A picture 3 feet across is hung in the center of a wall that is 19 feet wide. How many feet from the end of the wall is the nearest edge of the picture?
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The leftover wall is split evenly on both sides of the picture.
Show solution
Leftover wall = 19 − 3 = 16 feet, split into two equal margins.
The inner step has matching horizontal and vertical pieces that line up with the outer corners — so the perimeter equals that of the bounding rectangle.
Show hint (sharpest)
Bounding rectangle: 8 wide, 6 tall.
Show solution
Slide each piece of the inward step to the matching outer edge: the horizontal step segment fits onto the top edge, the vertical step segment onto the right edge.
Sale prices at the Ajax Outlet Store are 50% below original prices. On Saturdays an additional discount of 20% off the sale price is given. What is the Saturday price of a coat whose original price is $180?
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Apply each discount as a multiplier: 0.5 then 0.8.
Place the long 60 m side against the wall so the fence runs 36 + 60 + 36 = 132 m.
Show hint (sharpest)
Posts every 12 m on a 132 m path including both ends = 132⁄12 + 1.
Show solution
With the 60 m side along the wall, the fence has length 36 + 60 + 36 = 132 m. Posts every 12 m including both endpoints give 132⁄12 + 1 = 12 posts; the corners (at 36 m and 96 m) are multiples of 12, so no extras.
At the beginning of a trip, the mileage odometer read 56,200 miles. The driver filled the gas tank with 6 gallons of gasoline. During the trip, the driver filled his tank again with 12 gallons of gasoline when the odometer read 56,560. At the end of the trip, the driver filled his tank again with 20 gallons of gasoline. The odometer read 57,060. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?
Show hint (soft nudge)
The first 6-gallon fill-up only tops off the tank — gas used during the trip is what was refilled afterwards.
Show hint (sharpest)
Miles ÷ gallons used.
Show solution
Miles driven = 57,060 − 56,200 = 860. Gas used (the two refills) = 12 + 20 = 32.
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
Show hint (soft nudge)
Each statement says: A getting an A forces the next person to get one too.
Show hint (sharpest)
If Alan got an A, the chain would force at least three more A's; same for Beth.
Show solution
Alan's getting an A would force Beth, then Carlos, then Diana — 4 A's, too many. Beth's getting an A would force Carlos and Diana — 3 A's, still too many. So Alan and Beth don't get A's.
If Carlos gets an A, Diana must too — and that's exactly 2 A's: Carlos and Diana.
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately
Show hint (soft nudge)
Fix Al's group, then ask the chance Bob and Carol each land in the same one.
Show hint (sharpest)
Each independently lands in any group with chance ≈ 1⁄3.
Show solution
Whatever group Al is in, Bob lands there with chance ≈ 1⁄3 and Carol independently with chance ≈ 1⁄3.
