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2003 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2003 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Algebraic Manipulation — Each number on top is 2 times a counting number and the number right under it is 3 times the same counting number, so the fraction is 2/3. |
| 2 | D | Linear Equations & Word Problems — Divide by the 14 days first: one green plus one pink costs 39 dollars. Two prices that add to 39 and differ by 1 are 20 and 19. |
| 3 | B | Sequences & Series — The first 8 odd numbers add to 64, so the five even numbers add to 60. Their middle one is the average, 12, so the smallest is 8. |
| 4 | A | Quadrilaterals & Polygon Areas — Find the five areas from the figure (21, 20, 15, 6, 4). To spend the least, put the cheapest flower in the biggest region and the priciest in the smallest. |
| 5 | C | Linear Equations & Word Problems — Because of the 4-inch overlap, each pass mows only 24 inches = 2 feet of new grass. Walking 5000 feet an hour then clears 10000 square feet an hour. |
| 6 | D | Triangles: Area & Pythagorean — A screen 4 wide and 3 tall has a diagonal of 5, so the width is always 4/5 of the diagonal: 4/5 of 27 is 21.6. |
| 7 | B | Number Properties — The rounded-down square root stays the same between perfect squares: it is 1 three times, 2 five times, 3 seven times, then 4 once at 16. |
| 8 | B | Sequences & Series — From the 2nd term to the 4th you multiply by the ratio twice, so the ratio squared is 3. The ratio may be negative. |
| 9 | B | Exponents, Logarithms & Radicals — Write every number as a power of 5. Then the exponents must match: -4 = -12/x, so x = 3. |
| 10 | C | Basic Counting — Old plates: 26 times 10 four times. New plates: 26 three times times 10 three times. Divide new by old and cancel: 26 squared over 10. |
| 11 | A | Coordinate Geometry — Both lines drop 15 units to reach the x-axis: slope 3 needs 5 steps left, slope 5 needs 3 steps left. The crossings are 5 and 7. |
| 12 | C | Linear Equations & Word Problems — Treat Betty and Clare as one team whose money doubles. If Al starts with a, the team has 1000 - a, and the total 1500 gives a = 400. |
| 13 | E | Bases & Digits — A two-digit number's digits add to at most 18, and only 3 and 12 in that range have digit sum 3. Count both cases. |
| 14 | D | Exponents, Logarithms & Radicals — The exponent b must go evenly into both 8 and 2, so b is at most 2. Then a = 405 and a + b = 407. |
| 15 | E | Games & Processes — Every match knocks out exactly one player, and 99 players must be knocked out to leave one winner. So there are 99 matches, and 99 = 9 times 11. |
| 16 | E | Basic Counting — There are 6 times m times m dinners, which must reach 365. Seven main courses give 294, too few; eight give 384. |
| 17 | B | Solid Geometry — The melted ice cream is 3/4 of the sphere's volume, which simplifies to pi times r cubed. Setting that equal to the cone's volume gives height = 3 times radius. |
| 18 | D | Divisibility & Factors — Five odd numbers in a row always include a multiple of 3 and a multiple of 5, but not always a multiple of 11. |
| 19 | E | Circles — Neighboring small semicircles cross at a point 1 away from both centers, forming equilateral triangles. The white part is two triangles plus slices totaling 5/6 of a circle. |
| 20 | D | Similar & Congruent Triangles — FG = 2 is parallel to AB = 5, so the top triangle is a 2/5 copy of triangle AEB. Their heights differ by 3, so the height is 5. |
| 21 | C | Conditional Probability & States — All beads end red exactly when two of the three draws are green. Remember the bag holds fewer greens after each green draw. |
| 22 | B | Clocks, Calendars & Time — A full day has 180 chimes and the rest of February 26 has 91. The other 1912 chimes are 10 full days plus 112: March 9. |
| 23 | D | Quadrilaterals & Polygon Areas — Cut the octagon into 8 equal triangles from its center. The rectangle is made of 4 triangles, each equal in area to one of the 8. |
| 24 | E | Sequences & Series — Each step adds -2y, so the 3rd and 4th terms are x - 3y and x - 5y. Dividing xy = x - 3y by y gives y = -3/5. |
| 25 | B | Modular Arithmetic — The 90 numbers 1023, 1123, ..., 9923 go up by 100. Each step changes the remainder after dividing by 3, so every third number works. |
2003 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.