Mixed-topic practice

Work. The output rises 20 while the input rises 5, so the slope is 4. From input 8 to 11, add 3(4) = 12 to the output: f (11) = 34 + 12 = 46. Finding b = 2 is optional.

Answer: 46

Check. The rule f(x) = 4x + 2 fits both given pairs.

Key takeaway. The decisive clue is a constant rate of change, not the presence of function notation.

Work. The two multipliers are 1.25 and 0.80. Their product is 1, so the final price equals the original price. Equivalently, p(1.25)(0.80) = 84 gives p = 84.

Answer: $84

Check. 84 → 105 → 84. The percentages use different bases.

Key takeaway. Equal-looking percentages are not added; compose the multipliers.

Work. The area equation is x(x + 5) = 84, so x² + 5x − 84 = (x + 12)(x − 7) = 0. A positive width requires x = 7. The length is 12 and the perimeter is 2(7 + 12) = 38.

Answer: 38

Check. 7 · 12 = 84. The negative candidate −12 cannot be a length.

Key takeaway. A quadratic root is an intermediate result when the question asks for perimeter.

Work. Adding the equations eliminates y and produces 6x = 30 immediately. There is no need to compute x = 5 unless using it to check the result.

Answer: 30

Check. x = 5 gives y = 3 in either original equation.

Key takeaway. Compare the requested expression with sums, differences, and multiples of the given equations.

Work. The factor over 3 input units is 80/10 = 8. A positive one-unit base therefore satisfies b³ = 8, so b = 2. Two more input units multiply the value by 2² = 4: g(6) = 320.

Answer: 320

Check. The rule g(x) = 5 · 2^x reproduces both given values.

Key takeaway. For exponential models, compare ratios over equal input intervals rather than differences.

Work. Let the total be n > 0. The condition is 18/n ≤ 0.06. Multiplying by positive n gives 18 ≤ 0.06n, so n ≥ 300. Thus the least total is 300.

Answer: 300

Check. 18/300 = 0.06; any smaller positive total gives a larger defective fraction.

Key takeaway. Identify whether the variable is a part or the whole. Here 300 is the total, not the number of nondefective items.

Work. The axis of symmetry is x = 6. Input 10 is 4 units to the right, so the other input is 4 units to the left: a = 2. Algebraically, (a − 6)² = 16 gives a = 10 or 2.

Answer: 2

Check. f(2) = f (10) = 20.

Key takeaway. Equal quadratic outputs occur at inputs equally spaced from the axis, unless the two inputs coincide.

Work. The selected population contains only supporters, of whom there are 35 + 28 = 63. Among them, 35 are bicycle commuters. The probability is 35/63 = 5/9.

Answer: $\frac{\text{5}}{\text{9}}$

Check. 35/50 answers a different question: support given bicycle commuting.

Key takeaway. Reverse conditioning usually changes the denominator; the two conditional probabilities need not be equal.

Work. At the t-intercept, P = 0. Solve 480 − 12t = 0 to get t = 40. In context, this is the predicted number of hours needed to finish scanning the pages.

Answer: 40 hours, when no pages remain.

Check. The rate has units pages per hour, so 480 pages/(12 pages/hour) = 40 hours.

Key takeaway. An intercept is not merely a coordinate to compute; its interpretation depends on which variable is zero.

Work. Substitute y = 3: (x − 2)² + 16 = 25, so (x − 2)² = 9. Thus the intersections have x = −1 and x = 5, both with y = 3. Their horizontal separation is 5 − (−1) = 6.

Answer: 6

Check. The chord is 4 units from the center of a radius-5 circle; its half-length is $\sqrt{\text{25 − 16}}$ = 3.

Key takeaway. The requested distance can be simpler than a full system solution.

Work. Each B value is obtained by multiplying its corresponding A value by 2 and adding 8. Addition does not change spread; multiplication by 2 doubles every deviation from the mean. The ratio is 2.

Answer: 2

Check. The means are 5 and 18. Deviations (−3, −1, 1, 3) become (−6, −2, 2, 6).

Key takeaway. Compare transformations before calculating a standard deviation from scratch.

Work. A nonzero coefficient of x yields one solution. No solution is possible when k − 3 = 0 but the right side is nonzero. At k = 3, the equation becomes 0 = 7, a contradiction.

Answer: k = 3

Check. For every other k, x = (2k + 1)/(k − 3) exists.

Key takeaway. Before dividing by an expression containing a parameter, test when that expression is zero.

Work. Since 1 m = 100 cm, an area conversion uses 100² = 10,000. Thus 0.72 m² = 7200 cm².

Answer: 7200 cm²

Check. A 0.8 m by 0.9 m sheet has the same area as an 80 cm by 90 cm sheet.

Key takeaway. The exponent on a unit applies to the conversion factor too.

Work. Equating outputs gives x² − 6x + 7 − c = 0. Tangency requires one repeated real root, so (−6)² − 4(7 − c) = 0. Hence 36 − 28 + 4c = 0, and c = −2.

Answer: −2

Check. At c = −2, the intersection equation is (x − 3)² = 0; both graphs meet at (3, 10).

Key takeaway. Tangency is a solution-count statement. A graph can suggest it; a repeated-root calculation verifies it.

Work. The sample proportion is 90/250 = 0.36. Apply that rate to the population: 6000(0.36) = 2160. This is an estimate, not a known exact count.

Answer: An estimated 2160 members.

Check. The estimated count is between 0 and 6000 and corresponds to 36% of the population.

Key takeaway. Inference requires an appropriate sampling process and retains uncertainty even when the arithmetic is exact.