Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
Show hint
Read off the rate from the graph: 1 mile per 5 minutes, so 30 min gives 30/5 miles.
A sequence of numbers starts with 1, 2, and 3. The fourth number of the sequence is the sum of the previous three numbers in the sequence: 1 + 2 + 3 = 6. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
Show hint
Just iterate the rule: each new term = sum of the previous three.
Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
Show hint
Both C and D sit on the railroad; CD = 3. The plot's base is CD with the apex at A.
Show solution
Base CD = 3 (along the railroad). Height from A to the railroad = 3 (along Main Street).
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
Show hint
Each interior polygon loses 2 sides to the chain (one to the previous, one to the next). The first and last lose only 1.
Show solution
End polygons (triangle, octagon) contribute all but 1: (3 − 1) + (8 − 1) = 2 + 7 = 9.
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil?
Show hint
Working in cents, the price divides both 143 and 195. Compute gcd(143, 195).
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
Show hint
Odd + even = odd, so all 9 sums are odd. Just count which are prime.
Show solution
Sums: 3, 5, 7, 5, 7, 9, 7, 9, 11. Non-prime: the two 9s.
Austin and Temple are 50 miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
Show hint
Average speed = total distance / total time. Don't just average the two speeds — equal distances at different speeds gives the harmonic mean.
Show solution
Time out: 50/60 = 5/6 hr. Time back: 50/40 = 5/4 hr. Total time: 5/6 + 5/4 = 25/12 hr.
Total distance: 100. Average: 100 / (25/12) = 48 mph.
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, 1/4 cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water, and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?
Show hint
For each ingredient, compute how many recipes worth Jordan has, then multiply by 5 servings. The smallest result limits the answer.
Show solution
Chocolate: 5/2 recipes worth.
Sugar: 2 / (1/4) = 8 recipes.
Milk: 7/4 recipes.
Smallest is milk at 7/4. Servings: 5 · 7/4 = 35/4 = 8¾.
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y?
Show hint
Prime-factor 360 = 23 · 32 · 5. Make every exponent even (square) or a multiple of 3 (cube).
Show solution
360 = 23 · 32 · 51.
Square: bump 2's exponent to 4 and 5's to 2 ⇒ multiply by 2 · 5 = 10. So x = 10.
Cube: bump 2's to 3 (already), 3's to 3 (add one 3), 5's to 3 (add two 5s) ⇒ multiply by 3 · 25 = 75. So y = 75.
The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot light tiles and dark tiles. Notice that the corners have dark tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many dark tiles will be needed?
Show hint
Dark tiles sit at (odd row, odd column). In a 7×7: 4 odd rows × 4 odd cols = 16. In a 15×15: 8 × 8.
Show solution
7-foot floor: odd positions 1, 3, 5, 7 ⇒ 4 each direction; 4 × 4 = 16 (consistent with diagram).
Andy and Bethany have a rectangular array of numbers with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is A. Bethany adds the numbers in each column. The average of her 75 sums is B. What is the value of AB?
Show hint
Both Andy and Bethany sum the same array total. So 40A = sum of array = 75B.
On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
Show hint
Each boy gets b beans, so all boys get b2. Similarly girls get g2. b2 + g2 = 394.
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is 12 foot from the top face. The second cut is 13 foot below the first cut, and the third cut is 117 foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
Show hint (soft nudge)
Look from each of the 6 directions and sum the visible areas.
Show hint (more)
Top and bottom views are each four 1×1 squares (the bases of the four pieces) = 4 sq ft each.
Show hint (sharpest)
End views show the full height of the tallest piece (1/2 sq ft each); front and back show the staircase silhouette whose total height adds to 1.
Show solution
Top view: each piece's 1×1 top is visible ⇒ 4 sq ft. Same for bottom ⇒ 4 sq ft.
Front and back views: silhouette is 4 rectangles of width 1 with heights summing to 1 ⇒ 1 sq ft each, total 2 sq ft.
End views: in the monotone-staircase arrangement, each end view equals the cross section of the tallest piece A: 1 · (1/2) = 1/2. Two ends ⇒ 1 sq ft.