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2008 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2008 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Basic Counting — Each time a 2-point basket becomes a 3-point basket, the total goes up by exactly 1. Six choices for the number of 3-pointers give six totals. |
| 2 | B | Sets, Estimation & Miscellaneous — Write out the new block after reversing the two rows, then add each diagonal. The sums are 50 and 54. |
| 3 | D | Exponents, Logarithms & Radicals — Turn every root into a fraction exponent. x times the square root of x is x to the 3/2, and a cube root divides that exponent by 3. |
| 4 | C | Linear Equations & Word Problems — To make one salary as large as possible, pay everyone else the smallest amount allowed. The star gets whatever is left of the 700,000. |
| 5 | A | Functions — (x - y) squared and (y - x) squared are the same number. The operation squares the difference of its two inputs, and equal inputs have difference 0. |
| 6 | C | Ratios, Percents & Averages — Measure both points from D. BD is one fifth of AD and CD is one tenth of AD, so BC is the difference, one tenth of AD. |
| 7 | C | Similar & Congruent Triangles — Count the small triangles row by row: 1, 3, 5, and so on up to 19. The first ten odd numbers add to 10 times 10, which is 100. |
| 8 | C | Diophantine Equations — Carnations cost an even amount and 50 is even, so the roses must cost an even amount too. The number of roses is 0, 2, 4, up to 16. |
| 9 | A | Quadratics — Divide by a to get x squared minus 2x plus b over a. The two solutions add to 2, so their average is 1, no matter what b is. |
| 10 | A | Circles — Draw a line from the center through the chord's midpoint. The chord is 4 from the center and C is 1 beyond it, so AC has legs 3 and 1. |
| 11 | B | Sequences & Series — The missing term between the two known ones is the fourth term. Call it a, build the fifth and sixth terms from it, and set the sixth equal to 128. |
| 12 | A | Sets, Estimation & Miscellaneous — Each flip means 100,000 steps. Forty-four flips plus the final 50,000 give 4,450,000 steps, and dividing by 1800 gives about 2470 miles. |
| 13 | B | Sequences & Series — If the first n terms average n, they add up to n times n. Each term is the jump from one square to the next: the odd numbers. |
| 14 | B | Coordinate Geometry — A sits straight above B at height 5 divided by the square root of 3. A quarter turn counterclockwise sends (x, y) to (-y, x). |
| 15 | A | Diophantine Equations — The Pythagorean theorem simplifies to a squared equals 2b plus 1. So a is odd, and b stays below 100 only up to a = 13. |
| 16 | A | Basic Probability — Once at least one die is rolled, the sum is odd half the time. Only two tails is different: no dice, and 0 is even. |
| 17 | B | Basic Probability — One fixed order, like yes-no-no, has chance 0.7 times 0.3 times 0.3. The single yes can come on any of 3 occasions, so multiply by 3. |
| 18 | B | Linear Equations & Word Problems — Measure speed in bricks per hour. Brenda lays one ninth of the chimney each hour and Brandon one tenth; together, minus 10 bricks, they must lay one fifth. |
| 19 | E | Solid Geometry — Look at the tank's round end. The water is a one-third slice of the circle minus a triangle. Multiply that area by the length 9. |
| 20 | B | Basic Probability — Each die has six faces, so there are 36 equally likely outcomes. Count the face pairs for sums 5, 7 and 9, counting a repeated number once for each face. |
| 21 | C | Arrangements with Restrictions — Seat the men first. Each wife then has exactly two allowed chairs, and one wife's choice forces all the others, so the wives can sit in just 2 ways. |
| 22 | C | Basic Probability — Place the three red beads first. They cannot touch, so they fit only four ways in six spots. Then check where the white and blue beads can go. |
| 23 | B | Diophantine Equations — The painted part is 2 feet shorter in each direction and is half the floor. The equation rearranges to (a minus 4) times (b minus 4) equals 8. |
| 24 | C | Angles & Polygons — Cut angles B and C in half with lines meeting at E. This makes three matching triangles with 60-degree angles at E, so E lands on side AD. |
| 25 | B | Linear Equations & Word Problems — Track the truck's lead over Michael. Each 50-second cycle it gains 100 feet while driving and loses 150 while stopped, so the lead shrinks by 50. |
2008 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.