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2007 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2007 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Ratios, Percents & Averages — Turn each discount into a price per ticket first: 25% off 20 dollars is 15 dollars, and 30% off is 14 dollars. Then multiply by the number of tickets. |
| 2 | A | Functions — The strange symbols are just recipes. Put 6 in place of a and 2 in place of b in each recipe, work out the two numbers, then divide. |
| 3 | D | Solid Geometry — The brick sits fully under water and pushes up exactly its own volume of water. Spread that volume over the tank's floor to get the rise. |
| 4 | A | Linear Equations & Word Problems — Two odd numbers in a row are 2 apart. If the bigger one is three times the smaller, the difference of 2 must equal twice the smaller. |
| 5 | B | Systems of Equations — Work in cents. The two purchases give two equations for the pencil price and the notebook price; solve them, then price the new order. |
| 6 | A | Ratios, Percents & Averages — A percent increase compares the gain to the starting number. From 60 to 66 is exactly 10%, and every other year's gain is less than 10% of its start. |
| 7 | D | Ratios, Percents & Averages — The state tax is 10% of what is left, not of the whole. So the taxes are 20% plus 8%, or 28% of the inheritance. |
| 8 | D | Angles & Polygons — Both isosceles triangles have AC as the base. Find the base angle at A in each one (70 and 20 degrees); angle BAD is the gap between them. |
| 9 | E | Exponents, Logarithms & Radicals — Rewrite 81 as 3 to the 4th and 125 as 5 to the 3rd. Then each equation says two exponents are equal, giving two simple equations in a and b. |
| 10 | E | Ratios, Percents & Averages — Turn averages into totals. The family totals 20 times its size, the mother and children 16 times theirs; the gap is the father's 48. |
| 11 | C | Sets, Estimation & Miscellaneous — A face and its opposite face use each of the 8 corners once. Their sums add to 36, the total of 1 through 8, so each face sums to 18. |
| 12 | D | Basic Counting — Each tourist has 2 choices, giving 64 ways with no rule. Only 2 of them leave a guide with nobody, so the answer is 62. |
| 13 | B | Linear Equations & Word Problems — Set the walking speed to 1 and biking to 7. Walking straight takes as long as walking home plus biking the whole way from home to the stadium. |
| 14 | A | Circles — A 3-4-5 triangle has a right angle, so its longest side is a diameter of the circle, which is 6. Scale the 3-4-5 triangle up by 6/5. |
| 15 | B | Circles — Walk along a diagonal from the center to a corner: 2 + 1 = 3 to a small circle's center, then the diagonal of a 1-by-1 square to the corner. |
| 16 | E | Basic Probability — Only odd and even matter. A product of two random numbers is odd just 1/4 of the time, and ad - bc is even when both products match. |
| 17 | D | Number Properties — 75 is 3 times 5 times 5. To make a perfect cube, m must add two more 3s and one more 5, so m = 45 and n = 15. |
| 18 | C | Quadrilaterals & Polygon Areas — Put the shape on a grid. Then ABCM is triangle ABG with triangle CMG cut off, and M's position comes from where the two lines cross. |
| 19 | C | Quadrilaterals & Polygon Areas — Measure the unpainted half: four equal right triangles against the sides. What remains of each side is two short painted pieces set by the brush width. |
| 20 | D | Algebraic Manipulation — Square a + 1/a. Since a times 1/a is 1, the middle term is just 2. Square twice: 16 - 2 = 14, then 196 - 2 = 194. |
| 21 | C | Solid Geometry — One length is passed along: the outer cube's edge equals the sphere's diameter, which equals the long diagonal through the inner cube, corner to opposite corner. |
| 22 | D | Bases & Digits — The digits wrap around, so each digit appears once as a hundreds, once as a tens and once as a ones digit. The sum is 111 times the digit total. |
| 23 | B | Diophantine Equations — Write the difference of squares as (m + n) times (m - n) = 96. These two factors must both be even, which leaves only four factor pairs. |
| 24 | B | Circles — Radii to E and F make a rectangle. The shaded part is that rectangle minus two right triangles with legs 2 and two 45-degree circle slices. |
| 25 | D | Bases & Digits — Digit sums are small, so n must be between 1969 and 2007. A remainder argument with 9 shows n is a multiple of 3, leaving 13 numbers to test. |
2007 AMC 10A — 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.