GED Math Practice Test 3

1. The question asks how many \tfrac{3}{8}-ounce doses fit into 4\tfrac{1}{2} ounces, so divide.
2. Change the mixed number to an improper fraction: 4\tfrac{1}{2} = \dfrac{4 \times 2 + 1}{2} = \dfrac{9}{2}.
3. Dividing by a fraction means multiplying by its reciprocal (“keep, change, flip”): \dfrac{9}{2} \div \dfrac{3}{8} = \dfrac{9}{2} \times \dfrac{8}{3}
4. Simplify before multiplying: \dfrac{9}{3} = 3 and \dfrac{8}{2} = 4, so the product is 3 \times 4 = 12.

The nurse can give 12 doses.

Check: 12 \times \dfrac{3}{8} = \dfrac{36}{8} = 4\tfrac{4}{8} = 4\tfrac{1}{2} fluid ounces ✓

Tip: “How many groups of size b fit in a?” always means a \div b. If your answer is smaller than the starting amount when dividing by a number less than 1, you multiplied instead of dividing.

1. Square everything inside the parentheses (power of a product, power of a power): (2x^{3}y^{-2})^{2} = 2^{2}\,x^{3 \cdot 2}\,y^{-2 \cdot 2} = 4x^{6}y^{-4}
2. Divide the coefficients: \dfrac{4}{4} = 1.
3. Divide the x powers by subtracting exponents: x^{6 - (-1)} = x^{7}.
4. Divide the y powers: y^{-4 - 3} = y^{-7}.
5. Rewrite the negative exponent as a denominator: y^{-7} = \dfrac{1}{y^{7}}.

The expression equals

Check: Let x = 1 and y = 2. Original: \dfrac{(2 \cdot 1 \cdot \tfrac{1}{4})^{2}}{4 \cdot 1 \cdot 8} = \dfrac{\tfrac{1}{4}}{32} = \dfrac{1}{128}. Answer: \dfrac{1}{2^{7}} = \dfrac{1}{128} ✓

Tip: Subtracting a negative exponent adds: 6 - (-1) = 7. Write the subtraction out in full instead of doing it in your head.

1. Simple interest formula (on the formula sheet): I = prt, where p = principal, r = yearly rate as a decimal, t = years.
2. Convert the rate: 6.5\% = 0.065.
3. Interest: I = 4{,}800 \times 0.065 \times 3 = 312 \times 3 = 936
4. Total repaid = principal + interest: 4{,}800 + 936 = 5{,}736.

Dana will repay $5,736.

Check: One year of interest is 4{,}800 \times 0.065 = \$312; three years is \$936; \$4{,}800 + \$936 = \$5{,}736 ✓

Tip: Read the last line of the question: “interest” and “total amount repaid” are different answers, and both will appear among the choices. Simple interest grows by the same dollar amount every year — no compounding.

1. Rectangle area: A = lw = 24 \times 16 = 384\text{ ft}^2.
2. The semicircle's diameter is the 16-ft side, so its radius is r = 16 \div 2 = 8\text{ ft}.
3. Full circle area (formula sheet): \pi r^{2} = \pi(8)^{2} = 64\pi. Half of it: \tfrac{1}{2} \cdot 64\pi = 32\pi \approx 100.53\text{ ft}^2
4. Total: 384 + 100.53 = 484.53 \approx 485\text{ ft}^2.

The landscaper needs about 485 square feet of sod.

Check: The semicircle should be a bit less than half of a 16 \times 16 square (256 \div 2 = 128); 100.5 fits. 384 + 100.5 \approx 485 ✓

Tip: For composite figures, sketch the pieces separately and label each one. Always ask “is this length a radius or a diameter?” before using a circle formula.

1. Substitute h = 3, using parentheses: f(3) = -2(3)^{2} + 16(3)
2. Exponent first (order of operations): (3)^{2} = 9.
3. Multiply: -2 \times 9 = -18 and 16 \times 3 = 48.
4. Add: -18 + 48 = 30.

So f(3) = 30, meaning 30 mg of the drug is in the bloodstream 3 hours after the dose.

Check: Factor first: f(h) = 2h(8 - h), so f(3) = 2(3)(5) = 30 ✓

Tip: In -2h^{2} only h is squared — the -2 is not. On the TI-30XS, type (−) 2 × 3 x² + 16 × 3 and the calculator follows order of operations for you.

1. The hourly charge depends on time: 62h.
2. The service fee is charged once: +\,85.
3. The coupon is subtracted once: -\,20.
4. Put it together and combine the constant terms: 62h + 85 - 20 = 62h + 65

The cost is 62h + 65 dollars.

Check: A 2-hour job: 85 + 62(2) - 20 = 85 + 124 - 20 = 189, and 62(2) + 65 = 124 + 65 = 189 ✓

Tip: The rate (“per hour”, “per month”, “each”) is the number that goes with the variable. Fixed fees, coupons and starting amounts are constants. Test any expression by plugging in a small number like h = 2.

1. Slope of the line through (0, 2) and (40, 18): m = \dfrac{18 - 2}{40 - 0} = \dfrac{16}{40} = 0.4
2. The line crosses the y-axis at (0, 2), so b = 2. Equation: y = 0.4x + 2.
3. Substitute x = 50: y = 0.4(50) + 2 = 20 + 2 = 22.

The predicted drop is 22 mmHg.

Check: From (40, 18), go 10 mg farther right. At 0.4 mmHg per mg, that adds 10 \times 0.4 = 4: 18 + 4 = 22 ✓

Tip: Predictions come from the line, not from a single nearby dot. The relationship is linear but not proportional (the line does not pass through the origin), so you cannot simply scale 18 by \tfrac{50}{40}.

1. Let d = number of deposits. “At least $3,000” means \ge: 1{,}250 + 85d \ge 3{,}000
2. Subtract 1,250 from both sides: 85d \ge 1{,}750.
3. Divide by 85 (positive, so the sign stays): d \ge 20.588\ldots
4. d must be a whole number, and 20 deposits is not enough. Round up: d = 21.

He needs at least 21 deposits.

Check: After 20 deposits: 1{,}250 + 85(20) = \$2{,}950 (too little). After 21: 1{,}250 + 85(21) = \$3{,}035 \ge \$3{,}000 ✓

Tip: In “at least” problems with whole-number answers, round up even if the decimal is small — rounding down never quite reaches the goal. Always test the value just below your answer.

1. Identify a = 2, b = 4, c = -7.
2. Quadratic formula (formula sheet): x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.
3. Discriminant: b^{2} - 4ac = 16 - 4(2)(-7) = 16 + 56 = 72.
4. Simplify: \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}.
5. Substitute: x = \dfrac{-4 \pm 6\sqrt{2}}{4} = \dfrac{2(-2 \pm 3\sqrt{2})}{4} = \dfrac{-2 \pm 3\sqrt{2}}{2}

The solutions are x = \dfrac{-2 \pm 3\sqrt{2}}{2} (about 1.12 and -3.12).

Check: x \approx 1.121: 2(1.121)^{2} + 4(1.121) - 7 \approx 2.514 + 4.485 - 7 \approx 0 ✓

Tip: The most common slip is with -4ac when c is negative: -4(2)(-7) is +56. Put negative values in parentheses as you substitute, and divide every term of the numerator by 2a.

1. First vial: 3 of 12 vials are high-dose: P = \dfrac{3}{12}.
2. Second vial: now only 11 vials remain and only 2 are high-dose: P = \dfrac{2}{11}.
3. “Both” means the first and the second, so multiply: \dfrac{3}{12} \times \dfrac{2}{11} = \dfrac{6}{132} = \dfrac{1}{22}

The probability is \dfrac{1}{22} (about 4.5%).

Check: Count pairs: there are \binom{3}{2} = 3 high-dose pairs out of \binom{12}{2} = 66 possible pairs, and \tfrac{3}{66} = \tfrac{1}{22} ✓

Tip: “Without replacement” means both the top and the bottom of the second fraction go down by 1. “And” → multiply; “or” (for events that can't both happen) → add.

1. Label the points: (x_1, y_1) = (-3, 7) and (x_2, y_2) = (5, -5).
2. Slope formula (formula sheet): m = \dfrac{y_2 - y_1}{x_2 - x_1}.
3. Rise: y_2 - y_1 = -5 - 7 = -12.
4. Run: x_2 - x_1 = 5 - (-3) = 5 + 3 = 8.
5. Divide and simplify: m = \dfrac{-12}{8} = -\dfrac{3}{2}.

The slope is -\tfrac{3}{2}.

Check: Start at (-3, 7). Moving right 8 and down 12 lands at (5, -5) ✓. The line falls from left to right, so a negative slope makes sense.

Tip: Rise goes on top. Subtracting a negative (5 - (-3)) becomes addition. A quick sketch tells you the sign of the slope before you compute.

1. Each group's total = (number of patients) × (group mean):
   • A: 12 \times 182 = 2{,}184
   • B: 18 \times 205 = 3{,}690
   • C: 10 \times 230 = 2{,}300
2. Grand total: 2{,}184 + 3{,}690 + 2{,}300 = 8{,}174.
3. Number of patients: 12 + 18 + 10 = 40.
4. Overall mean: \dfrac{8{,}174}{40} = 204.35.

The mean is 204.35 mg/dL.

Check: The answer must lie between 182 and 230, and it should be pulled toward 205 because Group B is the largest group. 204.35 fits ✓

Tip: This is a weighted average: bigger groups count more. Averaging the three means (205.67) would only be correct if every group had the same number of people.

1. Points: (3, 540) and (7, 1{,}060).
2. Slope (cost per day): m = \dfrac{1{,}060 - 540}{7 - 3} = \dfrac{520}{4} = 130
3. Use C = 130d + b with the point (3, 540): 540 = 130(3) + b = 390 + b, so b = 150.
4. Equation: C = 130d + 150 (a $150 flat fee plus $130 per day).

The answer is C = 130d + 150.

Check: d = 7: 130(7) + 150 = 910 + 150 = 1{,}060 ✓

Tip: Never divide one total by its days (540 \div 3 = 180) unless the situation is proportional. Two points → slope → intercept. Test your equation with both points; a wrong equation usually fits only one.

1. Walking along two sides: 120 + 50 = 170 ft.
2. The diagonal is the hypotenuse of a right triangle with legs 120 and 50. Pythagorean theorem (formula sheet): c^{2} = 120^{2} + 50^{2} = 14{,}400 + 2{,}500 = 16{,}900
3. c = \sqrt{16{,}900} = 130 ft.
4. Difference: 170 - 130 = 40 ft.

The diagonal path is 40 feet shorter.

Check: 50-120-130 is the 5-12-13 right triangle multiplied by 10 ✓, and the hypotenuse (130) is longer than either leg but shorter than their sum.

Tip: Read what is asked: the length of the path (130) is a step, not the final answer. Recognizing the 3-4-5 and 5-12-13 families saves time.

1. Rectangle area: multiply the binomials (FOIL): (2x + 3)(x - 4) = 2x^{2} - 8x + 3x - 12 = 2x^{2} - 5x - 12
2. Square planter area: x \cdot x = x^{2}.
3. Paved area = rectangle − square: (2x^{2} - 5x - 12) - x^{2} = x^{2} - 5x - 12

The paved area is x^{2} - 5x - 12 square feet.

Check: Let x = 10: the patio is 23 \times 6 = 138, the planter is 100, so the paved part is 38. Formula: 100 - 50 - 12 = 38 ✓

Tip: In FOIL, the middle terms come from the outer and inner products and keep their signs: 2x(-4) = -8x and 3(x) = +3x. Plugging in a number such as x = 10 is a fast way to test every answer choice.

1. Let a = adult tickets and c = child tickets.
   • Number of tickets: a + c = 240
   • Money: 8a + 5c = 1{,}620
2. From the first equation, a = 240 - c. Substitute into the second: 8(240 - c) + 5c = 1{,}620
3. Distribute: 1{,}920 - 8c + 5c = 1{,}620, so 1{,}920 - 3c = 1{,}620.
4. Solve: -3c = -300, so c = 100. Then a = 240 - 100 = 140.

100 child tickets were sold.

Check: 100 + 140 = 240 tickets ✓ and 8(140) + 5(100) = 1{,}120 + 500 = \$1{,}620 ✓

Tip: After solving a system, reread which variable the question wants — the other value (140) is always a tempting choice. Checking in both original equations catches most errors.

1. Read the five-number summary: minimum = 5, Q_1 = 12, median = 18, Q_3 = 27, maximum = 45.
2. A box plot splits the data into four parts with about 25% of the values in each:
   • 5 to 12: 25%
   • 12 to 18: 25%
   • 18 to 27: 25%
   • 27 to 45: 25%
3. Everything from Q_1 = 12 up to the maximum is 25\% + 25\% + 25\% = 75\% of the patients.

So about 75% of patients waited 12 minutes or longer is supported.

Check the others: the IQR is 27 - 12 = 15 (not 40, which is the range 45 - 5); a box plot shows the median, not the mean; and about 50% (not 25%) waited longer than the median of 18 ✓

Tip: A long whisker or a wide section means the data are spread out, not that it holds more people — every section holds about a quarter of the data.

1. Road length in feet: 3.6 \times 5{,}280 = 19{,}008 ft.
2. Two lines: 19{,}008 \times 2 = 38{,}016 ft of line.
3. Gallons needed: 38{,}016 \div 320 = 118.8 gallons.
4. Buckets: 118.8 \div 5 = 23.76 buckets.
5. 23 buckets hold only 115 gallons — not enough — so round up to 24.

The crew must buy 24 buckets.

Check: 24 buckets = 120 gallons, which paints 120 \times 320 = 38{,}400 ft \ge 38{,}016 ft ✓; 23 buckets paint only 36{,}800 ft ✗

Tip: In “how many containers” problems, always round up. Underline every quantity in the question (here: 2 lines, miles, 5-gallon buckets) and check that you used each one.

1. Distribute: -4(x - 3) = -4x + 12. The inequality becomes -4x + 12 + 2 > 3x - 7.
2. Combine: -4x + 14 > 3x - 7.
3. Subtract 3x from both sides: -7x + 14 > -7.
4. Subtract 14: -7x > -21.
5. Divide by -7 and flip the sign: x < 3.

The answer is x < 3.

Check: Try x = 0 (which is less than 3): -4(-3) + 2 = 14 > -7 ✓. Try x = 5: -4(2) + 2 = -6 > 8? No ✓, so values above 3 do not work.

Tip: You can avoid dividing by a negative by moving the variable terms to the side where the coefficient is positive: 14 + 7 > 7x gives 3 > x — the same answer. Always test one number from your solution.

1. Cylinder (formula sheet): V = \pi r^{2} h = \pi (10)^{2}(30) = 3{,}000\pi.
2. Sphere (formula sheet): V = \tfrac{4}{3}\pi r^{3}. A hemisphere is half of that: \tfrac{1}{2} \cdot \tfrac{4}{3}\pi (10)^{3} = \tfrac{2}{3}\pi(1{,}000) \approx 666.67\pi
3. Total: 3{,}000\pi + 666.67\pi = 3{,}666.67\pi \approx 11{,}519.17.
4. Round: 11,519 cubic feet.

Check: Cylinder alone \approx 9{,}425; hemisphere \approx 2{,}094; 9{,}425 + 2{,}094 = 11{,}519 ✓

Tip: Composite solids = add (or subtract) the pieces. Watch for “hemisphere” (half), and use the π key instead of 3.14 so your rounding matches the answer choices.

1. Unit price = price ÷ cubic feet:
   • 1.5 cu ft: 3.69 \div 1.5 = \$2.46 per cu ft
   • 2 cu ft: 4.79 \div 2 \approx \$2.40 per cu ft
   • 3 cu ft: 6.89 \div 3 \approx \$2.30 per cu ft
   • 10 cu ft: 24.50 \div 10 = \$2.45 per cu ft
2. The smallest unit price is about $2.30.

The 3-cubic-foot bag is the best buy.

Check: Three 1-cubic-foot portions at $2.30 cost 3 \times 2.30 = 6.90 \approx 6.89 ✓

Tip: “Best buy” means lowest unit price, not lowest total price or largest package. Always divide price by quantity (dollars per cubic foot), never quantity by price — or if you do, pick the largest result.

1. Rewrite Main Street in slope-intercept form: -4y = -3x + 12, so y = \tfrac{3}{4}x - 3. Its slope is \tfrac{3}{4}.
2. Perpendicular slope = negative reciprocal: flip \tfrac{3}{4} to \tfrac{4}{3} and change the sign → -\tfrac{4}{3}.
3. Use y = mx + b with the point (12, 6): 6 = -\tfrac{4}{3}(12) + b = -16 + b
4. Solve: b = 6 + 16 = 22.

The new street is y = -\tfrac{4}{3}x + 22.

Check: The point (12, 6): -\tfrac{4}{3}(12) + 22 = -16 + 22 = 6 ✓. Slopes: \tfrac{3}{4} \times \left(-\tfrac{4}{3}\right) = -1 ✓ (perpendicular slopes multiply to -1).

Tip: Perpendicular = flip and change sign. Parallel = same slope. When solving for b, isolate it carefully: moving -16 to the other side makes it +16.

1. The walkway adds x on the left and x on the right, so the outer length is 20 + 2x. Likewise, the outer width is 12 + 2x.
2. The outer rectangle's area is 384: (20 + 2x)(12 + 2x) = 384
3. Expand: 240 + 40x + 24x + 4x^{2} = 384, so 4x^{2} + 64x + 240 = 384.
4. Subtract 384: 4x^{2} + 64x - 144 = 0.
5. Divide every term by 4: x^{2} + 16x - 36 = 0

The equation is x^{2} + 16x - 36 = 0.

Check: It factors as (x + 18)(x - 2) = 0, so x = 2 (a width can't be negative). Outer rectangle: 24 \times 16 = 384 ✓

Tip: A border of width x around a rectangle adds 2x to each dimension. To check an equation choice, solve it and test the answer in the real situation.

1. Public safety is 35% and parks is 15%. The difference is 35\% - 15\% = 20\%.
2. Convert to dollars: 20\% of $48 million = 0.20 \times 48 = 9.6 million.

The city plans to spend $9.6 million more on public safety.

Check (the long way): Public safety: 0.35 \times 48 = 16.8 million. Parks: 0.15 \times 48 = 7.2 million. Difference: 16.8 - 7.2 = 9.6 million ✓

Tip: A circle graph shows percents of a whole. To answer a dollar question you must multiply by the total. Subtracting percents first (then multiplying once) saves a step.

1. Real length: 3.2 \times 250 = 800 ft. Real width: 1.8 \times 250 = 450 ft.
2. Real area: 800 \times 450 = 360{,}000\text{ ft}^2.
3. Convert to acres (divide by the square feet in one acre): 360{,}000 \div 43{,}560 \approx 8.26\text{ acres}

The park covers about 8.3 acres.

Check: Map area 3.2 \times 1.8 = 5.76\text{ in}^2. Each square inch stands for 250 \times 250 = 62{,}500\text{ ft}^2, and 5.76 \times 62{,}500 = 360{,}000 ✓

Tip: A scale applies to lengths. For area, the scale factor is squared (250^{2}). The safest method is to convert each length first, then multiply.

1. Slope = change in balance ÷ change in months: m = \dfrac{1{,}300 - 400}{6 - 0} = \dfrac{900}{6} = 150
2. Units: dollars (the y units) per month (the x units).
3. In context, the balance goes up $150 each month. Since the balance grows by the same amount every month, that is the amount she adds.

The slope means Rosa's balance increases by $150 per month.

Check: 400 + 150 \times 6 = 400 + 900 = \$1{,}300 ✓

Tip: In a linear real-world graph, the slope is the rate (per month, per hour, per item) and the y-intercept is the starting amount (here, $400). Dividing the final balance by the time (1{,}300 \div 6) ignores the starting amount.

1. Plan A is in slope-intercept form y = mx + b: rate = \$45 per month, starting amount = \$300.
2. Plan B from the table:
   • Starting amount (at x = 0): $180.
   • Rate: from x = 0 to x = 2 the balance rises 290 - 180 = 110, so the rate is \dfrac{110}{2} = \$55 per month.
3. Compare: Plan B grows faster ($55 > $45 per month), but Plan A starts higher ($300 > $180).

The true statement is “Plan B has the greater rate of change.”

Check: The table is linear: every 2 months it rises by $110 (400 - 290 = 110, 510 - 400 = 110) ✓, so 55 per month holds everywhere.

Tip: When a table skips values of x, divide the change in output by the change in input — don't read the jump between columns as the rate.

1. Pump A: in V = 2.5t the unit rate is the coefficient: 2.5 mL per minute.
2. Pump B: \dfrac{12}{4} = 3 and \dfrac{30}{10} = 3, so the rate is 3 mL per minute (it is proportional, since both ratios match).
3. Pump B is faster by 3 - 2.5 = 0.5 mL per minute.
4. Over 1 hour (60 minutes): 0.5 \times 60 = 30 mL.

Pump B delivers 30 mL more in one hour.

Check: Pump A: 2.5 \times 60 = 150 mL. Pump B: 3 \times 60 = 180 mL. 180 - 150 = 30 ✓

Tip: To compare proportional relationships given in different forms, turn each into a unit rate (y per 1 x) first. Then watch the time unit: the question asks about an hour, but the rates are per minute.

1. Factor the first denominator (difference of squares): x^{2} - 4 = (x - 2)(x + 2).
2. The common denominator is (x - 2)(x + 2). Multiply the second fraction by \tfrac{x + 2}{x + 2}: \dfrac{1}{x - 2} = \dfrac{x + 2}{(x - 2)(x + 2)}
3. Subtract the numerators. The minus sign applies to both terms of x + 2: \dfrac{2x - (x + 2)}{(x - 2)(x + 2)} = \dfrac{2x - x - 2}{(x - 2)(x + 2)} = \dfrac{x - 2}{(x - 2)(x + 2)}
4. Cancel the common factor (x - 2): \dfrac{1}{x + 2}

The expression equals \dfrac{1}{x + 2}.

Check: Let x = 3. Original: \dfrac{6}{5} - \dfrac{1}{1} = \dfrac{1}{5}. Answer: \dfrac{1}{3 + 2} = \dfrac{1}{5} ✓

Tip: You can only add or subtract fractions with a common denominator — never subtract tops and bottoms straight across. Testing a value such as x = 3 quickly eliminates wrong choices.

1. Fixed weight: nurse + gurney = 180 + 310 = 490 pounds.
2. Boxes: b boxes at 45 pounds each weigh 45b pounds.
3. Total weight: 45b + 490.
4. “At most 2,500” means the total is less than or equal to 2,500: 45b + 490 \le 2{,}500

The inequality is 45b + 490 \le 2{,}500.

Check: Solving gives 45b \le 2{,}010, so b \le 44.67 — at most 44 boxes. Test: 45(44) + 490 = 2{,}470 \le 2{,}500 ✓, while 45(45) + 490 = 2{,}515 is too heavy ✓

Tip: Translate key phrases: “at most / no more than / maximum” → \le; “at least / no less than / minimum” → \ge. The per-item amount (45) goes with the variable.

1. Start with y = mx + b and m = -\tfrac{2}{5}: y = -\tfrac{2}{5}x + b.
2. Substitute the point (10, -3): -3 = -\tfrac{2}{5}(10) + b = -4 + b
3. Solve for b: add 4 to both sides: b = -3 + 4 = 1.
4. Equation: y = -\tfrac{2}{5}x + 1.

The answer is y = -\tfrac{2}{5}x + 1.

Check: At x = 10: -\tfrac{2}{5}(10) + 1 = -4 + 1 = -3 ✓. The slope matches ✓

Tip: Point-slope form y - y_1 = m(x - x_1) gives the same result: y + 3 = -\tfrac{2}{5}(x - 10), so y = -\tfrac{2}{5}x + 4 - 3. The y-coordinate of the point is not the y-intercept unless x = 0.