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2009 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2009 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | B | Linear Equations & Word Problems — Five muffins cost 2 dollars and 50 cents, and each swap to a bagel adds 25 cents, so two swaps reach a whole dollar amount. |
| 2 | C | Fractions & Decimals — Work out the top and the bottom separately first: the top is 1/12 and the bottom is 1/6, and 1/12 is half of 1/6. |
| 3 | C | Ratios, Percents & Averages — The three lost cans were exactly the paint for the 5 rooms she can no longer do, so 3 cans paint 5 rooms, and 25 rooms need 15 cans. |
| 4 | C | Quadrilaterals & Polygon Areas — The two triangles use up 25 - 15 = 10 meters of the top edge, so each has legs of 5, which is also the yard's width. |
| 5 | D | Ratios, Percents & Averages — Twenty percent less than 60 is 48. A number plus a third of itself is 4 thirds of it, so 4 thirds make 48. |
| 6 | D | Divisibility & Factors — The twins share an age, so 128 is Kiana's age times the twins' age twice. Try the few possible twin ages and keep the one where Kiana is younger. |
| 7 | C | Basic Counting — Only the single plus sign can be moved earlier. It can join the 3 with the left product, the right product, both, or neither, giving 26, 46, 50 and 70. |
| 8 | B | Ratios, Percents & Averages — Follow a price of 100 dollars month by month. It reaches 120 after March, and dropping back to 100 is 20 out of 120, about 17 percent. |
| 9 | A | Angles & Polygons — The equal sides meet at B, so triangle ABC has angles B, and twice five-halves of B, giving B = 30. Vertical angles carry C's angle across. |
| 10 | E | Triangles: Area & Pythagorean — The stump, the ground, and the broken top form a right triangle with legs x and 1 and long side 5 - x. When you expand, the x-squared terms cancel. |
| 11 | A | Basic Counting — The middle digit has no partner, so it must be the 5 that appears an odd number of times. The first three digits are 2, 3, 5 in any order. |
| 12 | A | Triangles: Area & Pythagorean — Every triangle has two corners on one line and one on the other, so its height never changes. Only the base length matters: 1, 2 or 3. |
| 13 | C | Modular Arithmetic — Rolling lays the sides down in order AB, BC, CD, DE, EA, repeating every 23 units. 2009 leaves remainder 8 when divided by 23, landing on CD. |
| 14 | D | Sequences & Series — After each refill there is always exactly 3/4 quart of other seeds, since the birds eat them all. Track only the millet until it passes 3/4 quart. |
| 15 | E | Linear Equations & Word Problems — Going from half full to two-thirds full adds one-sixth of the water, and that one-sixth weighs a - b. Add the last one-third (twice that) to the two-thirds weighing. |
| 16 | B | Circles — BO cuts the 60 degree angle at B in half, and radius OA meets the tangent at a right angle, so triangle OAB is 30-60-90 with BO twice the radius. |
| 17 | C | Coordinate Geometry — The shaded part is the big right triangle with corners (c,0), (3,0), (3,3), minus the one empty unit square in its corner. Set it equal to 5/2. |
| 18 | D | Similar & Congruent Triangles — Right triangles AME and ABC share the angle at A, so they have the same shape, and ME is 3/4 of AM just as BC is 3/4 of AB. |
| 19 | A | Basic Counting — The time is right only when no 1 appears. Multiply the share of good hours (8 of 12) by the share of good minutes (45 of 60). |
| 20 | B | Triangle Centers & Cevians — An angle bisector splits the opposite side in the same ratio as the two sides that form the angle: here 1 to the square root of 5. |
| 21 | D | Modular Arithmetic — Divided by 8, powers of 3 leave remainders 1, 3, 1, 3, and so on. Each pair adds to 4, and there are 1005 pairs. |
| 22 | B | Solid Geometry — The cuts are vertical, so the piece is triangle B stretched to height 2. Cake is twice B's area; icing is B's area plus a 2 by 2 side. |
| 23 | C | Geometric Probability — A runner is in the photo within one-eighth of a lap time of crossing the start line. Find each runner's time window and see where they overlap. |
| 24 | A | Angles & Polygons — Extend the slanted sides of all the trapezoids down to the center of the arch. The nine trapezoids split the 180 degree straight angle into nine 20 degree slices. |
| 25 | B | Basic Probability — A loop must run through four faces forming a belt. There are 3 belts; each needs its 4 faces right, the other 2 faces are free, and belts never overlap. |
2009 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.