GACE Mathematics (6–12) (711)

1. Question 1 · Numbers and Quantity

   A store raises the price of a calculator by 20%, then lowers the new price by 25% during a sale, and finally raises the sale price by 10%. Compared with the original price, which statement describes the final price?
   • A.The price is unchanged.
   • B.The price decreased by 1%.
   • C.The price increased by 5%.
   • D.The price increased by 1%.
   Show answer and explanation

   Answer: B. The price decreased by 1%.

   Successive percent changes multiply: 1.20 × 0.75 × 1.10 = 0.99. A multiplier of 0.99 means the final price is 99% of the original, a 1% decrease. Adding the percents (20 − 25 + 10 = 5) ignores that each change applies to a different base. The price decreased by 1%.

2. Question 2 · Algebra

   Which ordered triple (x, y, z) is the solution of the following system? x + y + z = 6 2x − y + z = 3 x + 2y − z = 2
   • A.(2, 1, 3)
   • B.(1, 3, 2)
   • C.(3, 2, 1)
   • D.(1, 2, 3)
   Show answer and explanation

   Answer: D. (1, 2, 3)

   Adding the first and third equations gives 2x + 3y = 8; adding the second and third gives 3x + y = 5, so y = 5 − 3x. Then 2x + 15 − 9x = 8 gives x = 1, y = 2, and z = 6 − 1 − 2 = 3. Each distractor satisfies the first equation but fails another; for example, (2, 1, 3) gives 2(2) − 1 + 3 = 6 ≠ 3. The solution is (1, 2, 3).

3. Question 3 · Geometry

   A triangle has side lengths 9, 12, and x, where x is an integer. For how many values of x is the triangle obtuse?
   • A.5
   • B.4
   • C.10
   • D.9
   Show answer and explanation

   Answer: D. 9

   The triangle inequality requires 3 < x < 21, so x = 4, 5, …, 20. If x is the longest side, the triangle is obtuse when x² > 9² + 12² = 225, so x = 16 to 20 (5 values). If 12 is the longest side, it is obtuse when 9² + x² < 144, so x² < 63 and x = 4 to 7 (4 values). Counting only the first case gives 5, and including x = 15 (a right triangle) gives 10. The answer is 9.

4. Question 4 · Functions

   The function f is defined by f(x) = (x² − 4)/(x − 2) for x < 2 and f(x) = kx + 1 for x ≥ 2. For what value of k is f continuous at x = 2?
   • A.−1/2
   • B.3/2
   • C.2
   • D.5/2
   Show answer and explanation

   Answer: B. 3/2

   For x < 2, (x² − 4)/(x − 2) = x + 2, so the left-hand limit at 2 is 4. The right-hand piece gives f(2) = 2k + 1. Setting 2k + 1 = 4 gives k = 3/2. Substituting x = 2 directly into the numerator and treating the left side as 0 leads to 2k + 1 = 0 and k = −1/2. The function is continuous at x = 2 when k = 3/2.

5. Question 5 · Financial Literacy

   Taxable income bracket	Rate
   $0 to $10,000	10%
   $10,000 to $40,000	20%
   Over $40,000	30%

   A state uses the hypothetical progressive income-tax schedule shown. Each rate applies only to the portion of taxable income that falls within its bracket. A teacher has taxable income of $60,000. How much state income tax does she owe?
   • A.$18,000
   • B.$15,000
   • C.$13,000
   • D.$16,000
   Show answer and explanation

   Answer: C. $13,000

   Tax each slice at its own rate: 10% × $10,000 = $1,000; 20% × $30,000 (the $10,000 to $40,000 slice) = $6,000; 30% × $20,000 (the amount over $40,000) = $6,000. Total = $13,000. The $18,000 choice applies the top marginal rate of 30% to all $60,000, which is not how a progressive schedule works. So she owes $13,000.

6. Question 6 · Numbers and Quantity

   A leaky faucet drips 1 drop every 2 seconds. If 20 drops make 1 milliliter, about how much water is wasted in a 30-day month?
   • A.64.8 L
   • B.129.6 L
   • C.6.48 L
   • D.2.16 L
   Show answer and explanation

   Answer: A. 64.8 L

   A day has 86,400 seconds, so the faucet drips 43,200 drops, or 43,200 ÷ 20 = 2,160 mL per day. Over 30 days that is 64,800 mL = 64.8 L. Using 1 drop per second doubles the result to 129.6 L, and 2.16 L is only one day's waste. About 64.8 L is wasted.

7. Question 7 · Algebra

   Which point lies in the solution region of the system y ≥ 2x − 3, y < −x + 6, x ≥ 0, y ≥ 0?
   • A.(2, 3)
   • B.(3, 3)
   • C.(4, 4)
   • D.(−1, 2)
   Show answer and explanation

   Answer: A. (2, 3)

   For (2, 3): 3 ≥ 1, 3 < 4, and both coordinates are nonnegative, so all four inequalities hold. The point (3, 3) is on the boundary y = −x + 6, but that inequality is strict, so the point is excluded. (4, 4) fails y ≥ 2x − 3 because 4 < 5, and (−1, 2) fails x ≥ 0. The point is (2, 3).

8. Question 8 · Geometry

   Lines ℓ and m pass through the center O of a regular octagon. Both are lines of symmetry of the octagon, and they form a 22.5° angle. A figure is reflected over ℓ, and the image is then reflected over m. Which single transformation is equivalent to this composition, and does it map the octagon onto itself?
   • A.A 45° rotation about O; yes, it maps the octagon onto itself
   • B.A 22.5° rotation about O; no, it does not map the octagon onto itself
   • C.A reflection over the line bisecting the angle between ℓ and m; yes, it maps the octagon onto itself
   • D.A translation by twice the distance between ℓ and m; no, it does not map the octagon onto itself
   Show answer and explanation

   Answer: A. A 45° rotation about O; yes, it maps the octagon onto itself

   Reflections over two lines that intersect at O are equivalent to a rotation about O by twice the angle between the lines: 2 × 22.5° = 45°. A regular octagon has rotational symmetry in multiples of 360°/8 = 45°, so the rotation maps it onto itself. Using 22.5° forgets the doubling, a composition of two reflections is never a single reflection, and a translation results only when the lines are parallel. The answer is a 45° rotation that maps the octagon onto itself.

9. Question 9 · Functions

   Evaluate lim(h→0) [sin(π/3 + h) − √3/2]/h.
   • A.1/2
   • B.√3/2
   • C.0
   • D.The limit does not exist.
   Show answer and explanation

   Answer: A. 1/2

   This is the limit definition of the derivative of f(x) = sin x at x = π/3, because sin(π/3) = √3/2. Since f′(x) = cos x, the limit equals cos(π/3) = 1/2. Answering √3/2 evaluates sine instead of its derivative, and calling the limit nonexistent misreads the 0/0 form. The limit is 1/2.

10. Question 10 · Financial Literacy

    Taxable income bracket	Rate
    $0 to $45,000	12%
    Over $45,000	22%

    A teacher with $44,000 of taxable income turns down a $3,000 raise, saying it would push her into the higher bracket of the hypothetical tax schedule shown and "cost her money." Each rate applies only to income within its bracket. How much of the raise would she actually keep after this income tax?
    • A.$2,340
    • B.$2,440
    • C.$2,640
    • D.−$2,060 (a net loss)
    Show answer and explanation

    Answer: B. $2,440

    Only the raise is newly taxed. Her income goes from $44,000 to $47,000: $1,000 stays in the 12% bracket ($120 tax) and $2,000 falls in the 22% bracket ($440 tax), so the extra tax is $560 and she keeps $3,000 − $560 = $2,440. The net-loss choice wrongly taxes all $47,000 at 22%; a higher bracket never applies to income below its threshold. She keeps $2,440.

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