Solving a system of linear equations means finding the values of the variables that satisfy all equations in the system simultaneously. Common methods include substitution and elimination.

• Substitution: Solve one equation for a variable, then plug into the other.
• Elimination: Multiply equations to make coefficients of one variable opposites, then add equations.
• A unique solution means the lines intersect at one point.

Memory trick: Two lines meet at one spot, find that shared location.

Calculating the value of an exponential function at a specific point, often used in scenarios like radioactive decay where a quantity decreases by a fixed factor over equal time intervals (half-life).

• Identify the base (often 1/2 for half-life) and the exponent.
• Substitute the given values for variables into the function.
• Perform the exponentiation first, then multiplication.

Memory trick: Plug and Chug, then Watch it grow/shrink!

Rewriting an algebraic expression in a simpler, equivalent form by combining like terms and applying the distributive property.

• Use the distributive property to remove parentheses.
• Identify and combine like terms (terms with the same variable and exponent).
• Combine constant terms.

Memory trick: Distribute, then Combine, make it shine!

For real-world problems modeled by quadratic functions, the maximum or minimum value often represents an optimal condition (e.g., maximum profit, minimum cost, maximum height). This value is found at the vertex of the parabola.

• If the quadratic coefficient 'a' is negative, the vertex gives a maximum value.
• If 'a' is positive, the vertex gives a minimum value.
• The x-coordinate of the vertex, -b/(2a), gives the input value (e.g., number of units) for the max/min.

Memory trick: Find the peak of the profit hill or the bottom of the cost valley.

For a quadratic function in the form f(x) = ax^2 + bx + c, the maximum or minimum value occurs at the vertex. If 'a' is negative, the parabola opens downwards, and the vertex is a maximum. If 'a' is positive, it opens upwards, and the vertex is a minimum.

• Vertex x-coordinate: x = -b / (2a)
• Vertex y-coordinate: Substitute x-coordinate into the function to find the maximum/minimum value.
• Coefficient 'a' determines if it's a maximum (a < 0) or minimum (a > 0).

Memory trick: Vertex Vibe: High point or Low point, always at the turn!

Calculating the value of an exponential function at a specific point, often used in scenarios like population growth or compound interest.

• Identify the initial amount, growth rate (or base), and time.
• Substitute the given values for variables into the function.
• Perform the exponentiation first, then multiplication.

Memory trick: Plug in time, watch it climb!

Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time. It is characterized by rapid acceleration of growth.

• Formula: Final Amount = Initial Amount × (Growth Factor)^Number of Periods.
• Growth factor for doubling is 2.
• The number of periods is the exponent.

Memory trick: Start small, multiply big, power up fast, like a growth-digit.

A proportion is an equation stating that two ratios are equal. To solve for an unknown in a proportion involving fractions, cross-multiplication is often used.

• Set up two equivalent ratios as a proportion.
• Place the unknown variable (x) in the appropriate position.
• Cross-multiply the terms.
• Solve the resulting linear equation for x.

Memory trick: Ratios balance like scales; cross-multiply to find the missing piece.

A unit rate expresses how much of one quantity there is per one unit of another quantity. Proportionality means that two quantities have a constant ratio.

• Unit Rate = Total Quantity / Total Units.
• Can be used to scale quantities up or down.
• Often involves division to find the rate, then multiplication to scale.

Memory trick: Rate first, then multiply, for how far or how much, don't be shy.

A map scale represents the ratio of a distance on the map to the corresponding distance on the ground. Converting between units (e.g., cm to km) requires knowing the standard conversion factors.

• Scale factor applies to linear distances.
• 1 meter = 100 centimeters.
• 1 kilometer = 1000 meters.
• Therefore, 1 kilometer = 100,000 centimeters.

Memory trick: Scale up first, then slide the decimal, from little map to real-world colossal.

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.

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.

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.

Percentages of a whole represent parts of a total quantity, where the sum of all parts expressed as percentages equals 100%.

• The total is always 100%.
• To find a missing percentage, subtract known percentages from 100%.
• To find the number corresponding to a percentage, multiply the total by the percentage (expressed as a decimal).
• Ensure all parts are accounted for to sum to 100%.

Memory trick: Whole pie is 100; subtract the slices to find what's left for you.

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.

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 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.

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.