Free Praxis 8003 Mathematics Practice Test

1. 0.7
2. 0.07 ✓
3. 0.007
4. 7

Answer: B. The places to the right of the decimal point are tenths, hundredths, and thousandths in that order. The 7 sits in the hundredths place, so its value is 7 × 1/100 = 0.07.

• 7: Treats the digit as its face value and ignores position entirely. Place value is set by the digit’s column, so a digit after the decimal point always names a part of one.
• 0.7: Reads the 7 as the tenths digit, giving it ten times its actual place value. The tenths place belongs to the 4, and the second place to the right of the decimal is hundredths.
• 0.007: Places the digit one column too far right, in the thousandths. That place is held by the 8, and the 7 is one column closer to the decimal point.

1. Have the student convert both fractions to decimals using a calculator
2. Model both fractions with fraction strips of the same whole and compare the shaded lengths ✓
3. Have the student memorize the rule that a larger denominator means a smaller fraction
4. Have the student practice finding common denominators for unrelated fraction pairs

Answer: B. The error is conceptual: the student treats the denominator as a count of size rather than as the number of equal parts in one whole. Comparing shaded strips of the same whole makes the part-whole relationship visible, so the student sees that eighths are smaller pieces than fourths and that the same whole holds fewer fourths than eighths.

• Have the student memorize the rule that a larger denominator means a smaller fraction: Substitutes a verbal rule for the underlying idea. A student can recite the rule and still miscompare fractions, because the whole-part reasoning the error comes from is never addressed.
• Have the student practice finding common denominators for unrelated fraction pairs: Targets a procedure the student did not misuse. Finding common denominators is a correct method that was not attempted here, and it does not surface the belief that more parts means a larger fraction.
• Have the student convert both fractions to decimals using a calculator: Uses a tool that reports the answer without confronting the belief. A display may show 0.125 against 0.25, but the original claim about 8 and 4 is left unexamined.

1. An area model split into 20 × 10, 20 × 4, 3 × 10, and 3 × 4 rectangles ✓
2. A number line showing twenty-three jumps of 14
3. A hundreds chart with 322 squares shaded in
4. Fourteen counters arranged in seven pairs

Answer: A. The area model decomposes each factor by place value and shows the four partial products as separate regions. Adding those regions (200 + 80 + 30 + 12) reproduces the 322 that the written algorithm records, so students see where each line of the algorithm comes from.

• A number line showing twenty-three jumps of 14: Represents repeated addition of one factor and hides the place-value decomposition. Jumps of 14 show 23 groups but never display the four partial products the algorithm adds.
• A hundreds chart with 322 squares shaded in: Shows the product rather than its structure. Shading 322 squares displays the result but gives no access to the place-value parts that the written algorithm combines.
• Fourteen counters arranged in seven pairs: Models a grouping task unrelated to the two factors. Grouping counters into pairs illustrates an even number, not the splitting of 23 and 14 into tens and ones.

1. It is invalid, because only one addend may be rounded in an estimate
2. It is reasonable, because 5,000 + 3,000 = 8,000 is within 100 of the exact sum ✓
3. It is too low, because both addends were rounded down
4. It is exact, because rounding to the nearest thousand preserves the sum

Answer: B. The exact sum is 8,089. Rounding 4,987 up to 5,000 adds 13, and rounding 3,102 down to 3,000 subtracts 102, so the two adjustments nearly cancel and the estimate lands within about 1 percent of the exact value.

• It is too low, because both addends were rounded down: Misreads which way each addend moved. 4,987 rounded up to 5,000, so the two adjustments offset each other rather than both reducing the total.
• It is invalid, because only one addend may be rounded in an estimate: States a rule that does not exist. Rounding one addend, both addends, or neither is a methodological choice, and no estimation convention forbids rounding both.
• It is exact, because rounding to the nearest thousand preserves the sum: Claims a property rounding does not have. The exact sum is 8,089, so the rounded estimate differs from it by 89.

1. 5 remainder 7 vans, because the division result should be stated exactly
2. 7 vans, because the remainder is rounded up to the next multiple of 8
3. 5 vans, because a remainder is not a whole van
4. 6 vans, because the 7 students left over still need seats ✓

Answer: D. Five vans hold 40 students, which leaves 7 students unseated. Because every student must travel, the quotient has to be rounded up to 6, and the remainder is what signals that an extra van is needed rather than being discarded.

• 5 vans, because a remainder is not a whole van: Drops the remainder as if those students did not exist. The 7 students are unseated, so reporting 5 vans understates what the situation requires.
• 5 remainder 7 vans, because the division result should be stated exactly: Reports the raw division result without interpreting it. A remainder answers the calculation but not the question of how many whole vans must be reserved.
• 7 vans, because the remainder is rounded up to the next multiple of 8: Rounds the remainder up to a full group before adding it. The remainder is 7, not 8, so this overcounts by one van.

1. Prime factorization of the sum
2. Least common denominator
3. Greatest common factor
4. Least common multiple ✓

Answer: D. The bells next ring together at the smallest number of minutes that is a multiple of both 6 and 8. That value is the least common multiple, which is 24 minutes.

• Greatest common factor: Finds the largest shared divisor instead of the first shared event. The question asks for the earliest time both intervals meet, which is a multiple, not a factor.
• Least common denominator: Names a fraction procedure the problem does not contain. No fractions are being added, so no shared denominator is being sought.
• Prime factorization of the sum: Factors a total that the problem never forms. The meeting time comes from the two intervals and their first common multiple, not from factoring 14.

1. Divide the total amount by the serving size: 2 ÷ 1/4 ✓
2. Divide the serving size by the total amount: 1/4 ÷ 2
3. Subtract one serving from the total amount: 2 − 1/4
4. Multiply the total amount by the serving size: 2 × 1/4

Answer: A. A serving-count question asks how many groups of one fourth fit into the total, which is measurement division. Dividing 2 by 1/4 gives 8, so eight quarter-cup servings fit in the two cups of juice.

• Multiply the total amount by the serving size: 2 × 1/4: Multiplies the total by the serving size instead of dividing. The product, half a cup, is a volume of juice rather than the count of servings the question asks for.
• Divide the serving size by the total amount: 1/4 ÷ 2: Reverses the two quantities, dividing the serving size by the total. That result is smaller than one serving and cannot count how many servings fit.
• Subtract one serving from the total amount: 2 − 1/4: Subtracts a single serving and reports the juice remaining. The question asks how many servings fit, not how much juice is left over.

1. Associative property
2. Distributive property ✓
3. Identity property of multiplication
4. Commutative property

Answer: B. The teacher decomposes 7 into 5 + 2 and multiplies each part by 6, then adds the two partial products. Distributing a factor over a sum is exactly what the distributive property describes, and 30 + 12 = 42 checks the result.

• Commutative property: Concerns the order of two factors rather than the splitting of one factor. Reordering would give 7 × 6, a different move from the one the teacher made.
• Associative property: Concerns how three factors are grouped, as in (6 × 5) × 2. Regrouping multiplies all three numbers instead of adding two partial products.
• Identity property of multiplication: Concerns multiplying by one, which leaves a number unchanged. No factor of one appears in the decomposition of 7.

1. The student ignored the subtraction; the correct value is 6
2. The student multiplied before subtracting; the correct value is 10
3. The student subtracted before multiplying; the correct value is 2 ✓
4. The student subtracted before multiplying; the correct value is 13

Answer: C. Multiplication is performed before subtraction, so 3 × 2 = 6 and 8 − 6 = 2. The student found 8 − 3 first, which reverses the agreed order of operations.

• The student multiplied before subtracting; the correct value is 10: Names the correct order of operations but keeps the student’s wrong result. Multiplying first gives 8 − 6 = 2, not 10.
• The student subtracted before multiplying; the correct value is 13: Identifies the right error and then misstates the outcome. Once 3 × 2 is resolved to 6, the expression is 8 − 6, which equals 2.
• The student ignored the subtraction; the correct value is 6: Describes an error the student did not make and drops a term. Even setting the subtraction aside leaves 6 from 3 × 2, which is not the value of the expression.

1. 4/3
2. 0.75 ✓
3. 0.34
4. 3.4
5. 75% ✓

Answers: B and E. Dividing 3 by 4 gives 0.75, so the decimal form is 0.75. Multiplying a decimal by 100 gives its percent form, and 0.75 × 100 = 75, so the percent form is 75%. Both name the same part of one whole as the fraction 3/4.

• 3.4: Reads the fraction bar as a decimal point and writes the digits in order. 3.4 places the 3 in the ones place, producing a number greater than one.
• 4/3: Reciprocates the fraction by swapping numerator and denominator. The result is greater than one, while 3/4 is less than one.
• 0.34: Appends the numerator and denominator as decimal places. The 3 and 4 are a numerator and a denominator, not tenths and hundredths.

Answer: 12. The discount is 15% of 80. Convert the percent to a decimal and multiply: 0.15 × 80 = 12, so the discount is $12. Subtract to check the sale price: 80 − 12 = $68, which is a different quantity from the one the item asks for.

Answer: 0.875. Rewrite 3/4 with a denominator of 8 by multiplying numerator and denominator by 2, which gives 6/8. Adding 6/8 + 1/8 = 7/8. Dividing 7 by 8 gives the decimal form 0.875.

1. Multiply by 2 and add 3
2. Multiply by 5
3. Add 4
4. Multiply by 3 and add 2 ✓

Answer: D. Each output is the input multiplied by 3 with 2 added: 3(1) + 2 = 5, 3(2) + 2 = 8, and 3(3) + 2 = 11. The rule holds for every row, and the outputs increase by 3 each time the input increases by 1.

• Multiply by 5: Matches only the first row and fails the rest. An input of 2 would give 10 rather than 8, so the rule breaks on the very next entry.
• Add 4: Uses a constant addition that fits no more than one row. Input 1 does give 5, but 2 + 4 = 6 rather than 8.
• Multiply by 2 and add 3: Fits the first row and then drifts away. Input 2 gives 7 instead of 8, and input 3 gives 9 instead of 11.

1. The student has not yet learned to subtract within ten
2. The student does not apply the commutative property when adding numbers
3. The student cannot yet recall the addition facts within ten
4. The student reads the equals sign as an instruction to compute rather than as a statement that both sides balance ✓

Answer: D. The student treated everything to the left of the equals sign as the problem and wrote its answer in the box, which leaves 5 + 2 on the right and breaks the balance. The box must make both sides equal, so it holds 5.

• The student cannot yet recall the addition facts within ten: Assumes a computation gap where the arithmetic is correct. The sum 3 + 4 really is 7, so the addition facts are not what is missing here.
• The student has not yet learned to subtract within ten: Attributes the error to a skill the task does not require. Nothing in the item calls for a difference, and the student never attempts one.
• The student does not apply the commutative property when adding numbers: Names a property unrelated to the error. The order of the addends is not what the student misread; the meaning of the equals sign is.

1. Remove one bag from each side
2. Remove four unit blocks from each side ✓
3. Add four unit blocks to each side
4. Split both sides into three equal groups of blocks

Answer: B. Undoing the addition isolates the term holding the unknown. Removing four blocks from both sides keeps the scale balanced and leaves 3n = 15, after which dividing both sides by 3 gives n = 5.

• Split both sides into three equal groups of blocks: Divides before the constant has been cleared. Nineteen blocks do not split into three equal groups while four extra blocks still sit on the left.
• Add four unit blocks to each side: Moves the constant in the wrong direction. Adding blocks enlarges both sides instead of isolating the three bags.
• Remove one bag from each side: Removes an unknown that only one side has. The right side holds no bag, so this would unbalance the scale rather than simplify it.

1. 5n, which multiplies the unknown number by five
2. n − 5, which subtracts five from the unknown number ✓
3. n + 5, which adds five to the unknown number
4. 5 − n, which subtracts the unknown number from five

Answer: B. The phrase names the unknown first and then removes five from it, so the unknown is written before the subtraction. In n − 5 the value is five smaller than n for every n, which is what "five less than a number" describes.

• 5 − n, which subtracts the unknown number from five: Reverses the order of the subtraction. Putting five first makes the result shrink as n grows, the opposite of the relationship the phrase describes.
• 5n, which multiplies the unknown number by five: Reads "less than" as multiplication. The phrase describes removing five from a number, not taking five copies of it.
• n + 5, which adds five to the unknown number: Reverses the direction of the change. Five less than a number is smaller than that number, while adding five makes the result larger.

1. $3.50, found by subtracting $1.00 from the total cost
2. $0.75, found by dividing the number of notebooks by the total cost
3. $27.00, found by multiplying the total cost by the number of notebooks
4. $0.75, found by dividing the total cost by the number of notebooks ✓

Answer: D. A unit rate divides the quantity of money by the number of items so that the result describes one item. Here 4.50 ÷ 6 = 0.75, so each notebook costs $0.75.

• $0.75, found by dividing the number of notebooks by the total cost: Pairs the right value with an operation that cannot produce it. Dividing 6 by 4.50 gives about 1.33, so the method and the stated result contradict each other.
• $27.00, found by multiplying the total cost by the number of notebooks: Compounds the cost instead of distributing it. Multiplying dollars by notebooks produces a larger total, not the price of one.
• $3.50, found by subtracting $1.00 from the total cost: Subtracts an amount the problem never supplies. No relationship in the item justifies removing a dollar from the total.

1. 18 ✓
2. 9
3. 36
4. 12

Answer: A. The exponent applies only to x, so substitute first and square: 3² = 9. The coefficient 2 then multiplies that value, giving 2 × 9 = 18.

• 9: Squares x correctly and then drops the coefficient. The 2 in front still multiplies the squared value, doubling 9 to 18.
• 12: Multiplies the coefficient by the exponent and by x, treating the exponent as a factor of the coefficient. The exponent attaches to x only.
• 36: Squares the whole product instead of the variable alone. The expression means 2 · (x²), so the 2 stays outside the exponent.

1. 5n + 4
2. 5n − 1 ✓
3. n + 5
4. 4n

Answer: B. Consecutive terms increase by 5, so the expression needs a coefficient of 5 on n. Testing n = 1 gives 5(1) − 1 = 4, which matches the first term, and the rule continues to produce 9, 14, and 19.

• 4n: Multiplies the first term by n, which fits only the first position. At n = 2 the rule gives 8 rather than 9.
• 5n + 4: Uses the correct common difference but adds the first term instead of the right constant. At n = 1 the rule gives 9, one term ahead of the sequence.
• n + 5: Treats the sequence as growing by one each step. The given terms grow by 5, so adding n cannot keep pace.

1. (x + 3) + (x + 3) + (x + 3) + (x + 3) ✓
2. 4x + 12 ✓
3. 4x + 3
4. 7x
5. 4x + 7

Answers: A and B. Multiplying each term inside the parentheses by 4 gives 4x + 12, so the expanded form is 4x + 12. The same value can be written by repeating the factor four times and adding the copies, which collects to 4x + 12 as well.

• 4x + 3: Distributes the 4 to the variable only and leaves the 3 untouched. The factor multiplies the whole sum, so 3 becomes 12.
• 7x: Combines unlike terms by adding the coefficient and the constant. The 3 is not a term in x, so 4x and 3 cannot merge into one x term.
• 4x + 7: Adds the 3 to the coefficient rather than multiplying it. Distribution multiplies every term inside the parentheses by 4, and 4 × 3 = 12.

Answer: 9. Add 7 to both sides to undo the subtraction: 5x = 45. Divide both sides by 5 to undo the multiplication: x = 9. Substituting back checks the work: 5(9) − 7 = 45 − 7 = 38.

1. The student computed the perimeter; the area is 6 + 4 = 10 square centimeters
2. The student computed the perimeter; the area is 6 × 4 = 24 square centimeters ✓
3. The student used the wrong unit; the area is 20 square centimeters
4. The student should have doubled the sum; the area is 40 square centimeters

Answer: B. Adding the four side lengths measures the distance around the rectangle, which is its perimeter of 20 cm. Area counts the unit squares that cover the interior, so it multiplies length by width: 6 × 4 = 24 square centimeters.

• The student computed the perimeter; the area is 6 + 4 = 10 square centimeters: Identifies the right error and then applies the wrong operation. One length plus one width is half the perimeter, and labeling 10 square centimeters describes no region of the rectangle.
• The student used the wrong unit; the area is 20 square centimeters: Treats a wrong operation as a labeling problem. The total 20 is a correct perimeter, and renaming it in square units does not turn a distance into an area.
• The student should have doubled the sum; the area is 40 square centimeters: Doubles the perimeter and reports the result as area. Doubling 20 gives 40 centimeters around two rectangles, not the space inside one.

1. 18 cubic centimeters
2. 9 cubic centimeters
3. 54 cubic centimeters
4. 27 cubic centimeters ✓

Answer: D. Volume of a cube is the edge length used as a factor three times: 3 × 3 × 3 = 27. The result is reported in cubic centimeters because three lengths are multiplied together.

• 9 cubic centimeters: Squares the edge instead of cubing it. Nine square centimeters is the area of one face, and volume multiplies that face by the third edge.
• 18 cubic centimeters: Multiplies the edge by six, the number of faces, and reports the product as volume. Counting faces describes the surface, not the space inside.
• 54 cubic centimeters: Computes surface area in place of volume. Six faces of 9 square centimeters total 54 square centimeters, which measures the outside of the cube.

1. The conversion is correct, because 10 centimeters make 1 meter
2. The conversion is wrong, because centimeters are smaller than meters
3. The conversion is wrong, because 1 meter contains 1,000 centimeters
4. The conversion is correct, because 100 centimeters make 1 meter ✓

Answer: D. One meter equals 100 centimeters, so converting a smaller unit to a larger one divides by 100. Dividing 150 by 100 gives 1.5, so 150 centimeters is 1.5 meters.

• The conversion is wrong, because 1 meter contains 1,000 centimeters: Applies the millimeter conversion factor to centimeters. A meter holds 100 centimeters, and 1,000 centimeters would be 10 meters.
• The conversion is correct, because 10 centimeters make 1 meter: Reaches the right verdict from a wrong factor. If 10 centimeters made a meter, then 150 centimeters would be 15 meters.
• The conversion is wrong, because centimeters are smaller than meters: Draws the wrong conclusion from an irrelevant premise. Relative unit size does not set the numeric factor, and this conversion is accurate.

1. Every square is a rectangle, but not every rectangle is a square ✓
2. Squares and rectangles are separate categories that do not overlap
3. Neither squares nor rectangles are required to have four right angles
4. Every rectangle is a square, but not every square is a rectangle

Answer: A. A rectangle is defined as a quadrilateral with four right angles, and a square meets that definition because it also has four right angles. The square adds a further condition, four equal sides, which narrows the category, so the rectangle family is the wider one.

• Every rectangle is a square, but not every square is a rectangle: Reverses the inclusion relationship. Requiring four equal sides makes the square the narrower category, so a 6 by 4 rectangle is not a square.
• Squares and rectangles are separate categories that do not overlap: Treats the two categories as disjoint when one sits inside the other. A square satisfies every property that defines a rectangle.
• Neither squares nor rectangles are required to have four right angles: Denies the property that defines a rectangle in the first place. Four right angles is exactly the condition both shapes share.

1. The mode, because it reports the most common price
2. The median, because it is not pulled away by the single extreme value ✓
3. The range, because it shows how spread out the prices are
4. The mean, because it uses every value in the data set

Answer: B. One very large value moves the mean toward it, so the mean stops describing where most prices sit. The median depends only on position in the ordered list, so a single extreme price shifts it by at most one place and it stays close to the typical value.

• The mean, because it uses every value in the data set: Chooses the measure most affected by the extreme value. Using every value is exactly why the outlier drags the mean upward, away from the typical price.
• The mode, because it reports the most common price: Selects a count of frequency rather than a measure of center. When each price appears once there is no single most common value, and frequency says nothing about the outlier.
• The range, because it shows how spread out the prices are: Offers a measure of spread in place of a measure of center. The range is fixed by the two most extreme prices, so it is defined entirely by the outlier.

1. It shows the range of the data, from the fewest to the most books read
2. It shows the mean number of books read by the class
3. It shows the most frequent value, which may describe fewer than half of the students ✓
4. It shows the total number of students who were surveyed

Answer: C. Each mark in a line plot stands for one student, so the height of a stack counts how many students reported that value. The tallest stack is therefore the mode, and with 20 students spread across several values it can hold as few as 4 students.

• It shows the mean number of books read by the class: Reads a frequency as an average. The stack height counts students at one value, while the mean requires adding every value and dividing by 20.
• It shows the total number of students who were surveyed: Reads one stack as the whole sample. That stack counts only the students at that value, and the 20 students are spread across the whole plot.
• It shows the range of the data, from the fewest to the most books read: Confuses a count with a measure of spread. The range is the difference between the largest and smallest values, which no single stack displays.

1. 145°, because complementary angles add to 180°
2. 55°, because complementary angles add to 90° ✓
3. 35°, because complementary angles are equal in measure
4. 55°, because complementary angles add to 180°

Answer: B. Complementary angles are defined by a sum of 90°, so the unknown angle is 90° − 35° = 55°. Checking the definition, 35° + 55° = 90°, which confirms the pair is complementary.

• 145°, because complementary angles add to 180°: Applies the supplementary sum to a complementary pair. Angles adding to 180° are supplementary, and 180° − 35° = 145°, so this answer belongs to a different relationship.
• 55°, because complementary angles add to 180°: Reaches the right measure through the wrong definition. The complement is found by subtracting from 90°, not 180°, which would give 145°.
• 35°, because complementary angles are equal in measure: Assumes the two angles must be congruent. The definition requires their measures to total 90°, and 35° + 35° = 70°, which falls short.

1. 2 feet
2. 36 centimeters
3. 3 feet ✓
4. 1 meter
5. 1 yard ✓

Answers: C and E. There are 12 inches in a foot, so 3 × 12 = 36 inches. There are also 3 feet in a yard, and since 3 feet is 36 inches, 1 yard is 36 inches as well. Both entries name the same length in different customary units.

• 2 feet: Counts only two of the three feet needed. At 12 inches per foot, 2 feet is 24 inches, a foot short of the target.
• 36 centimeters: Keeps the number and changes the unit. Inches and centimeters are different lengths, and 36 centimeters is about 14 inches.
• 1 meter: Substitutes a metric length for a customary one. A meter is a little over 39 inches, so it is longer than 36 inches.

Answer: 60. Volume of a rectangular prism is length × width × height. Substituting the three dimensions gives 4 × 3 × 5 = 60, so the volume is 60 cubic centimeters. The unit is cubic because three lengths are multiplied together.