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2017 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2017 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | B | Bases & Digits — Undo the steps from the end. Switched back, the result was 17, 27, 37, 47 or 57, and only 47 = 3 times 12 plus 11 works. |
| 2 | C | Linear Equations & Word Problems — Find the time for each part of a lap: 100 meters takes 25 seconds and 300 meters takes 60 seconds, so one lap is 85 seconds. |
| 3 | E | Absolute Value & Inequalities — Add the two facts: y is more than -1 and z is more than 1, so y + z is more than 0. Every other choice can be negative. |
| 4 | D | Algebraic Manipulation — Clear the fraction in the given equation: it simplifies to x = y. Then the fraction asked for is 4x over 2x, which is 2. |
| 5 | D | Linear Equations & Word Problems — Call the cherry count c, so blueberry is 2c. After eating 10 of each, 2c - 10 = 3(c - 10), so c = 20 and blueberry is 40. |
| 6 | B | Solid Geometry — Volume allows at most 4 blocks, since 5 blocks need 20 cubic inches and the box has 18. Three flat blocks plus one on its side fit. |
| 7 | C | Linear Equations & Word Problems — Both halves have the same length d, and the walking distance is exactly d. Biking time d/17 plus walking time d/5 equals 44 minutes, written in hours. |
| 8 | C | Coordinate Geometry — Because AB = AC, the altitude from A lands at the middle of BC. So D is the midpoint, and C is the step from B to D taken twice. |
| 9 | D | Basic Probability — There are 27 equally likely ways to answer. Exactly two right happens in 3 times 2 = 6 ways, all three right in 1 way: 7 out of 27. |
| 10 | E | Coordinate Geometry — The slopes are a/2 and -2/b, and perpendicular slopes multiply to -1, so a = b. Putting (1, -5) into both equations then gives c. |
| 11 | D | Ratios, Percents & Averages — Picture 100 students. 12 like dancing but say they dislike it, and 36 dislike it and say so. Of the 48 who say dislike, 12 actually like it. |
| 12 | A | Ratios, Percents & Averages — Try real numbers: a trip needing 3 liters in the old car needs 2 in the new one. At 1.20 dollars instead of 1, cost drops from 3.00 to 2.40. |
| 13 | C | Inclusion-Exclusion — Sort students by how many classes they take. 20 students in all, 9 take two or more, so 11 take one; the class lists add to 32 sign-ups. |
| 14 | D | Modular Arithmetic — The fourth powers of 1, 2, 3, 4 all leave remainder 1 when divided by 5. So N to the 16th leaves 1 unless N is a multiple of 5. |
| 15 | E | Similar & Congruent Triangles — Similar right triangles give AE = 9/5, which is 9/25 of the diagonal AC = 5. Triangle AED is then 9/25 of triangle ACD, whose area is 6. |
| 16 | A | Bases & Digits — Count the numbers with no 0, where every digit is 1 to 9: 9 + 81 + 729 + 729 = 1548. The other 469 contain a 0. |
| 17 | B | Basic Counting — Choosing a set of different digits gives exactly one increasing number and one decreasing number. Count digit sets: 511 increasing, 1022 decreasing, minus the 9 one-digit numbers counted twice. |
| 18 | D | Arrangements with Restrictions — Put the single green disk at the top corner or the bottom middle. Only a left-right flip is left, and the 10 red placements give 6 paintings each. |
| 19 | E | Triangles: Area & Pythagorean — Cut the big triangle into ABC plus three outside triangles. Each has sides 4s and 3s with a 120-degree angle, so area 12 times ABC. |
| 20 | B | Divisibility & Factors — 21! has eighteen factors of 2. Divisors split into 19 equal groups by how many 2s they contain, and the odd ones are one group. |
| 21 | D | Triangle Centers & Cevians — The right angle at A makes AD = 5. Each half-triangle has area 12, and each radius is its area divided by half its perimeter. |
| 22 | D | Circles — AB is a diameter, so the angle at C is a right angle. Then triangle ACB has the same shape as right triangle ADE, which has area 35/2. |
| 23 | C | Modular Arithmetic — Split 45 into 5 and 9. N ends in 4, so its remainder by 5 is 4; its digit sum matches 1 through 44 added up, 990. |
| 24 | C | Coordinate Geometry — The centroid (1, 1) is also the center of the triangle's circle. Mirror symmetry across y = x forces one corner to be (-1, -1). |
| 25 | E | Modular Arithmetic — Subtract 90 from each score. The 7-test total is a multiple of 7 from 28 to 49, and minus 5 it must be a multiple of 6: only 35. |
2017 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.