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2014 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2014 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Fractions & Decimals — Add the three fractions first to get 4/5. The power -1 means flip that single fraction to 5/4, and only then multiply by 10. |
| 2 | C | Linear Equations & Word Problems — One day uses 7/12 of a can. Ten days use 70/12 of the 72/12 in the box, so the last bit is eaten on the 11th morning. |
| 3 | E | Linear Equations & Word Problems — Follow the loaves in three groups, 24, 16 and 8, to get 88 dollars of sales. Then subtract what all 48 loaves cost to make, 36 dollars. |
| 4 | B | Arrangements with Restrictions — Place blue and yellow first, since they have the most rules. Only three spot pairs keep blue first with a gap; orange and red then fit in exactly one way. |
| 5 | C | Statistics & Data — Pretend there are 100 students. The mean is 87. Counting up from the lowest score, the 50th student falls in the group that scored 90, so the median is 90. |
| 6 | A | Ratios, Percents & Averages — First find how much one cow gives in one day, b/(ac) gallons. Then multiply by the number of cows d and the number of days e. |
| 7 | B | Absolute Value & Inequalities — Adding two inequalities that point the same way is always safe. Subtracting, multiplying and dividing are not, and one example with negative numbers breaks them. |
| 8 | D | Number Properties — The two factorials share a big common part, so each choice is a perfect square times (n+1)/2. Only 18/2 = 9 is a perfect square. |
| 9 | C | Triangles: Area & Pythagorean — The missing altitude is the one drawn to the longest side. Find the area using the two legs, then use that same area with the longest side as the base. |
| 10 | B | Ratios, Percents & Averages — The average of five consecutive whole numbers is the middle one. So b is a + 2, and the second list averages b + 2, which is a + 4. |
| 11 | C | Ratios, Percents & Averages — Write each coupon's savings for a price P. Beating the 20-dollar coupon needs P above 200; beating the 18 percent coupon needs P below 225. Only 219.95 fits. |
| 12 | C | Circles — Each sector opens as wide as the hexagon's corner, 120 degrees, which is one third of a circle. Six thirds make two whole circles of radius 3. |
| 13 | C | Quadrilaterals & Polygon Areas — Cut the hexagon into the triangle, the three squares, and three thin gap triangles. Each gap triangle has two sides of length 1 with a 120-degree angle between them. |
| 14 | D | Coordinate Geometry — Find the two slopes: one is m, the other is -1/m. Making the two y-intercepts add to zero gives m = 3, so the intercepts are 10 and -10. |
| 15 | C | Linear Equations & Word Problems — Look only at the distance left after the first hour. Driving it at 35 instead of 50 miles per hour takes exactly one and a half hours longer. |
| 16 | E | Quadrilaterals & Polygon Areas — Put the rectangle on a grid. The shaded shape is a kite with a tall diagonal of length 1 and a crossing diagonal of length 1/3. |
| 17 | D | Basic Probability — Choose which die shows the total (3 ways). The other two dice must add up to at most 6, and there are 15 ordered ways to do that. |
| 18 | B | Coordinate Geometry — Opposite corners of a square have heights adding to the same total, so 0 and 5 are opposite. Each side is the long side of a 1-by-4 right triangle. |
| 19 | A | Solid Geometry — XY runs corner to corner through the stack and has length 2 root 33. The edge-3 cube covers heights 4 to 7, which is 3/10 of the drop. |
| 20 | D | Bases & Digits — Work out 8 times 88, 888, 8888. The answer is always a 7, then a run of ones, then 04, so the digit sum is k + 9. |
| 21 | E | Diophantine Equations — Setting the two x-intercepts equal gives a times b = 15. So a is 1, 3, 5 or 15, and the crossing points are -5, -5/3, -1 and -1/3. |
| 22 | E | Triangles: Area & Pythagorean — Draw a 30-degree line from B; BE cuts that angle in half. The angle bisector rule gives CE, then DE is 10 root 3 and AE is 20. |
| 23 | C | Transformations & Symmetry — The fold line is tilted at 60 degrees, so the folded corner lands exactly on the opposite corner. The doubled part is an equilateral triangle, one third of the paper. |
| 24 | A | Sequences & Series — The 500,000th listed number is 500,000 plus how many numbers were skipped before it. Up to that point, 1 + 2 + ... + 996 = 496,506 numbers were skipped. |
| 25 | B | Exponents, Logarithms & Radicals — Since 5 is between 4 and 8, each gap between neighbouring powers of 5 holds two or three powers of 2. Share 2013 powers of 2 among 867 gaps. |
2014 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.