Free Praxis 5003 Exam Flashcards

In 3,472 the digit 4 sits in the hundreds PLACE, so its VALUE is 400, not 4. Place names the position; value equals the digit times the place value. Expanded form records both: 3,472 = 3,000 + 400 + 70 + 2.

A digit represents 10 times what the same digit represents one place to its right, and one-tenth of what it represents one place to its left. Extend it across several places: in 23.12 the 2 in the tens place is worth 20 and the 2 in the hundredths place is worth 0.02, so the first is 1,000 times the second.

234 = 2 x 10^2 + 3 x 10^1 + 4 x 10^0 = 200 + 30 + 4. Whole-number exponents on 10 encode place value: 10^0 = 1, 10^1 = 10, 10^2 = 100, 10^3 = 1,000. The same number can also be read as 23 tens and 4 ones - all these forms name one quantity.

Look at the digit one place to the RIGHT of the rounding place. Nearest hundred: the tens digit 8 is 5 or more, so round up to 4,700. Nearest thousand: the hundreds digit 6 is 5 or more, so round up to 5,000. Rounding never depends on digits farther right than that one place.

168 students riding buses that hold 45 gives 168 / 45 = 3 remainder 33. Because the leftover 33 students still need a ride, round UP to 4 buses. A different question drops the remainder (3 buses can be filled completely) or keeps it as a fraction (3 and 33/45 = 3 and 11/15). The context, not the arithmetic, decides.

Multiplying by a factor between 0 and 1 gives a product SMALLER than the other factor: 12 x 3/4 = 9. Dividing by a number between 0 and 1 gives a quotient LARGER than the dividend: 12 / (3/4) = 16. 'Multiplication makes bigger, division makes smaller' is a misconception that holds only for factors and divisors greater than 1.

Parentheses first: 10 - 6 = 4. Exponent next: 4^2 = 16. Then multiplication and division left to right: 3 x 16 = 48, then 48 / 4 = 12. Finally addition: 8 + 12 = 20. Multiplication and division share equal priority, as do addition and subtraction - work left to right, not multiplication before division.

Multiply by the reciprocal of the divisor: 3/4 x 3/2 = 9/8 = 1 and 1/8. The division also asks 'how many 2/3s fit inside 3/4?' Since 3/4 is larger than 2/3, the answer must exceed 1 - a fast reasonableness check that catches a flipped reciprocal.

Commutative changes ORDER: a + b = b + a and ab = ba. Associative changes GROUPING: (a + b) + c = a + (b + c). Distributive spreads multiplication over addition: 4(x + 3) = 4x + 12. Identity leaves the number unchanged: a + 0 = a and a x 1 = a. Subtraction and division are neither commutative nor associative.

Adding a positive moves right; adding a negative moves left. So -7 + 4 = -3 and -7 - 4 = -11. Subtracting is adding the opposite: 5 - (-3) = 5 + 3 = 8. On a number line, |a - b| is the DISTANCE between a and b regardless of order.

Fraction to decimal: divide numerator by denominator, so 3/8 = 0.375. Decimal to percent: multiply by 100, so 0.375 = 37.5%. Percent to fraction: write over 100 and simplify, so 37.5% = 375/1000 = 3/8. Every percent is a fraction with denominator 100.

Convert to one common form: 2/3 is about 0.667, 5/8 = 0.625, and 0.7 stays 0.7. Order: 5/8 < 2/3 < 0.7. Comparing numerators or denominators alone fails. A common denominator works too - with 24ths they are 16/24, 15/24, and 16.8/24.

A unit fraction has numerator 1. As the denominator grows the piece shrinks: 1/4 > 1/8 > 1/12. Any fraction decomposes into unit fractions: 3/5 = 1/5 + 1/5 + 1/5. Fraction comparisons are only valid when both fractions refer to the SAME whole - half a large pizza is not half a small one.

A $40 item marked up 15% costs 40 x 1.15 = $46. The same item discounted 15% costs 40 x 0.85 = $34. Percent change = (amount of change / ORIGINAL amount) x 100, so the original value is always the denominator. A 15% rise followed by a 15% drop does not return to $40 - it lands at $39.10.

A ratio compares two quantities (3 cups flour to 2 cups sugar). A rate compares quantities with different units (120 miles in 2 hours). A unit rate has a denominator of 1 (60 miles per hour). Divide to get one: a 20-ounce box costing $6.99 is 6.99 / 20 = $0.3495 per ounce.

Cross multiply: 3 x 20 = 8x, so 60 = 8x and x = 7.5. Scale factor works too: 20 / 8 = 2.5, so 3 x 2.5 = 7.5. Set proportions up so matching units occupy matching positions - miles over hours on both sides, never miles over hours equal to hours over miles.

Percent means per hundred, so 24% = 24/100 = 0.24. To find what percent 18 is of 45, divide: 18/45 = 0.4 = 40%. To find 24% of 75, multiply: 0.24 x 75 = 18. Identify which number is the whole first - it is the one following 'of'.

84 = 2^2 x 3 x 7 and 60 = 2^2 x 3 x 5. The GCF multiplies only the SHARED primes at their LOWEST powers: 2^2 x 3 = 12. The LCM multiplies ALL primes at their HIGHEST powers: 2^2 x 3 x 5 x 7 = 420.

A prime has exactly two distinct factors, 1 and itself. The number 1 is NEITHER prime nor composite because it has only one factor. 2 is the only even prime. Composite numbers have three or more factors - 9 has 1, 3, and 9. Word clue: splitting into the largest equal groups needs the GCF; finding when two repeating cycles next align needs the LCM.

Round to numbers that are easy to compute with. For 48 x 21, use 50 x 20 = 1,000, so the exact product 1,008 is reasonable. For 3/8 + 5/9, note both addends are near 1/2, so the sum should be near 1 - the exact value 67/72 (about 0.93) passes, while an answer of 8/17 does not.

An expression has no relation symbol; it can be simplified or evaluated but not solved (x + 6). An equation sets two expressions equal and has solutions (x + 6 = 11 gives x = 5). An inequality uses <, >, less-than-or-equal, or greater-than-or-equal and usually has infinitely many solutions (x + 6 > 11 gives x > 5).

It has three TERMS: 7x^2, -3x, and 5. The coefficients are 7 and -3, the constant term is 5, and the leading coefficient is 7 because it belongs to the highest-power term. A term carries its own sign, so the coefficient of the middle term is -3, not 3.

(4x + 5y) + (3x - 2y) = 7x + 3y. Subtraction distributes the minus sign to EVERY term inside: (4x + 5y) - (3x - 2y) = 4x + 5y - 3x + 2y = x + 7y. Only like terms - same variable raised to the same power - can be combined.

Substitute using parentheses: 3(-4)^2 - 2(5) = 3(16) - 10 = 48 - 10 = 38. Apply the exponent before multiplying, and note that (-4)^2 = 16 while -4^2 means -(4^2) = -16. Dropping the parentheses is the most common error on this type.

2(5y + 8) - 6y = 10y + 16 - 6y = 4y + 16. Factoring reverses the process: 4y + 16 = 4(y + 4). Equivalent expressions produce the same value for EVERY substitution - test y = 1 and both 2(5 + 8) - 6 and 4(1) + 16 give 20.

'Six less than twice a number is 20' becomes 2n - 6 = 20, not 6 - 2n = 20. The phrases 'less than' and 'subtracted from' REVERSE the written order, while 'is', 'are', and 'was' become the equal sign. Solving gives 2n = 26 and n = 13.

Collect variables on one side and constants on the other: 8x - 3x = 13 + 17, so 5x = 30 and x = 6. Always verify by substituting back: 8(6) - 17 = 31 and 3(6) + 13 = 31, so both sides agree.

Multiplying or dividing BOTH sides by a negative number reverses the inequality symbol: -3x < 12 becomes x > -4. Adding or subtracting any number never flips it. Solving 2(5y + 8) - 6y < 36 needs no flip: 4y + 16 < 36, so 4y < 20 and y < 5.

Strict inequalities use an OPEN circle at the boundary because that value is excluded; 'or equal to' inequalities use a CLOSED, filled circle because it is included. Both of these shade to the right of 2. The matching 'less than' forms shade to the left with open and closed circles respectively.

A rectangle has area A = lw. Given A = 84 square feet and length 12 feet, solve for width: w = A / l = 84 / 12 = 7 feet. You may rearrange the formula first or substitute the known values first - both routes give the same width, so choose whichever leaves less algebra.

If y changes by a constant amount for each 1-unit change in x, the relationship is linear. For (1, 5), (2, 8), and (3, 11), y rises 3 per step, so y = 3x + 2, where 2 is the value at x = 0. Check the rule against every row, not just the first.

For 7, 11, 15, 19, ..., the common difference is 4, so the nth term is 4n + 3. The 20th term is 4(20) + 3 = 83. Build the rule as (common difference) x n + (first term minus the common difference), then confirm it reproduces the first two terms.

An arithmetic pattern ADDS a constant: 3, 8, 13, 18 adds 5 each time. A geometric pattern MULTIPLIES by a constant ratio: 3, 6, 12, 24 doubles each time. Test the differences first; if they are not constant, test the ratios before assuming a more complicated rule.

Inputs 2, 4, and 6 give outputs 7, 13, and 19. Outputs rise 6 for every input rise of 2, so the rate is 3 per input and the rule is y = 3x + 1. Verify with EVERY row - a rule that fits only the first pair is the classic wrong answer choice.

In c = 2b, giving the cost c in dollars of b bottles at $2 each, b is INDEPENDENT because you choose it and c is DEPENDENT because it follows from b. The dependent variable goes on the vertical axis when the relationship is graphed.

Acute is less than 90 degrees, right is exactly 90, obtuse is between 90 and 180, straight is 180, and reflex is between 180 and 360. Complementary angles sum to 90 and supplementary angles sum to 180, so an angle supplementary to 118 degrees measures 62 degrees.

A line extends forever in both directions, a ray has one endpoint, and a segment has two endpoints. Parallel lines never intersect and stay a constant distance apart. Perpendicular lines intersect at 90 degrees. Intersecting alone does not mean perpendicular - the right angle must be given or provable.

Every square is both a rectangle and a rhombus, and all three are parallelograms - but the converses are false, since a rectangle need not be a square. Triangles carry two labels at once: by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse), so a triangle can be isosceles AND right.

Perimeter is the distance around, in linear units (in, cm). Area is the surface covered, in square units (in^2, cm^2). Volume is the space filled, in cubic units (in^3, cm^3). Equal perimeters do not force equal areas: a 4 by 6 and a 2 by 8 rectangle both have perimeter 20, but areas of 24 and 16.

Triangle: A = (1/2)bh. Parallelogram: A = bh. Trapezoid: A = (1/2)(b1 + b2)h. In all three, h is the PERPENDICULAR height, never the slanted side. A triangle with base 9 cm and height 6 cm has area (1/2)(9)(6) = 27 square centimeters.

Decompose the figure into rectangles and triangles, find each area, and add. An L-shape built from a 10 by 4 rectangle and a 6 by 3 rectangle has area 40 + 18 = 58 square units. Alternatively, subtract the missing corner from a bounding rectangle - choose the route needing fewer unknown side lengths.

Volume V = lwh; surface area SA = 2(lw + lh + wh), which a net shows as six rectangles in three matching pairs. For a 5 by 3 by 2 prism, V = 30 cubic units and SA = 2(15 + 10 + 6) = 62 square units. Doubling every dimension multiplies area by 4 and volume by 8.

Quadrant I is (+, +), II is (-, +), III is (-, -), and IV is (+, -), numbered counterclockwise starting at the upper right. The origin is (0, 0), and points sitting on an axis belong to no quadrant. In (x, y) the x-coordinate moves horizontally first, so (-3, 5) lies in Quadrant II.

Length: 12 in = 1 ft, 3 ft = 1 yd, 5,280 ft = 1 mi. Capacity: 8 fl oz = 1 c, 2 c = 1 pt, 2 pt = 1 qt, 4 qt = 1 gal. Weight: 16 oz = 1 lb, 2,000 lb = 1 ton. Converting to a SMALLER unit multiplies, so 3.5 gallons = 14 quarts; converting to a larger unit divides.

From largest to smallest: kilo (1,000), hecto (100), deka (10), the base unit, deci (0.1), centi (0.01), milli (0.001). Each step is a factor of 10, so conversion just shifts the decimal point: 2.4 km = 2,400 m, and 350 mL = 0.35 L.

Count up in chunks instead of subtracting clock times like decimals. From 10:45 a.m. to 2:20 p.m.: 15 minutes reaches 11:00, then 3 hours reaches 2:00, then 20 minutes finishes - a total of 3 hours 35 minutes. Because 1 hour = 60 minutes, 3.5 hours is 3 h 30 min, not 3 h 50 min.

Mean = 40 / 5 = 8. Median = the middle value of the ordered list = 7. Mode = the most frequent value = 7. Range = 13 - 4 = 9. Order the data before finding the median, and with an EVEN number of values average the two middle ones.

Outliers drag the mean but barely move the median. For salaries of $30k, $32k, $35k, $36k, and $300k, the mean is $86.6k - higher than four of the five values - while the median of $35k represents the group. Use the median for skewed data and the mean for roughly symmetric data.

Circle graphs show parts of a whole, bar graphs compare categories, line graphs show change over time, histograms show frequency across numeric intervals, and scatterplots show the relationship between two numeric variables. Describe a scatterplot trend as positive or negative and linear or nonlinear - and remember association is not causation.

P(event) = favorable outcomes / total equally likely outcomes, always between 0 and 1. From a bag of 4 red and 6 blue marbles, P(red) = 4/10 = 2/5 = 0.4. A probability of 0 is impossible, 1 is certain, and 1/2 is equally likely, so 0.4 is 'less likely than not'. The complement is P(not red) = 1 - 2/5 = 3/5.