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2011 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2011 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Fractions & Decimals — Add up the tops and bottoms first. The problem becomes 12/9 minus 9/12, which is 4/3 minus 3/4. |
| 2 | E | Ratios, Percents & Averages — Her average now is 77, so she wants 80. Six tests averaging 80 must add up to 480, and she already has 385. |
| 3 | A | Absolute Value & Inequalities — Each side could be up to half an inch shorter than reported. The smallest area uses the shortest possible width and the shortest possible length together. |
| 4 | C | Linear Equations & Word Problems — Each person should pay half of the total. LeRoy gives Bernardo the difference between that half and what he already paid. |
| 5 | E | Divisibility & Factors — Break 161 into factors: it is 7 times 23. The reversed number has two digits, so it was 23. That makes the real number 32 and the other number 7. |
| 6 | A | Linear Equations & Word Problems — Undo the story backwards from the end. Add back what he gave away, then remember what is left after eating a third is two thirds of what he had. |
| 7 | B | Angles & Polygons — The two angles add to 108 degrees, so they are 39 and 69. The third angle is 180 minus 108, which is 72, and it is the biggest. |
| 8 | B | Logic Puzzles — Hot and sunny always means crowded. So if the beach was not crowded, the day was not both hot and sunny: at least one of the two was missing. |
| 9 | D | Similar & Congruent Triangles — Triangle EBD is a smaller copy of ABC. One third of the area means sides shrink by the square root of 3, and BD matches BC = 4. |
| 10 | B | Sequences & Series — The ten smaller numbers add up to 1,111,111,111. Nine times that is 9,999,999,999, just 1 short of the largest number, so the ratio is almost exactly 9. |
| 11 | D | Sets, Estimation & Miscellaneous — If every month had at most 4 birthdays, the room would hold at most 48 people. With 52, some month has 5, and 6 can fail. |
| 12 | A | Circles — The straight parts are equal on both edges. The two round ends form a circle whose radius grows by 6, adding exactly 12 pi meters. |
| 13 | D | Geometric Probability — The product is positive when both numbers are negative or both are positive. A number is negative 2/3 of the time and positive 1/3 of the time. |
| 14 | C | Algebraic Manipulation — The diagonal gives l squared plus w squared = 625, and the area gives lw = 168. So (l + w) squared = 625 + 336 = 961. |
| 15 | E | Functions — Replace each @ by 'add, then divide by 2' and simplify. Law I gets an extra half of x; laws II and III come out equal. |
| 16 | A | Geometric Probability — Let the octagon's side be the square root of 2. The corner triangles then have legs 1, and the octagon is a square minus four corners. |
| 17 | C | Circles — Angle A is 90 degrees because EB is a diameter, so angle ABE is 50. Parallel lines copy that 50 to angle BED, and angle BCD is 180 minus 50. |
| 18 | E | Triangles: Area & Pythagorean — Parallel sides make angle MDC equal angle AMD, so triangle MDC has two equal sides: CM = CD = 6. With CB = 3, angle CMB is 30. |
| 19 | A | Absolute Value & Inequalities — Let u stand for |x|. The equation becomes u squared minus 5u minus 24 = 0, so |x| = 8. The roots are 8 and -8. |
| 20 | C | Quadrilaterals & Polygon Areas — The short diagonal BD splits the rhombus into two equilateral triangles of side 2. In each, the part closer to B than to the other corners is exactly one third. |
| 21 | B | Number Properties — The biggest difference, 9, is w minus z. It is made of three gaps between neighbors, which must be 1, 3 and 5, with 1 in the middle. |
| 22 | A | Solid Geometry — The pyramid's height is half the square root of 2. The level slice at the cube's height is a square, and it is exactly the cube's top face. |
| 23 | D | Modular Arithmetic — Only the last three digits matter, and they match those of 11 to the 2011th power. For 11 to the n, they come from 1 + 10n + 50n(n-1). |
| 24 | B | Coordinate Geometry — The line hits a lattice point at x exactly when the slope times x is whole. So a is the smallest fraction above 1/2 with bottom at most 100. |
| 25 | D | Circles — Each new side is half the perimeter minus an old side. Three sides in a row stay three sides in a row, and the middle side halves each time. |
2011 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.