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2015 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2015 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Exponents, Logarithms & Radicals — A negative power means one over the positive power, and squaring -2 gives positive 4. So the expression is 2 minus 1/4. |
| 2 | B | Clocks, Calendars & Time — From 1:00 to 2:40 is 100 minutes, and that covers two tasks, not three. So each task takes 50 minutes. |
| 3 | A | Linear Equations & Word Problems — If 28 is written twice, the other number is 44/3, not a whole number. So 28 is written three times and the other is 8. |
| 4 | C | Fractions & Decimals — Write every share out of 60: Alex 12, Beth 20, Cyril 15. Dan gets the rest, 60 - 47 = 13, which is more than Alex's 12. |
| 5 | B | Logic Puzzles — Only Marta's place is given as an exact number, 6th. Start there and follow the clues one by one: Jack 5th, Todd 3rd, Rand 2nd, Hikmet 8th. |
| 6 | E | Logic Puzzles — Tuesday must be a running day, since four days in a row hold at most two runs. The tennis rule then rules out all but one schedule. |
| 7 | A | Functions — Work from the inside parentheses out. The left side is 1/2 - 1/3 = 1/6, the right side is 1 - 3/5 = 2/5, and 1/6 - 2/5 = -7/30. |
| 8 | E | Transformations & Symmetry — Follow one point through all three moves: (x, y) ends up at (y, x). So the three moves together just flip the F across the line y = x. |
| 9 | B | Circles — The shaded fin is a quarter of the big circle with half of the small circle cut out: 9pi/4 - 9pi/8 = 9pi/8. |
| 10 | C | Modular Arithmetic — There are 1007 negative numbers, an odd count, so the product is negative. All the numbers are odd and one of them is -5, so the product ends in 5. |
| 11 | B | Primes — Only the digits 2, 3, 5, 7 are allowed, so there are just 4 + 16 = 20 numbers to check. Eight of them are prime. |
| 12 | A | Coordinate Geometry — Put (x, -x) into the distance formula from (5, 5). The x terms cancel and the condition becomes x squared at most 25, so x goes from -5 to 5. |
| 13 | E | Triangles: Area & Pythagorean — The triangle is a right triangle with sides 5, 12, 13. The two short sides are already altitudes, and the third altitude comes from the area: 60/13. |
| 14 | D | Quadratics — Pull out the shared factor (x - b). The roots are b and the average of a and c, so put 9 in b and 8, 7 in a, c. |
| 15 | B | Diophantine Equations — With h horses and c cows the total is 13h + 5c. Take 13s away from each choice and look for a multiple of 5. |
| 16 | C | Basic Probability — Each number must be a bigger multiple of the next, so Cal gets 1 or 2. There are 9 good outcomes out of 720. |
| 17 | B | Solid Geometry — The side-face centers make a diamond in the middle layer, half of 5 by 4. The octahedron is two pyramids on that diamond. |
| 18 | D | Expected Value — A coin ends tails only after three tails in a row, chance 1/8. So each coin ends heads with chance 7/8, and 64 times 7/8 = 56. |
| 19 | C | Circles — The circle's center is equally far from both ends of every chord, which puts it at the middle of AB. Then equal distances force equal legs. |
| 20 | A | Paths & Grids — Color corners in two colors so each edge joins different colors. After 7 moves Erin must end on the corner opposite the start. |
| 21 | D | Number Properties — Jump counts are steps divided by 2 or 5, rounded up. Their difference stays close to 3/10 of the steps, so only 62 to 66 need checking. |
| 22 | D | Angles & Polygons — Every triangle in the figure has angles 36, 72, 108 degrees, so lengths grow by the golden ratio: FG = 1/phi, JH = 1, CD = phi. |
| 23 | B | Divisibility & Factors — End zeros come from factors of 5. Doubling n usually just doubles them, so list small n and find the few that triple: 8, 9, 13, 14. |
| 24 | D | Sequences & Series — At square step numbers the ant sits at a spiral corner. Step 1936 is at (-22, 22); then 44 steps down and 35 steps right. |
| 25 | B | Diophantine Equations — Dividing by 2abc gives 1/a + 1/b + 1/c = 1/2. Since a is smallest, a is 3 to 6; try each and solve for b and c. |
2015 AMC 10B — 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.