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2014 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2014 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Linear Equations & Word Problems — One more nickel would make the two piles equal, so right now there is one more penny than nickel: 7 pennies and 6 nickels. |
| 2 | E | Exponents, Logarithms & Radicals — The top is 8 + 8 = 16 and the bottom is 1/8 + 1/8 = 1/4; dividing by 1/4 is multiplying by 4. |
| 3 | E | Ratios, Percents & Averages — Gravel and dirt together are 1/3 + 1/5 = 8/15 of the trip, so the 20 paved miles are the remaining 7/15 of it. |
| 4 | B | Linear Equations & Word Problems — Calvin's bill is twice Susie's, so 2 muffins and 16 bananas cost the same as 8 muffins and 6 bananas; cancel what they share. |
| 5 | A | Linear Equations & Word Problems — Call the panes 2x wide and 5x tall. Across there are 4 panes and 5 borders, down there are 2 panes and 3 borders; the two totals must match. |
| 6 | C | Ratios, Percents & Averages — Measure money in regular balloon prices. On sale, a pair costs 1 + 2/3 = 5/3 prices, so 30 prices buy 18 pairs, which is 36 balloons. |
| 7 | A | Ratios, Percents & Averages — Percent greater is measured against the number you start from, B: find how much bigger A is (A - B), divide by B, and multiply by 100. |
| 8 | E | Linear Equations & Word Problems — Two conversions are needed: 3 minutes is 180 seconds, and every 3 feet is one yard. Find feet per second first, then scale. |
| 9 | A | Algebraic Manipulation — Multiply top and bottom by wz: the given fraction becomes (z + w)/(z - w), which is the asked fraction with its sign flipped. |
| 10 | C | Bases & Digits — Columns where the answer digit repeats an added digit force C = 0 and no carrying, so D = A + B for different nonzero digits A and B. |
| 11 | C | Ratios, Percents & Averages — Discounts one after another multiply what is left of the price. The three deals take off 27.75%, 27.1% and 28.75%, so n must be more than 28.75. |
| 12 | C | Divisibility & Factors — Divisors come in pairs that multiply to N, so the fifth largest divisor is N divided by the fifth smallest. The smallest divisors are 1, 2, 4, 5, 8. |
| 13 | B | Triangles: Area & Pythagorean — Measure from the center of the middle hexagon: each corner of triangle ABC is 2 units away, and the three corners are evenly spaced, so the triangle is equilateral. |
| 14 | D | Bases & Digits — Reversing a 3-digit number changes it by 99 times (c - a). That must equal 55 times the hours, forcing c - a = 5. |
| 15 | A | Triangles: Area & Pythagorean — Splitting the right angle at D into three equal parts gives 30 and 60 degree angles, so the 30-60-90 triangles fix where E and F land. |
| 16 | B | Basic Probability — Count two separate cases: all four dice match (6 ways), and exactly three match (6 values, 4 spots for the odd die, 5 values for it). |
| 17 | D | Divisibility & Factors — Pull out the obvious 2^1002. Then count the 2s hiding in 5^1002 - 1 by splitting it into (5^501 - 1)(5^501 + 1). |
| 18 | E | Statistics & Data — To make the biggest number large, make the other ten as small as possible. Using only three 8s lets two 1s into the low spots. |
| 19 | D | Geometric Probability — Fix the first point. The chords from it that just touch the inner circle end 120 degrees away on each side, leaving a 120-degree arc that works. |
| 20 | C | Quadratics — Treat x^2 as a single number. The expression factors as (x^2 - 1)(x^2 - 50), which is negative only when x^2 is between 1 and 50. |
| 21 | B | Quadrilaterals & Polygon Areas — Drop both heights. The two end pieces of the long side add to 12; the Pythagorean theorem on each side gives pieces 2 and 10, height squared 96. |
| 22 | B | Circles — The middle circle sits at the square's center. Its radius plus 1/2 equals the distance from the center to a semicircle's center, which is half of root 5. |
| 23 | E | Solid Geometry — Slice through the middle: a circle inside a trapezoid. Equal touching segments make the slanted side R + r, so sphere radius squared equals Rr. |
| 24 | B | Arrangements with Restrictions — What is left after removing a run of neighbors is also a run, and the two sums add to 15. So only sums 6 and 7 can fail. |
| 25 | C | Conditional Probability & States — Look at the gaps between escape chances on neighboring pads: each gap is the previous one times N/(10 - N), and all gaps add up to 1. |
2014 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.