101 questions
What is 15% of 240?
- a.45
- b.36✓
- c.24
- d.360
To find 15% of 240, multiply 240 by 0.15, which equals 36. Percent means 'per hundred,' so 15% is 0.15 in decimal form. This is a basic percentage calculation.
Solve for x: 3x + 7 = 22.
- a.15
- b.3
- c.5✓
- d.7
Subtract 7 from both sides to get 3x = 15, then divide by 3 to find x = 5. Isolating the variable step by step solves a linear equation. Checking: 3(5) + 7 = 22, which is correct.
A rectangle has a length of 12 cm and a width of 5 cm. What is its area?
- a.60 cm²✓
- b.34 cm²
- c.120 cm²
- d.17 cm²
Area of a rectangle is length times width: 12 × 5 = 60 square centimeters. Area is always expressed in square units. The perimeter, by contrast, would be 2(12 + 5) = 34 cm.
What is the value of 2³ + 4²?
- a.24✓
- b.20
- c.14
- d.18
2³ equals 8 and 4² equals 16, and their sum is 24. An exponent tells how many times to multiply the base by itself. Evaluating exponents before adding follows the order of operations.
Simplify the fraction 18/24 to lowest terms.
- a.6/8
- b.9/12
- c.2/3
- d.3/4✓
The greatest common factor of 18 and 24 is 6, and dividing both by 6 gives 3/4. A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. Forms like 6/8 and 9/12 are equal in value but not fully reduced.
If a car travels 180 miles in 3 hours, what is its average speed?
- a.540 mph
- b.63 mph
- c.50 mph
- d.60 mph✓
Average speed equals distance divided by time: 180 ÷ 3 = 60 miles per hour. This uses the relationship speed = distance/time. The units confirm the answer is in miles per hour.
What is the mean (average) of 4, 8, 10, and 14?
- a.36
- b.10
- c.8
- d.9✓
Add the numbers to get 36, then divide by the count of 4, giving 9. The mean is the sum of values divided by how many values there are. The total, 36, is not the average itself.
Solve for y: y/4 = 9.
- a.5
- b.36✓
- c.13
- d.2.25
Multiply both sides by 4 to undo the division, giving y = 36. Since y divided by 4 equals 9, y must be 4 times 9. Checking: 36 ÷ 4 = 9, which is correct.
A shirt originally costs $40 and is on sale for 25% off. What is the sale price?
- a.$30✓
- b.$10
- c.$15
- d.$35
A 25% discount on $40 is $10, so the sale price is $40 − $10 = $30. You can also multiply $40 by 0.75 to get $30 directly. The discount is subtracted from the original price.
What is the slope of the line passing through the points (1, 2) and (3, 8)?
- a.6
- b.4
- c.3✓
- d.2
Slope equals the change in y over the change in x: (8 − 2) / (3 − 1) = 6/2 = 3. Slope measures how steep a line is. A positive slope means the line rises from left to right.
The ratio of boys to girls in a class is 3:5. If there are 24 students, how many are girls?
- a.12
- b.9
- c.15✓
- d.18
The ratio 3:5 has 8 parts total, so each part equals 24 ÷ 8 = 3 students. Girls make up 5 parts: 5 × 3 = 15. Ratios divide a whole into proportional shares.
What is 3/4 written as a decimal?
- a.0.75✓
- b.0.34
- c.1.33
- d.0.43
Dividing the numerator 3 by the denominator 4 gives 0.75. A fraction represents division of the top number by the bottom number. This is a common fraction-to-decimal conversion.
A circle has a radius of 6 cm. What is its circumference? (Use π ≈ 3.14.)
- a.18.84 cm
- b.37.68 cm✓
- c.113.04 cm
- d.12 cm
Circumference equals 2πr: 2 × 3.14 × 6 = 37.68 cm. The radius is the distance from center to edge. Using the area formula πr² would instead give about 113 cm².
Evaluate the expression 5(x − 2) when x = 6.
- a.20✓
- b.28
- c.10
- d.40
Substitute x = 6 to get 5(6 − 2) = 5(4) = 20. Following order of operations, solve inside the parentheses first, then multiply. Substituting values into expressions is a key algebra skill.
What is the next number in the pattern: 3, 6, 12, 24, ___?
- a.48✓
- b.30
- c.72
- d.36
Each term is double the one before it, so 24 × 2 = 48. Recognizing that the pattern multiplies by 2 reveals the rule. This is a geometric sequence.
A recipe needs 2 cups of flour for 12 cookies. How many cups are needed for 30 cookies?
- a.2.5 cups
- b.4 cups
- c.5 cups✓
- d.6 cups
Set up a proportion: 2/12 = x/30, so x = (2 × 30)/12 = 5 cups. Proportions keep the ratio of flour to cookies constant. Scaling recipes uses this direct relationship.
What is the area of a triangle with a base of 10 cm and a height of 6 cm?
- a.24 cm²
- b.30 cm²✓
- c.16 cm²
- d.60 cm²
Area of a triangle is one-half the base times the height: (1/2)(10)(6) = 30 cm². The factor of one-half distinguishes a triangle from a rectangle of the same base and height. Area is measured in square units.
Solve the inequality: 2x − 3 < 7.
- a.x > 5
- b.x < 2
- c.x < 5✓
- d.x < 10
Add 3 to both sides to get 2x < 10, then divide by 2 to get x < 5. When dividing an inequality by a positive number, the direction of the sign stays the same. The solution is all values less than 5.
What is 0.6 written as a percent?
- a.0.6%
- b.600%
- c.6%
- d.60%✓
To convert a decimal to a percent, multiply by 100, so 0.6 becomes 60%. This moves the decimal point two places to the right. Percent means 'out of one hundred.'
A box contains 3 red, 4 blue, and 5 green marbles. What is the probability of drawing a blue marble at random?
- a.1/2
- b.1/4
- c.5/12
- d.1/3✓
Probability is favorable outcomes over total outcomes. There are 4 blue marbles out of 3 + 4 + 5 = 12, so the probability is 4/12, which reduces to 1/3. The 1/4 figure is the probability of drawing a red marble (3 out of 12), and 5/12 is the probability of drawing a green one — both are answers to a question that was not asked. And 1/2 would require six blue marbles rather than four.
If 5 pencils cost $1.25, how much do 8 pencils cost at the same rate?
- a.$1.85
- b.$1.60
- c.$2.00✓
- d.$2.50
Each pencil costs $1.25 ÷ 5 = $0.25, so 8 pencils cost 8 × $0.25 = $2.00. Finding the unit rate lets you scale to any quantity. Unit pricing is a common real-world calculation.
What is the value of the expression -4 + 9 − 3?
- a.2✓
- b.10
- c.8
- d.-10
Working left to right: -4 + 9 = 5, and 5 − 3 = 2. Adding a positive to a negative moves toward zero and beyond. Careful sign handling gives the correct result of 2.
A square has a perimeter of 36 cm. What is the length of one side?
- a.12 cm
- b.9 cm✓
- c.18 cm
- d.6 cm
A square has four equal sides, so divide the perimeter by 4: 36 ÷ 4 = 9 cm. Perimeter is the total distance around a shape. Each side of this square measures 9 cm.
Which value of x makes the equation 2x = 5x − 12 true?
- a.6
- b.4✓
- c.3
- d.2
Subtract 2x from both sides to get 0 = 3x − 12, then add 12 and divide by 3 to find x = 4. Gathering variables on one side isolates x. Checking: 2(4) = 8 and 5(4) − 12 = 8, which matches.
The temperature rose from -5°F to 12°F. By how many degrees did it rise?
- a.17°F✓
- b.12°F
- c.7°F
- d.-17°F
The change equals 12 − (−5) = 12 + 5 = 17 degrees. Subtracting a negative is the same as adding its positive. The temperature climbed 17°F.
What is 3/5 + 1/5?
- a.4/5✓
- b.2/5
- c.3/25
- d.4/10
When fractions share a denominator, add the numerators: 3 + 1 = 4, giving 4/5. The denominator stays the same at 5. This result is already in lowest terms.
A phone plan costs $30 per month plus $0.10 per text. Which expression gives the total monthly cost for t texts?
- a.30 + 0.10t✓
- b.30t + 0.10
- c.30.10t
- d.30 + 0.10 + t
The $30 fee is charged once whatever happens, so it stands alone; the 10 cents is charged per text, so it is multiplied by the number of texts. That gives 30 + 0.10t. Writing 30t + 0.10 swaps the two roles, charging $30 per text and a flat dime a month. Writing 30 + 0.10 + t adds the number of texts to the money instead of multiplying, which mixes a count with dollars. And 30.10t charges the whole $30.10 for every single text.
What is the median of the data set 7, 3, 9, 5, 11?
- a.35
- b.5
- c.9
- d.7✓
First order the numbers: 3, 5, 7, 9, 11; the middle value is 7. The median is the center of an ordered list. With five values, the third number is the median.
If a triangle has angles measuring 40° and 75°, what is the third angle?
- a.75°
- b.55°
- c.65°✓
- d.115°
The angles of a triangle sum to 180°, so 180 − 40 − 75 = 65°. This triangle-angle-sum rule holds for every triangle. The missing angle measures 65 degrees.
Expand the expression 4(2x + 3).
- a.6x + 7
- b.8x + 12✓
- c.8x + 7
- d.8x + 3
Use the distributive property to multiply 4 by each term: 4 × 2x = 8x and 4 × 3 = 12. The result is 8x + 12. Distribution multiplies the outside factor across everything inside the parentheses.
A worker earns $18 per hour and works 35 hours in a week. What are the weekly earnings?
- a.$720
- b.$53
- c.$530
- d.$630✓
Multiply the hourly rate by hours worked: 18 × 35 = 630 dollars. Total pay equals rate times time. The worker earns $630 that week.
What is the square root of 144?
- a.11
- b.72
- c.12✓
- d.14
The square root of 144 is the number that, multiplied by itself, gives 144; that number is 12 because 12 × 12 = 144. Square roots reverse the squaring operation. Note that 72 is half of 144, not its root.
A number increased by 20% becomes 60. What was the original number?
- a.48
- b.50✓
- c.40
- d.72
If the original number is x, then 1.20x = 60, so x = 60 ÷ 1.20 = 50. Increasing by 20% multiplies by 1.20. Reversing the increase means dividing by 1.20.
Two angles are complementary. If one measures 35°, what is the other?
- a.45°
- b.145°
- c.55°✓
- d.65°
Complementary angles add up to 90°, so the other angle is 90 − 35 = 55°. Complementary differs from supplementary, which sums to 180°. The missing angle is 55 degrees.
What is 3/8 + 1/4?
- a.4/12
- b.3/4
- c.5/8✓
- d.1/2
Rewrite 1/4 as 2/8 so the denominators match, then add the numerators: 3/8 + 2/8 = 5/8. Adding numerators and denominators straight across gives 4/12, which is what makes fractions look harder than they are. One half is 4/8 and falls just short. And 3/4 is 6/8, one eighth too much.
What is 2/3 × 3/5?
- a.5/8
- b.2/5✓
- c.6/5
- d.1/5
Multiply straight across to get 6/15, then divide both parts by 3 for 2/5. Adding across instead of multiplying produces 5/8. Inverting the second fraction gives 6/5, which is what division would call for. And 1/5 drops a factor somewhere in the reduction.
What is 5/6 ÷ 1/3?
- a.2 1/2✓
- b.5/18
- c.2/9
- d.1 1/2
Dividing by a fraction means multiplying by its reciprocal: 5/6 × 3/1 = 15/6, which reduces to 5/2, or 2 1/2. Multiplying by 1/3 instead of dividing gives 5/18. The answer 1 1/2 would come from a different dividend altogether. And 2/9 does not arise from any correct step here.
Write 0.45 as a fraction in lowest terms.
- a.45/10
- b.4/5
- c.1/45
- d.9/20✓
The digits end in the hundredths place, so 0.45 is 45/100, and dividing both parts by 5 gives 9/20. Writing 45/10 misplaces the decimal by a factor of ten. Four fifths is 0.8, nearly twice as large. And 1/45 inverts the number entirely.
Which list puts 5/8, 0.6 and 58% in order from least to greatest?
- a.58%, 5/8, 0.6
- b.5/8, 0.6, 58%
- c.0.6, 58%, 5/8
- d.58%, 0.6, 5/8✓
Convert everything to decimals: 58% is 0.58, 0.6 stays as it is, and 5/8 is 0.625, so the order runs 0.58, 0.60, 0.625. Reversing that order puts the largest first. Placing 0.6 below 58% mistakes 0.6 for 0.06. And putting 5/8 before 0.6 treats eight parts as smaller than they are.
What is −7 − (−3)?
- a.4
- b.−10
- c.−4✓
- d.10
Subtracting a negative is the same as adding its opposite, so −7 − (−3) becomes −7 + 3, which is −4. Treating the two minus signs as one gives −10. Changing the sign of the first number as well gives 10. And 4 has the right size but the wrong sign, since the result stays below zero.
What is the product of −3 and −5?
- a.8
- b.15✓
- c.−15
- d.−8
Two negative factors give a positive product, so (−3)(−5) = 15. Keeping the answer negative ignores that rule. The value −8 comes from adding rather than multiplying. And 8 is that same sum with the sign dropped.
Evaluate |−9| − |4|.
- a.13
- b.5✓
- c.−13
- d.−5
Absolute value strips the sign, so this is 9 − 4, which equals 5. Adding the two values after stripping the signs gives 13. Keeping the first number negative and then subtracting gives −13. And −5 reverses the order of the subtraction.
A 12-ounce bottle of juice costs $3.60. What is the price per ounce?
- a.$0.30✓
- b.$3.33
- c.$0.33
- d.$4.32
A unit rate divides cost by quantity: $3.60 ÷ 12 = $0.30 per ounce. Dividing the ounces by the dollars gives 3.33 and answers a question nobody asked. The figure $0.33 comes from dividing by 11 or from a slip in the decimal. And multiplying rather than dividing gives $4.32.
A price rises from $40 to $50. What is the percent increase?
- a.25%✓
- b.20%
- c.125%
- d.10%
Percent change divides the increase by the original amount: 10 ÷ 40 = 0.25, or 25%. Dividing by the new price instead gives 20%, which is the classic trap in these problems. The raw increase of 10 dollars is not itself 10 percent. And 125% is the new price as a share of the old, not the increase.
8 is what percent of 32?
- a.4%
- b.40%
- c.400%
- d.25%✓
Divide the part by the whole and convert: 8 ÷ 32 = 0.25, which is 25%. The value 4 is the result of dividing 32 by 8, the reverse comparison. Forty percent of 32 would be 12.8. And 400% would make 8 four times the whole rather than a quarter of it.
An item costs $24 and sales tax is 7%. What is the total?
- a.$24.07
- b.$25.68✓
- c.$1.68
- d.$31.00
The tax is 0.07 × $24 = $1.68, and adding it to the price gives $25.68; multiplying $24 by 1.07 does both steps at once. Adding seven cents treats the percent as a flat amount. Adding seven dollars treats it as dollars rather than percent. And $1.68 is the tax alone, not the total.
A meal costs $45 and the diner leaves an 18% tip. How much is the tip?
- a.$81.00
- b.$0.81
- c.$5.31
- d.$8.10✓
Eighteen percent of $45 is 0.18 × 45 = $8.10. Moving the decimal one place too far left gives $0.81. Moving it the other way gives $81, which is more than the meal. And $5.31 would be about 11.8 percent, a rate the question did not give.
How much simple interest does $500 earn at 4% per year for 3 years?
- a.$60✓
- b.$600
- c.$20
- d.$540
Simple interest is principal times rate times time: 500 × 0.04 × 3 = $60. Leaving out the three years gives $20, the first year only. The figure $600 misplaces a decimal. And $540 is the balance after interest rather than the interest itself, and even then only if the interest were $40.
Write 3.2 × 10⁴ in standard form.
- a.320,000
- b.3,200
- c.32,000✓
- d.12.8
An exponent of 4 moves the decimal point four places to the right, turning 3.2 into 32,000. Moving it three places gives 3,200. Moving it five gives 320,000. And 12.8 is 3.2 multiplied by 4 rather than by ten to the fourth.
What is the value of (2³)²?
- a.512
- b.12
- c.32
- d.64✓
Work from the inside out: 2³ is 8, and 8² is 64; multiplying the exponents to get 2⁶ gives the same result. Multiplying the base by the exponents gives 12. Adding the exponents gives 2⁵, or 32. And 512 is 2⁹, which would come from raising 8 to the third power.
Between which two whole numbers does the square root of 50 fall?
- a.24 and 26
- b.8 and 9
- c.6 and 7
- d.7 and 8✓
Seven squared is 49 and eight squared is 64, so the square root of 50 sits just above 7. Six squared is only 36, too small to bracket 50 from above. Eight and nine bracket numbers from 64 upward. And 24 to 26 comes from halving 50 rather than taking a root.
A box measures 4 cm by 3 cm by 5 cm. What is its volume?
- a.12 cm³
- b.94 cm³
- c.60 cm³✓
- d.24 cm³
Volume of a rectangular solid is length times width times height: 4 × 3 × 5 = 60 cubic centimeters. Multiplying only two of the three dimensions gives 12. Adding the dimensions and doubling gives 24. And 94 is the surface area of this box, a different measurement in square units.
A cylinder has a radius of 3 cm and a height of 10 cm. What is its volume? (Use π ≈ 3.14.)
- a.188.4 cm³
- b.30 cm³
- c.94.2 cm³
- d.282.6 cm³✓
Volume of a cylinder is the base area times the height: 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cubic centimeters. Using the circumference instead of the area gives 188.4. Squaring nothing at all and multiplying 3.14 by 3 by 10 gives 94.2. And 30 is simply radius times height, with π forgotten.
What is the total surface area of a cube with edges 4 cm long?
- a.96 cm²✓
- b.24 cm²
- c.64 cm²
- d.16 cm²
A cube has six identical square faces, each 4 × 4 = 16 square centimeters, so the surface area is 6 × 16 = 96. The value 64 is the cube's volume. One face alone is 16. And 24 is the total edge length rather than an area.
What is the area of a circle with a radius of 5 cm? (Use π ≈ 3.14.)
- a.157 cm²
- b.15.7 cm²
- c.78.5 cm²✓
- d.31.4 cm²
Area is π times the radius squared: 3.14 × 25 = 78.5 square centimeters. The value 31.4 is the circumference, which uses 2πr instead. Half of that circumference is 15.7. And 157 doubles the correct area, as if the diameter had been squared in place of the radius.
A right triangle has legs of 6 cm and 8 cm. How long is the hypotenuse?
- a.7 cm
- b.14 cm
- c.48 cm
- d.10 cm✓
By the Pythagorean theorem, 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10. Adding the legs gives 14, which is longer than any side of this triangle can be. Multiplying them gives 48, which is twice the area. And 7 is the average of the legs, not a side length.
A rectangle is 15 m long and 8 m wide. What is its perimeter?
- a.120 m
- b.46 m✓
- c.60 m
- d.23 m
Perimeter adds all four sides: 2(15 + 8) = 46 meters. The product 120 is the area, measured in square meters rather than meters. Adding the length and width once gives 23, which is only half the way around. And 60 would be the perimeter of a 15-by-15 square.
How many inches are in 2.5 feet?
- a.24
- b.36
- c.25
- d.30✓
There are 12 inches in a foot, so 2.5 × 12 = 30 inches. The value 25 comes from treating the conversion as a decimal shift. Twenty-four inches is exactly two feet and leaves out the half. And 36 inches is three feet, half a foot too many.
A container holds 2,500 milliliters. How many liters is that?
- a.2.5✓
- b.25
- c.0.25
- d.250
A liter is 1,000 milliliters, so dividing 2,500 by 1,000 gives 2.5 liters. Dividing by 100 gives 25. Dividing by 10 gives 250. And 0.25 divides by 10,000, one decimal place too far.
A parallelogram has a base of 12 cm and a height of 7 cm. What is its area?
- a.38 cm²
- b.84 cm²✓
- c.42 cm²
- d.19 cm²
Area of a parallelogram is base times height: 12 × 7 = 84 square centimeters, with no factor of one half. Halving the product gives 42, which is the area of a triangle with the same base and height. Adding and doubling gives 38. And 19 is just the base plus the height.
A rectangular patio measures 10 ft by 6 ft. A 3 ft by 4 ft rectangular planter is cut out of it. What area of patio remains?
- a.60 square feet
- b.48 square feet✓
- c.72 square feet
- d.12 square feet
The whole rectangle is 10 × 6 = 60 square feet and the planter is 3 × 4 = 12, so 60 − 12 = 48 square feet remain. Reporting 60 forgets to remove the planter. Adding rather than subtracting gives 72. And 12 is the planter itself rather than what is left.
On a floor plan, 1 inch represents 4 feet. A wall drawn 3.5 inches long is actually—
- a.7 feet
- b.12 feet
- c.1.14 feet
- d.14 feet✓
Each inch stands for 4 feet, so 3.5 × 4 = 14 feet. Doubling instead of quadrupling gives 7. Ignoring the half inch gives 12. And dividing the drawing length by the scale gives 1.14, which inverts the relationship.
A trapezoid has parallel sides of 8 cm and 12 cm and a height of 5 cm. What is its area?
- a.50 cm²✓
- b.60 cm²
- c.25 cm²
- d.100 cm²
Average the parallel sides and multiply by the height: (8 + 12) ÷ 2 = 10, and 10 × 5 = 50 square centimeters. Forgetting to average gives 100. Halving twice gives 25. And 60 multiplies 12 by 5 while ignoring the shorter side.
A circular tabletop has a diameter of 14 inches. What is its circumference? (Use π ≈ 22/7.)
- a.44 inches✓
- b.22 inches
- c.154 inches
- d.88 inches
Circumference equals π times the diameter: 22/7 × 14 = 44 inches. Using the radius in place of the diameter gives 22. Doubling the diameter before multiplying gives 88. And 154 is the area in square inches, which uses the radius squared.
A wheelchair ramp rises 2 feet over a horizontal run of 24 feet. What is its slope?
- a.1/24
- b.1/12✓
- c.1/2
- d.12
Slope is rise over run: 2 ÷ 24 = 1/12. Dividing the run by the rise gives 12, which inverts the ratio and describes an impossibly steep ramp. One half ignores the run almost entirely. And 1/24 uses a rise of one foot rather than the two feet the ramp actually climbs.
Solve for x: 4x − 9 = 2x + 7.
- a.4
- b.16
- c.−1
- d.8✓
Subtract 2x from both sides to get 2x − 9 = 7, add 9 to get 2x = 16, then divide by 2 for x = 8. Forgetting to divide leaves 16. Subtracting 9 from 7 instead of adding gives −1. And 4 solves a different equation entirely; substituting it gives 7 on the left and 15 on the right.
Solve for x: 3(x + 2) = 21.
- a.5✓
- b.7
- c.19
- d.9
Divide both sides by 3 to get x + 2 = 7, then subtract 2 for x = 5. Stopping after the division leaves 7. Subtracting 2 from 21 without dividing gives 19. And 9 comes from dividing 21 by 3 and then adding 2 rather than subtracting.
Solve for x: x/3 + 4 = 10.
- a.42
- b.18✓
- c.2
- d.6
Subtract 4 from both sides to get x/3 = 6, then multiply by 3 for x = 18. Dividing instead of multiplying gives 2. Adding 4 before multiplying gives 42. And 6 is the value of x/3, not of x.
Solve for y: 2y − 5 = 13.
- a.9✓
- b.6.5
- c.18
- d.4
Add 5 to both sides to get 2y = 18, then divide by 2 for y = 9. Subtracting 5 instead of adding gives 2y = 8 and y = 4. Forgetting the division leaves 18. And 6.5 divides 13 by 2 without dealing with the 5 at all.
Simplify: 3x + 5 − x + 2.
- a.2x + 7✓
- b.4x + 7
- c.2x + 3
- d.9x
Combine the x terms, 3x − x = 2x, and the constants, 5 + 2 = 7, giving 2x + 7. Adding the x terms instead of subtracting gives 4x + 7. Subtracting the constants gives 2x + 3. And 9x lumps unlike terms together, which is the error the whole exercise guards against.
Multiply: (x + 3)(x + 4).
- a.2x + 7
- b.x² + 7x + 12✓
- c.x² + 12x + 7
- d.x² + 12
Multiply every term by every term: x·x gives x², the outer and inner products give 4x and 3x for 7x, and 3·4 gives 12. Swapping the middle and last terms gives x² + 12x + 7. Multiplying only the first and last terms gives x² + 12. And 2x + 7 is what you get by adding the two binomials instead of multiplying.
Factor completely: x² − 9.
- a.(x − 3)(x − 3)
- b.(x + 9)(x − 1)
- c.(x + 3)(x − 3)✓
- d.x(x − 9)
A difference of two squares factors into a sum times a difference, and here the square roots are x and 3. Using two identical factors would give x² − 6x + 9, which has a middle term this expression does not. The pair 9 and −1 multiplies to −9 but leaves a middle term of 8x. And factoring out an x is impossible when the second term has no x in it.
Factor: x² + 5x + 6.
- a.(x − 2)(x − 3)
- b.(x + 5)(x + 1)
- c.(x + 1)(x + 6)
- d.(x + 2)(x + 3)✓
Look for two numbers that multiply to 6 and add to 5, which are 2 and 3. The pair 1 and 6 multiplies to 6 but adds to 7. Two negatives would multiply to 6 but add to −5, giving the wrong middle sign. And 5 and 1 add to 6 and multiply to 5, exactly reversing what is needed.
If x² = 49, what are the possible values of x?
- a.24.5 only
- b.7 and −7✓
- c.7 only
- d.−7 only
Both 7 and −7 square to 49, so a quadratic of this form has two solutions. Reporting only the positive root is the most common omission on this kind of problem. Halving 49 gives 24.5, which is not a square root at all. And the negative root alone is just as incomplete as the positive one.
Solve the inequality: −3x > 12.
- a.x > −4
- b.x > 4
- c.x < −4✓
- d.x < 4
Dividing both sides by a negative number reverses the direction of the inequality, so −3x > 12 becomes x < −4. Keeping the sign as it was gives x > −4, the single most common error here. Dropping the negative from the answer gives x < 4. And x > 4 gets both the sign and the direction wrong.
Solve the system: x + y = 10 and x − y = 4.
- a.x = 6, y = 4
- b.x = 7, y = 3✓
- c.x = 3, y = 7
- d.x = 14, y = 4
Adding the two equations eliminates y and gives 2x = 14, so x = 7 and then y = 3. Swapping the values gives x = 3 and y = 7, which satisfies the first equation but makes the second −4. The pair 6 and 4 sums to 10 but differs by 2. And 14 and 4 satisfies neither equation.
Four times a number, decreased by 6, is 18. What is the number?
- a.48
- b.4.5
- c.6✓
- d.3
Write 4n − 6 = 18, add 6 to get 4n = 24, then divide by 4 for n = 6. Subtracting 6 instead of adding gives 4n = 12 and n = 3. Multiplying 18 by 4 and adding gives 48. And 4.5 divides 18 by 4 and never handles the 6.
Which expression means 'five less than twice a number n'?
- a.5n − 2
- b.2(n − 5)
- c.5 − 2n
- d.2n − 5✓
Twice the number is 2n, and five less than that subtracts 5 from it. Writing 5 − 2n reverses the subtraction, which changes the value at every n but zero. The form 2(n − 5) subtracts first and doubles afterward, giving 2n − 10. And 5n − 2 swaps the roles of the two numbers.
Evaluate 2a² − b when a = 3 and b = 5.
- a.1
- b.18
- c.13✓
- d.31
Square first: a² is 9, so 2 × 9 = 18, and 18 − 5 = 13. Squaring the product 2a instead of a alone gives 36 − 5 = 31. Stopping before subtracting b leaves 18. And treating a² as a gives 6 − 5 = 1.
Solve the proportion: 3/4 = x/20.
- a.12
- b.26.7
- c.15✓
- d.5
Cross multiply to get 4x = 60, then divide by 4 for x = 15. Multiplying 3 by 4 gives 12 and ignores the 20 entirely. Dividing 80 by 3 gives about 26.7, which crosses the wrong pair. And 5 comes from dividing 20 by 4 without using the 3.
The area of a rectangle is A = lw. Solved for w, this becomes—
- a.w = Al
- b.w = l/A
- c.w = A/l✓
- d.w = A − l
To isolate w, divide both sides by l, which leaves w = A divided by l. Multiplying instead of dividing gives Al, which has the units of a volume. Writing l/A inverts the fraction. And subtracting l treats a product as a sum.
Solve for x: 0.5x = 7.
- a.7.5
- b.14✓
- c.3.5
- d.0.07
Dividing by 0.5 is the same as multiplying by 2, so x = 14. Multiplying by 0.5 instead gives 3.5, which is half the target rather than double. Adding 0.5 gives 7.5. And 0.07 misplaces the decimal by two places.
A rectangle has perimeter P = 2l + 2w. If P = 30 and l = 9, what is w?
- a.6✓
- b.12
- c.21
- d.3
Substitute to get 30 = 18 + 2w, so 2w = 12 and w = 6. Forgetting the final division leaves 12. Subtracting only one length gives 21. And 3 halves 12 twice.
Simplify: 7x + 3(x − 2).
- a.10x + 6
- b.10x − 2
- c.10x − 6✓
- d.7x + 3x − 2
Distribute the 3 across both terms to get 7x + 3x − 6, then combine like terms for 10x − 6. Distributing to the x only and keeping +6 gets the sign wrong. Leaving −2 undistributed drops the multiplication. And the last choice is an incomplete step rather than a simplified expression.
Solve for x: 5 − 2x = 11.
- a.8
- b.−3✓
- c.−8
- d.3
Subtract 5 from both sides to get −2x = 6, then divide by −2 for x = −3. Dropping the negative gives 3, which substituted back yields −1 rather than 11. Adding 5 to 11 and halving gives 8. And −8 keeps that same error and adds a sign.
In the equation y = 3x − 2, the y-intercept is—
- a.−2✓
- b.−3
- c.3
- d.2
In slope-intercept form the constant term is the value of y when x is zero, so the line crosses the y-axis at −2. The 3 is the slope, which describes steepness rather than a crossing point. Positive 2 has the right size and the wrong sign. And −3 confuses the slope with the intercept and flips its sign.
What is the slope of the line y = −(1/2)x + 4?
- a.−1/2✓
- b.−4
- c.4
- d.1/2
The coefficient of x is the slope, and here it is negative one half, meaning the line falls as it moves right. Dropping the negative gives a line that rises instead. The 4 is the y-intercept. And −4 is that intercept with the sign changed.
Which point lies on the line y = 2x + 1?
- a.(1, 4)
- b.(3, 5)
- c.(3, 7)✓
- d.(2, 3)
Substituting x = 3 gives y = 2(3) + 1 = 7, so the point (3, 7) satisfies the equation. The point (3, 5) would need the equation y = 2x − 1. Substituting x = 1 gives 3, not 4. And x = 2 gives 5, not 3.
A table shows that x = 1 gives y = 5, x = 2 gives y = 8, and x = 3 gives y = 11. Which rule fits?
- a.y = x + 4
- b.y = 5x
- c.y = 3x + 2✓
- d.y = 2x + 3
Each step of 1 in x raises y by 3, so the slope is 3, and working back from x = 1 gives an intercept of 2. The rule y = 5x fits the first row only and fails the second. The rule y = x + 4 rises too slowly. And y = 2x + 3 fits the first row and then falls behind.
What is the slope of the line through (−2, 3) and (2, 11)?
- a.2✓
- b.−2
- c.4
- d.1/2
Slope is the change in y over the change in x: (11 − 3) ÷ (2 − (−2)) = 8 ÷ 4 = 2. Inverting the fraction gives 1/2. Mishandling the negative x-value gives −2. And 4 comes from dividing by 2 instead of by 4.
What is the slope of a horizontal line?
- a.1
- b.−1
- c.0✓
- d.undefined
A horizontal line has no rise over any run, so the slope is zero. A slope of 1 describes a line rising at forty-five degrees. Undefined is the slope of a vertical line, where the run is zero and division is impossible. And −1 describes a line falling at forty-five degrees.
Which set of ordered pairs represents a function?
- a.(1, 2), (1, 5), (3, 6)
- b.(1, 2), (2, 4), (3, 6)✓
- c.(2, 3), (2, 3), (2, 7)
- d.(4, 1), (4, 2), (4, 3)
A function assigns exactly one output to each input, so the set in which every x-value appears once and only once is the function. A set pairing x = 1 with both 2 and 5 gives one input two outputs and fails. A set in which x = 4 appears three times with different partners fails for the same reason. And repeating x = 2 with conflicting y-values is that same failure once more, whatever else the set contains.
If f(x) = 2x + 5, what is f(4)?
- a.13✓
- b.11
- c.14
- d.9
Substitute 4 for x: 2(4) + 5 = 8 + 5 = 13. Adding 4 and 5 gives 9 and skips the multiplication. Using 2 + 4 + 5 gives 11. And 14 would require the rule 2x + 6.
A line on a graph falls steadily from left to right. Its slope is—
- a.positive
- b.negative✓
- c.zero
- d.undefined
Falling to the right means y decreases as x increases, which makes the ratio of rise to run negative. A positive slope rises to the right. A zero slope is flat. And an undefined slope belongs to a vertical line, which neither rises nor falls across the page.
On a graph of distance against time, a flat horizontal segment means the object is—
- a.moving at a steady speed
- b.moving backward
- c.speeding up
- d.not moving✓
If distance does not change while time passes, the object has stopped. Speeding up would show as a curve growing steeper. Steady speed would show as a straight line sloping upward. And moving back toward the start would show as a line sloping down.
Where does the line y = 4x − 8 cross the x-axis?
- a.(0, 2)
- b.(−2, 0)
- c.(2, 0)✓
- d.(0, −8)
The x-axis is where y = 0, so 4x − 8 = 0 gives x = 2 and the point (2, 0). Writing (0, 2) puts the value on the wrong axis. The point (0, −8) is the y-intercept, the crossing on the other axis. And (−2, 0) has the sign of x reversed.
A line is parallel to y = 5x + 1. Its slope must be—
- a.5✓
- b.1
- c.−1/5
- d.−5
Parallel lines never meet, which requires identical slopes, so the slope is 5. The 1 is the intercept of the given line, not its slope. The value −5 describes a line falling as steeply as this one rises. And −1/5 is the slope of a line perpendicular to it.
Which equation describes a line with slope 3 that passes through (0, −4)?
- a.y = −4x + 3
- b.y = 4x − 3
- c.y = 3x + 4
- d.y = 3x − 4✓
Because the point given has x = 0, it is the y-intercept, so the equation is y = 3x − 4. Writing +4 puts the crossing above the origin instead of below. The equation y = −4x + 3 swaps the slope and the intercept. And y = 4x − 3 swaps them and changes the signs.
A scatterplot shows points rising from lower left to upper right. The two variables show—
- a.no association at all
- b.a negative association
- c.a perfect causal link
- d.a positive association✓
Points that rise together mean that as one variable increases so does the other, which is a positive association. A negative association would show points falling to the right. No association would show a shapeless cloud. And no scatterplot by itself establishes that one variable causes the other.
A repair shop charges $25 plus $5 for each hour of labor, so the cost is C = 25 + 5h. In this model the 5 represents—
- a.the total cost of a short job
- b.the fixed charge for any visit
- c.the cost of each additional hour✓
- d.the number of hours worked
The 5 multiplies the hours, so it is the rate at which cost grows with each hour, the slope of the line. The fixed charge that applies before any labor is the 25. A short job costs 25 plus whatever labor it takes, which is not 5 by itself. And the number of hours is h, the variable the 5 is multiplied by.
The graph of the equation y = x² is—
- a.a pair of parallel lines
- b.a U-shaped curve✓
- c.a straight line
- d.a circle
Squaring makes negative and positive x-values give the same positive y, so the graph is a parabola opening upward, symmetric about the y-axis. A straight line comes from an equation where x appears to the first power only. A circle requires both variables squared and summed. And parallel lines come from a pair of separate linear equations.