Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks she helps around the house for 8, 11, 7, 12 and 10 hours. How many hours must she work for the final week to earn the tickets?
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Target total = 6 · 10 = 60. Subtract what she's already done.
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?
Chandler wants to buy a 500 dollar mountain bike. For his birthday, his grandparents send him 50 dollars, his aunt sends him 35 dollars and his cousin gives him 15 dollars. He earns 16 dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
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Subtract birthday total from 500, then divide by 16.
The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square?
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Column 4 already has a 4. Row 1 col 4 must be 3 (not 4, not 1 or 2 elsewhere). Then row 2 col 4 must be 1. So col 4 needs a 2 in row 4.
Show solution
Column 4 already contains 4 in row 3. Row 1's missing digits are 3 and 4 (col 2 and col 4), and col 4 can't take 4 ⇒ row 1 col 4 = 3.
Row 2 needs 1 and 4 in cols 3 and 4. Col 4 can't take 4 ⇒ row 2 col 4 = 1.
Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles A, B, C and D. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle C?
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Look for unique numbers that anchor placement. Tile III is the only tile with 0 and 5 ⇒ it must be on an edge of the arrangement and matched accordingly.
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Then match adjacent tiles by their shared edge numbers.
Show solution
Tile III is the only tile with 0 and 5. These numbers must be on the outer perimeter (no other tile has them). The arrangement forces III into rectangle D.
Tile IV's left edge (9) matches III's right edge (5)? No — check edges. By matching: IV's edges (top 2, right 1, bottom 6, left 9) match III's (top 7, right 5, bottom 0, left 1) where IV's left 9 ... actually the matching is best traced from LIVE: IV ends up in rectangle C.
A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?
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A regular hexagon of side 1 decomposes into 6 equilateral triangles of side 1 — the same triangles as the extensions.
Show solution
Hexagon = 6 equilateral triangles of side 1.
Extensions = 6 equilateral triangles of side 1 (one on each edge).
Sets A and B, shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.
Amanda draws five circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C, A), where C is its circumference and A is its area. Which of the following could be her graph?
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C = 2πr, A = πr2. Eliminating r: A = C2/(4π). Look for a graph that's an increasing concave-up curve.
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
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Yellow: 0.30 · 30 = 9 L. After adding 5: 14 L out of 35 L.
The product of the two 99-digit numbers 303,030,303,…,030,303 and 505,050,505,…,050,505 has thousands digit A and units digit B. What is the sum of A and B?
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The last 4 digits of each factor are 0303 and 0505. Multiply those mod 10000.
Show solution
303 · 505 = 153015.
Last 4 digits: 3015 ⇒ thousands digit A = 3, units digit B = 5.
Pick two consecutive positive integers whose sum is less than 100. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
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(x+1)2 − x2 = 2x + 1 = sum of the two integers. The sum < 100 and is odd.
Show solution
(x+1)2 − x2 = 2x + 1 = (x) + (x+1).
Difference equals the sum of the two integers — less than 100, and odd.
Before district play, the Unicorns had won 45% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
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Let pre-district games be x. Pre-district wins: 0.45x. Final wins: 0.45x + 6 = (x + 8)/2.
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0.45x + 6 = (x + 8)/2.
Multiply by 10: 4.5x + 60 = 5x + 40 ⇒ 0.5x = 20 ⇒ x = 40.
Two cards are dealt from a deck of four red cards labeled A, B, C, D and four green cards labeled A, B, C, D. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
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Fix the first card. Of the 7 remaining cards, count those that win against it: 3 of the same color and 1 of the same letter (different color).
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Same color: 3 of the remaining 7.
Same letter (different color): 1 of the remaining 7.
A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90° right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
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For any point inside a square of side 10, distance to opposite sides always sums to 10.
Show solution
Lemming stays inside the square.
Distance to left + right walls = 10. Distance to top + bottom walls = 10. Total: 20.
A bag contains four pieces of paper, each labeled with one of the digits 1, 2, 3, or 4, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3?
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Divisible by 3 iff digit-sum divisible by 3. The chosen 3 digits' sum is what matters; the order is irrelevant for divisibility.
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Subsets of size 3 from {1, 2, 3, 4}: 4 total ({1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}).
Digit sums: 6 ✓, 7, 8, 9 ✓ ⇒ 2 subsets give a multiple of 3.
On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has a radius of 3. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?
Show hint (soft nudge)
Score is odd iff exactly one dart hits a 1 and the other hits a 2.
Show hint (sharpest)
Find probability of hitting a 1 and probability of hitting a 2, then 2 · P(1) · P(2).
Show solution
Outer ring area: 36π − 9π = 27π. Each outer sector: 9π ⇒ prob 9π/36π = 1/4.
Inner sectors: 3π each ⇒ prob 1/12 each.
Inner has one 1 and two 2s. Outer has two 1s and one 2.