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2008 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2008 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Clocks, Calendars & Time — One third of the job took 2 hours 40 minutes, so the whole job takes three times as long: 8 hours after the 8:30 AM start. |
| 2 | A | Ratios, Percents & Averages — Only ratios are given, so give the square a side of 1. Then the rectangle is 2 wide and 4 long, and the square fills 1 out of 8. |
| 3 | A | Divisibility & Factors — The divisors of 6 other than 6 are 1, 2 and 3, which add up to 6. So each step turns 6 back into 6. |
| 4 | C | Ratios, Percents & Averages — Two thirds of 10 bananas is 20/3 bananas, and that equals 8 oranges. So one banana is worth 6/5 of an orange, and 5/2 bananas are worth 3 oranges. |
| 5 | B | Algebraic Manipulation — The top of each fraction equals the bottom of the next one, so almost everything cancels, leaving the last top over the first bottom: 2008 over 4. |
| 6 | D | Ratios, Percents & Averages — Average speed is total distance over total time. With 60 km legs, the times are 20, 3 and 6 hours, so the average is 180 over 29, about 6. |
| 7 | E | Exponents, Logarithms & Radicals — Top and bottom have the same shape, but every power of 3 on top is 9 times the matching one below. So the whole fraction equals 9. |
| 8 | A | Ratios, Percents & Averages — Store B takes off 10% more than store A, but A gives 90 dollars back. The 90 beats that 10% by 15, so 10% of the price is 75. |
| 9 | B | Divisibility & Factors — Over a common bottom of 6 the expression becomes x over 2. So x must be even, and x = 2 shows it need not be a multiple of 3. |
| 10 | E | Quadrilaterals & Polygon Areas — Joining the midpoints of a square cuts off four corner triangles worth half its area. So each new square is half the last: 16, 8, 4. |
| 11 | D | Linear Equations & Word Problems — The trip to shore takes 15 minutes, so the boat may gain at most 30 ÷ 15 = 2 gallons each minute. LeRoy must bail the other 8. |
| 12 | C | Ratios, Percents & Averages — 25% more red than blue means red is 1.25 times blue, so blue is red divided by 1.25, which is 0.8r. Green is 1.6r, and the total is 3.4r. |
| 13 | D | Linear Equations & Word Problems — Together they paint 1/5 + 1/7 of the room each hour, but they only paint for t minus 1 hours, because one hour of the total is lunch. |
| 14 | D | Triangles: Area & Pythagorean — A 4 to 3 screen with a 27-inch diagonal is a 3-4-5 triangle scaled up: 21.6 wide and 16.2 tall. The movie is 10.8 tall, leaving 5.4 for two strips. |
| 15 | D | Linear Equations & Word Problems — Expanding Han's distance shows Ian's speed plus 5 times Ian's time is 65. Jan's extra distance is twice that amount plus 20, which is 150. |
| 16 | B | Circles — The small center lies on the line halving the 60 degree angle. A 30-60-90 triangle puts it 2r from O, and 2r + r = R, so r = R/3. |
| 17 | B | Circles — The band around the triangle is three 6 by 3 rectangles, one along each side, plus three rounded corner pieces that fit together into one full circle of radius 3. |
| 18 | B | Triangles: Area & Pythagorean — Square the sum of the legs: it equals the hypotenuse squared plus 4 times the area. Since that sum is 32 minus the hypotenuse, the squares cancel. |
| 19 | C | Transformations & Symmetry — Each turn moves P along a quarter circle. Its radius is the diagonal PR, root 40, for the first turn and the side PS = 6 for the second. |
| 20 | D | Similar & Congruent Triangles — The diagonals cut each other in the ratio 9 to 12, or 3 to 4. Triangles sharing a height have areas in that ratio: 18, 24, 24, 32. |
| 21 | A | Solid Geometry — All four sides of ABCD are equal, so it is a rhombus. Its diagonals are the cube diagonal, root 3, and a face diagonal length, root 2. |
| 22 | D | Basic Probability — Only three coin flips matter, so list all eight paths and follow the actual numbers. Five of the eight fourth terms are whole numbers. |
| 23 | B | Basic Counting — Each letter is in both subsets, only the first, or only the second. Choose the 2 shared letters, split the other 3, then halve. |
| 24 | D | Modular Arithmetic — k and its square end in 0. Powers of 2 end in 2, 4, 8, 6 repeating. Since k is a multiple of 4, 2^k ends in 6. |
| 25 | C | Circles — The touching inner corners form a regular hexagon with side x, so the inner edge is known from the center; one right triangle to the table edge gives x. |
2008 AMC 10A — answer key. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.