Inscribe the polygon in a 6-by-9 rectangle and subtract the missing notch.
Show hint (sharpest)
Notch is 2 wide × 4 tall.
Show solution
The shape fits inside a 6 × 9 = 54 rectangle. The notch removed is 2 × 4 = 8 (from AB length 6, FE = 6−4 = 2 wide, and AF = 5 minus the matching segment to the bottom run leaves a 4-tall step).
With V as the front face, fold the four neighbors of V into the four sides.
Show hint (sharpest)
Y sits across the strip from X, so they end up on opposite faces.
Show solution
Make V the front. U folds left, W folds right, X folds down to become the bottom, Z (below X) wraps around to become the back. That puts Y (the square attached to W's top) onto the top.
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.2 cm, 8.3 cm, and 9.5 cm. The area of the square is
Show hint
Triangle perimeter ÷ 4 = square side.
Show solution
Triangle perimeter = 6.2 + 8.3 + 9.5 = 24, so square side = 24 ⁄ 4 = 6.
If you walk for 45 minutes at a rate of 4 mph and then run for 30 minutes at a rate of 10 mph, how many miles will you have gone at the end of one hour and 15 minutes?
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Convert each leg's minutes into a fraction of an hour, then multiply by the speed.
If your average score on your first six mathematics tests was 84 and your average score on your first seven mathematics tests was 85, then your score on the seventh test was
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Convert each average to a sum, then subtract.
Show solution
Total of first 7 = 7 × 85 = 595. Total of first 6 = 6 × 84 = 504. Seventh test = 595 − 504.
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost?
Show hint (soft nudge)
9p < 10 and 10p > 11 give a narrow interval for p.
Show hint (sharpest)
10⁄9 ≈ 1.111 and 11⁄10 = 1.10.
Show solution
9p < 10 → p < 10⁄9 ≈ 1.111. 10p > 11 → p > 1.10. So 1.10 < p < 1.111.
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January 1 fall that year?
Show hint (soft nudge)
31 days = 4 weeks + 3 days. Three weekdays appear 5 times, the other four appear 4 times.
Show hint (sharpest)
For Tuesday and Saturday both to land at 4 times, neither can be among the first 3 days of the month.
Show solution
The three weekdays starting on Jan 1 each appear 5 times in a 31-day January. For Tue and Sat both to appear only 4 times, neither can be in those first three weekdays.
Only starting on Wednesday (Wed, Thu, Fri) leaves both Tue and Sat out of that group.
Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1. What fraction of telephone numbers begin with 9 and end with 0?
Show hint (soft nudge)
Count valid telephone numbers and those starting with 9 ending in 0, then take the ratio.
Show hint (sharpest)
Valid first digit: 8 choices. Last digit fixed: 1 choice. Middle five: 10 choices each.
Show solution
Total: 8 · 10⁶. Starting with 9 and ending in 0: 1 · 10⁵ · 1 = 10⁵.
King Middle School has 1200 students. Each pupil takes 5 classes a day. Each teacher teaches 4 classes. Each class has 30 students and 1 teacher. How many teachers are there at King Middle School?
Show hint (soft nudge)
Count student-class slots two ways.
Show hint (sharpest)
1200 students × 5 classes = total student-class slots; that also equals (# classes) × 30.
Show solution
Student-class slots: 1200 × 5 = 6000. Each class holds 30 students, so # classes = 6000 ⁄ 30 = 200.
Total of all 6 numbers 10 + 11 + ⋯ + 15 = 75. Adding the three side-sums gives each vertex twice and each midpoint once, so 3S = (vertex sum) + 75. Maximize by putting the three biggest at the vertices: 13 + 14 + 15 = 42.
3S = 42 + 75 = 117 → S = 39 (with midpoints 12, 10, 11 between the matching vertex pairs).
