Free TABE Math Exam Flashcards

Work in this order: Parentheses, Exponents, Multiplication and Division left to right, then Addition and Subtraction left to right. The trap is treating multiply and divide as two separate steps. 8 - 2 * 3 = 8 - 6 = 2, not 18. And 12 / 2 * 3 = 6 * 3 = 18, not 12 / 6 = 2.

Same signs: add the absolute values and keep that sign, so -7 + (-3) = -10. Different signs: subtract the smaller absolute value from the larger and keep the sign of the number that was larger, so -7 + 3 = -4 but -3 + 7 = 4.

Two negatives give a positive, one negative gives a negative: -6 * -4 = 24 but -6 * 4 = -24. Subtracting a negative is adding: 5 - (-3) = 5 + 3 = 8. In a long product, count the negative factors - an even count is positive, an odd count is negative.

You need a common denominator first, and the denominators themselves are never added. For 1/4 + 1/6 the LCD is 12, so 3/12 + 2/12 = 5/12. Add only the numerators and keep the shared denominator. Writing 1/4 + 1/6 = 2/10 is the classic error.

Multiply straight across with no common denominator needed: 2/3 * 3/5 = 6/15 = 2/5. To divide, multiply by the reciprocal (flip the second fraction): (3/4) / (2/5) = 3/4 * 5/2 = 15/8. Change mixed numbers to improper fractions before multiplying or dividing.

Fraction to decimal: divide top by bottom, 3/8 = 0.375. Decimal to percent: move the point 2 places right, 0.375 = 37.5%. Percent to decimal: move 2 places left, 37.5% = 0.375. Percent to fraction: put it over 100 and reduce, 25% = 25/100 = 1/4.

Multiply by a fraction equal to 1, arranged so the unwanted unit cancels. To change 5 feet to inches: 5 ft * (12 in / 1 ft) = 60 in. If the answer is wildly too big or too small, the factor was flipped. Know 1 ft = 12 in, 1 mi = 5,280 ft, 1 m = 100 cm, 1 hr = 60 min.

A rational number can be written as a fraction of integers, which includes terminating and repeating decimals such as 0.75, 2/3, and -4. An irrational number never terminates or repeats, such as pi or sqrt(2). To place sqrt(50), bracket it with perfect squares: 49 < 50 < 64, so it sits between 7 and 8, just above 7.

Undo the operations in reverse order - first undo addition or subtraction, then undo multiplication or division - doing the same thing to both sides. For 3x + 7 = 22, subtract 7 to get 3x = 15, then divide by 3 to get x = 5. Substitute back to check.

Distributing multiplies the outside term by every term inside, sign included: 4(2x - 3) = 8x - 12. Like terms share the same variable and exponent, so only their coefficients combine: 8x - 12 + 5x = 13x - 12. You cannot combine 3x with 3x^2.

Collect variable terms on one side and numbers on the other. For 5x - 4 = 2x + 11, subtract 2x to get 3x - 4 = 11, add 4 to get 3x = 15, so x = 5. Moving the smaller variable term first keeps the coefficient positive and prevents sign mistakes.

Solve an inequality exactly like an equation, with one extra rule: multiplying or dividing both sides by a NEGATIVE reverses the direction. So -2x < 10 becomes x > -5 after dividing by -2. Adding or subtracting a negative number never flips the symbol.

Same base multiplied: add the exponents, x^3 * x^4 = x^7. Same base divided: subtract them, x^7 / x^2 = x^5. Power raised to a power: multiply them, (x^3)^2 = x^6. These apply only when the bases match, so x^3 * y^4 will not simplify.

Any nonzero base to the 0 power equals 1, so 7^0 = 1 and (-5)^0 = 1. A negative exponent means take the reciprocal - it does not make the answer negative: 2^-3 = 1/2^3 = 1/8. Combining both rules, 3^2 * 3^-5 = 3^-3 = 1/27.

Write the value as a number from 1 up to 10 times a power of 10: 47,000 = 4.7 * 10^4 and 0.00062 = 6.2 * 10^-4. A positive exponent means a large number, a negative exponent a small one. To compare two values, look at the exponents first, then the leading numbers.

Raising a value by 5% is multiplying by 1.05, because a + 0.05a = 1.05a. A 20% discount means multiplying by 0.80, not by 0.20, so a $60 item marked down 20% costs 60 * 0.80 = $48. Percent change = (new - old) / old * 100.

Match the wording to operations: 'more than' and 'increased by' mean +; 'less than' means - and reverses the order, so 5 less than x is x - 5; 'of' means multiply; 'per' or 'each' signals a rate or division; and 'is' means =. Name the unknown with a letter before writing anything else.

Solve one equation for a variable, substitute it into the other, then back-substitute. Given y = 2x + 1 and 3x + y = 11: 3x + (2x + 1) = 11, so 5x = 10 and x = 2, which gives y = 5. The solution (2, 5) is the point where the two lines cross.

Treat the letter you want as the unknown and undo the operations surrounding it, doing the same to both sides. Simple interest I = P * r * t solved for r gives r = I / (P * t). The triangle area A = (1/2)bh solved for h gives h = 2A / b.

Set the equation equal to zero first. If it factors, set each factor to zero: x^2 + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, so x = -2 or x = -3. Otherwise use the quadratic formula for ax^2 + bx + c = 0: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Expect two solutions.

Write two equal ratios with matching units in matching positions, then cross multiply. If 3 pounds cost $7.50, then 8 pounds cost x: 3/7.50 = 8/x, so 3x = 60 and x = $20. The usual mistake is mixing the order - pounds over dollars on one side and dollars over pounds on the other.

A unit rate is the amount per one unit, so 180 miles in 3 hours is 60 miles per hour. In a proportional relationship y = kx, that rate is the constant k = y / x and it is identical for every pair in the table. Use it to compare buys: $6.00 for 4 lb is $1.50/lb, which beats $8.75 for 5 lb at $1.75/lb.

Every input has exactly one output. On a graph use the vertical line test: if any vertical line crosses the graph twice, it is not a function. In a table, the same x paired with two different y values disqualifies it. Repeated y values are perfectly fine.

f(x) names the output for input x; it does not mean f times x. To evaluate, substitute the value everywhere x appears. If f(x) = 3x - 4, then f(5) = 3(5) - 4 = 11 and f(-2) = 3(-2) - 4 = -10. Solving f(x) = 11 asks the reverse question: which input produces 11?

Slope m = (y2 - y1) / (x2 - x1), the rise over the run. Through (1, 3) and (5, 11): m = (11 - 3) / (5 - 1) = 8/4 = 2. Subtract the coordinates in the same order top and bottom. A horizontal line has slope 0; a vertical line has an undefined slope.

m is the slope, the rate of change, and b is the y-intercept, the value when x = 0. In y = 25x + 100 the line rises 25 for each 1 across and crosses the y-axis at 100. To graph it, plot (0, b) first and then count the rise and run from that point.

The slope is the per-unit rate of change and the y-intercept is the starting amount before anything happens. If cost C = 15h + 50, the $50 is a fixed fee charged for zero hours and the $15 is the charge per hour. Always attach units - dollars per hour, not just 15.

Divide the change in outputs by the change in inputs. If x runs 0, 2, 4, 6 while y runs 5, 11, 17, 23, then y rises 6 for every 2 of x, a rate of 3 per unit. Equal differences over equal steps mean the function is linear; if the differences vary, it is not.

The y-intercept is where the graph meets the y-axis, so set x = 0. The x-intercept is where it meets the x-axis, so set y = 0. For 2x + 5y = 20, setting x = 0 gives y = 4 and setting y = 0 gives x = 10. Zeroing out the wrong variable is the common slip.

Linear growth adds the same amount each step, a constant difference: 5, 8, 11, 14 rises by 3. Exponential growth multiplies by the same factor each step, a constant ratio: 5, 10, 20, 40 doubles. Given enough steps, exponential growth overtakes any linear growth no matter how steep the line.

It is simply the slope between the two endpoints: (f(b) - f(a)) / (b - a). If f(2) = 7 and f(6) = 23, the average rate of change on the interval from 2 to 6 is (23 - 7) / (6 - 2) = 4 per unit. For a straight line this matches the slope everywhere; for a curve it is only an average.

Find the slope first, then solve for b using either point. Through (2, 5) and (4, 11): m = (11 - 5) / (4 - 2) = 3. Substituting gives 5 = 3(2) + b, so b = -1 and the line is y = 3x - 1. Verify with the other point: 3(4) - 1 = 11.

Perimeter is the distance around, in linear units such as ft. Area is the surface covered, in square units such as sq ft. Volume is the space inside, in cubic units such as cu ft. Reading 'how much fencing' as area, or 'how much carpet' as perimeter, is the most common geometry error.

Rectangle: A = l * w. Parallelogram: A = b * h. Triangle: A = (1/2) * b * h. In each case h is the perpendicular height, never the slanted side. A triangle with base 10 and height 6 has area (1/2)(10)(6) = 30 square units.

Circumference C = 2 * pi * r, which is the same as pi * d. Area A = pi * r^2. The radius is half the diameter, and substituting the diameter for r is the top circle mistake. A circle of diameter 10 has r = 5, so C = 10pi (about 31.4) and A = 25pi (about 78.5).

Rectangular prism (box): V = l * w * h. Cylinder: V = pi * r^2 * h, which is the circular base area times the height. A can with radius 3 in and height 10 in holds pi(9)(10) = 90pi, about 283 cubic inches. Convert every measurement to the same unit before multiplying.

Cone: V = (1/3) * pi * r^2 * h. Pyramid: V = (1/3) * B * h, where B is the area of the base. Sphere: V = (4/3) * pi * r^3. A cone holds one third of the cylinder with the same base and height, so dropping the 1/3 triples the answer.

For right triangles only: a^2 + b^2 = c^2, where c is the hypotenuse, the side opposite the right angle and always the longest. Legs 6 and 8 give 36 + 64 = 100, so c = 10. To find a leg instead, subtract: if c = 13 and a = 5, then b^2 = 169 - 25 = 144, so b = 12.

Build a right triangle from the horizontal and vertical gaps and apply the Pythagorean theorem: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). From (1, 2) to (4, 6) the gaps are 3 and 4, so d = sqrt(9 + 16) = 5. Squaring erases negatives, so the order of subtraction does not matter.

Complementary angles add to 90 degrees and supplementary angles add to 180. Vertical angles, formed opposite each other where two lines cross, are equal. The three angles of any triangle add to 180, so if a right triangle has a 35 degree angle, the third is 180 - 90 - 35 = 55.

Mean = the sum of the values divided by how many there are, so 4, 8, 9, 15 averages 36/4 = 9. Reverse it by multiplying: averaging 85 across 5 tests requires 425 total points, so after scoring 82, 90, 78, and 88 (338 points), the fifth test must be 87.

Put the values in order first, then take the middle one: for 3, 7, 9, 12, 20 the median is 9. With an even count, average the two middle values, so 4, 6, 10, 14 has median (6 + 10)/2 = 8. Forgetting to sort the list first is the most common median mistake.

The mode is the value appearing most often, and a data set can have several modes or none. The range is the largest value minus the smallest, a single number describing spread. For 2, 5, 5, 9, 14 the mode is 5 and the range is 14 - 2 = 12.

The mean gets pulled toward an extreme value while the median barely shifts. For 20, 22, 24, 26 both the mean and median are 23. Adding a 200 sends the mean to 58.4 but moves the median only to 24. For skewed data such as incomes, the median describes the center better.

A box plot displays five numbers: minimum, Q1, median, Q3, and maximum. The box runs from Q1 to Q3 and contains the middle 50% of the data. IQR = Q3 - Q1, so Q1 = 30 and Q3 = 46 gives an IQR of 16. One long whisker signals a skewed distribution, not a mistake.

P(event) = favorable outcomes / total outcomes, always between 0 and 1 (0% to 100%). Drawing a red marble from 5 red and 15 others is 5/20 = 1/4 = 0.25 = 25%. The complement rule saves work: P(not the event) = 1 - P(event), so P(not red) = 0.75.

For independent events, where one does not affect the other, multiply: two coin flips both landing heads is (1/2)(1/2) = 1/4. Without replacement the second denominator drops, so drawing 2 red from 5 red out of 20 is (5/20)(4/19). For 'at least one', compute 1 - P(none).

Check the axis labels, the units, and the scale before comparing anything. A y-axis starting at 90 instead of 0 makes small gaps look enormous, and a scale counting by 25 means one gridline is not one unit. Read values off the axis rather than judging bar heights by eye.

Positive association means y rises as x rises, negative means y falls as x rises, and a shapeless cloud means no association. Also look for clusters and outliers. A line of best fit models a roughly linear pattern, but even a strong association is not proof that one variable causes the other.

In a fitted line y = mx + b, the slope is the predicted change in y for each 1-unit increase in x, and the intercept is the predicted y when x = 0. For plant height h = 1.5s + 4, each extra hour of sunlight predicts 1.5 cm more growth. Predictions far outside the data range are unreliable.