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2007 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2007 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Solid Geometry — The four walls unfold into one strip 44 feet long and 8 feet tall. Take away the doors and windows, then multiply by three rooms. |
| 2 | E | Functions — The new symbol is just a recipe: add the two numbers, then multiply by the second one. Swapping the order changes which number you multiply by. |
| 3 | B | Ratios, Percents & Averages — Average mileage is all the miles divided by all the gallons: 240 miles on 4 + 6 = 10 gallons. |
| 4 | D | Circles — The three center angles add to 360, so angle AOC is 100. The angle at B looks at the same arc, so it is half of that. |
| 5 | D | Logic Puzzles — The facts stack up like circles inside circles: Crups inside Dramps inside Arogs inside Brafs. So every Crup is also an Arog and a Braf. |
| 6 | D | Linear Equations & Word Problems — The three blank problems give 4.5 points for free, so the right answers must bring at least 95.5 more. Fifteen right is 90, too few; sixteen right is 96, enough. |
| 7 | E | Angles & Polygons — Draw the line from C to E. It cuts the pentagon into a square and an equilateral triangle, so the angle at E is 90 plus 60. |
| 8 | D | Basic Counting — Once a and c are picked, b is their average, so it is fixed. It is a whole number exactly when a and c are both even or both odd. |
| 9 | D | Modular Arithmetic — The last s is the 12th s, so it moves 1 + 2 + ... + 12 = 78 places, exactly three trips around the alphabet. |
| 10 | A | Triangles: Area & Pythagorean — The base BC is fixed, so area 1 fixes the height. Points at that distance from line BC form two parallel lines, one on each side. |
| 11 | C | Circles — The center lies on the triangle's line of symmetry. Setting its distance to the top corner equal to its distance to a bottom corner gives one equation. |
| 12 | D | Linear Equations & Word Problems — Going back N years, Tom loses N years but his three children together lose 3N. Writing 'Tom then = twice the children then' gives T = 5N. |
| 13 | D | Circles — Both circles pass through (0,0) and (2,2). The overlap is split by the line between those points into two equal pieces, each a quarter circle with a right triangle removed. |
| 14 | C | Ratios, Percents & Averages — The group size never changes, so the drop from 40 percent to 30 percent girls is the 2 girls who left: 10 percent of the group is 2. |
| 15 | D | Angles & Polygons — Write each angle as a piece of A: B is half of A, C a third, D a quarter. The four angles add to 360. |
| 16 | C | Ratios, Percents & Averages — Pretend the class has 10 students: 1 junior and 9 seniors. The class total is 840 and the seniors earned 747, so the junior scored the other 93. |
| 17 | D | Triangles: Area & Pythagorean — Join P to the three corners. The three small triangles fill the big one, and adding their areas shows the triangle's height equals 1 + 2 + 3 = 6. |
| 18 | B | Circles — Join the middle center to two neighboring outer centers: a right triangle with legs 1 + r and long side 2r. |
| 19 | C | Basic Probability — In every column the shaded squares are rows 1 and 3, or rows 2 and 4, and each of those row pairs comes up exactly half the time. |
| 20 | C | Basic Counting — Choose which 3 rows (10 ways) and which 3 columns (10 ways) the blocks use, then decide which row goes with which column (6 ways): 10 times 10 times 6. |
| 21 | B | Similar & Congruent Triangles — The small triangle above the square has the same shape as the big one, so compare width to height: s over 5 equals (height minus s) over height. |
| 22 | B | Expected Value — Of the 16 equal rolls, your number shows on both dice in 1, on exactly one die in 6, and on neither in 9. |
| 23 | E | Solid Geometry — The top piece is a shrunken copy of the pyramid. Half the area means every length, including the height, shrinks by a factor of the square root of 2. |
| 24 | C | Bases & Digits — A multiple of 4 must end in 44; a multiple of 9 needs the number of 4's to be a multiple of 9. Use nine 4's and one 9. |
| 25 | A | Diophantine Equations — Over the common bottom 9ab: since a and b share no factor, b divides 9 and a divides 14, and the 9 forces b to be a multiple of 3. |
2007 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.