GED — All Questions

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.