Printable paper
2012 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2012 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Linear Equations & Word Problems — Each classroom has 16 more students than rabbits. There are 4 classrooms, so there are 4 times 16, which is 64, more in all. |
| 2 | E | Quadrilaterals & Polygon Areas — The circle touches the top and bottom of the rectangle, so the width equals the circle's diameter, 10. The length is twice that, 20. |
| 3 | B | Coordinate Geometry — The mirror line y = 2000 is flat, so the x-coordinate stays 1000. The point is 12 above the line, so its reflection is 12 below it. |
| 4 | A | Modular Arithmetic — Put the two leftover piles together: 4 + 3 = 7 marbles. Seven marbles fill one more bag of 6, leaving 1. |
| 5 | D | Ratios, Percents & Averages — The tax and the tip are both taken from the meal price, so every dollar of meal costs 1.25 dollars in total. Divide 27.50 by 1.25. |
| 6 | A | Absolute Value & Inequalities — Making the first number bigger and the subtracted number smaller both make the difference bigger. The two roundings add up instead of cancelling. |
| 7 | D | Linear Equations & Word Problems — Both animals hid the same total. Write the chipmunk's total as 3 times its holes and the squirrel's as 4 times 4 fewer holes, and set them equal. |
| 8 | B | Absolute Value & Inequalities — The square must be 4, 9 or 16, so x - 2 is 2, 3 or 4 or their negatives. The six answers pair up around 2. |
| 9 | A | Number Properties — Split the totals into three pairs that add to 26, 15 and 16. Only the pair adding to 15, an odd number, is forced to contain an odd integer. |
| 10 | D | Divisibility & Factors — Cross-multiply to get M times N = 36. Every divisor of 36 chosen as M gives exactly one N, and 36 has 9 divisors. |
| 11 | A | Arrangements with Restrictions — Fill in Friday (cake) first, then work outward from it. Every other day then has 3 choices, since only the neighbor already filled in is ruled out. |
| 12 | B | Triangles: Area & Pythagorean — The 45-degree angle makes triangle ABC a right triangle with two equal legs of 10. Then AD is the long side of a right triangle with legs 10 and 30. |
| 13 | B | Linear Equations & Word Problems — Speeds add, not times. Pretend the escalator has 120 steps: Clea walks 2 steps per second, and with the escalator she moves 5, so the escalator alone moves 3. |
| 14 | D | Quadrilaterals & Polygon Areas — Each triangle is 3 tall, so the two tips are 6 - 2√3 apart. The rhombus has a 60-degree angle at each tip, so it is two equilateral triangles. |
| 15 | D | Games & Processes — There are 15 games, so 15 wins in total. Six teams can't share 15 wins equally, but five teams can each win 3 if they all beat the sixth team. |
| 16 | A | Circles — Connect the three centers to make an equilateral triangle of side 4. The triangle is the middle gap plus three 60-degree slices of circle that together make half a circle. |
| 17 | C | Solid Geometry — When a sector rolls into a cone, its curved edge becomes the base circle and its radius 12 becomes the slant side. That gives base radii 4 and 8. |
| 18 | C | Conditional Probability & States — Imagine 50,000 people: 100 are sick and test positive, but 2% of the 49,900 healthy people, 998 people, also test positive. So about 1 in 11 positives is real. |
| 19 | C | Quadrilaterals & Polygon Areas — BFDG is a trapezoid: BF and GD are both vertical, 6 apart. Similar triangles give BF = 30/4 = 7.5, and GD is half of 30, so 15. |
| 20 | A | Games & Processes — Silvia only adds 50, so Bernardo wins exactly when he hands her a number from 950 to 999. Find the smallest start that lands there. |
| 21 | A | Triangles: Area & Pythagorean — Two points 2a apart, each exactly a from a third point, force that third point to be the midpoint. The fourth point then makes an equilateral triangle. |
| 22 | B | Arrangements with Restrictions — The numbers written so far always form a run of consecutive numbers. Reading the list backward, each number removed must be an end of the run: 2 choices, nine times. |
| 23 | D | Solid Geometry — The line from the removed corner to the opposite corner stands straight up from the cut face. Take the cut-off corner's height, 1/√3, away from √3. |
| 24 | B | Basic Counting — Give each song a label: the group of girls who like it. There are 7 allowed labels, and the three two-girl labels must each be used at least once. |
| 25 | E | Paths & Grids — After the bug crosses an arrow in the 1st, 3rd, 5th or 7th column it can never go back. So count routes one stage at a time and multiply. |
2012 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.