Lower Elementary:
Question: Gabriella is selling lemonade. On Monday, she sold 11 glasses of lemonade. On Tuesday, she sold twice as many glasses as she did on Monday. On Wednesday, she sold twice as many glasses as she did on Tuesday. How many total glasses of lemonade did Gabriella sell from Monday to Wednesday?
Answer: 77 glasses of lemonade
Solution: On Monday, Gabriella sold 11 glasses. On Tuesday, she sold twice as many glasses of lemonade as she did on Monday. 11 doubled = 10 doubled + 1 doubled = 20 + 2 = 22. She sold 22 glasses of lemonade on Tuesday. On Wednesday, she sold twice as many glasses as she did on Tuesday. 22 doubled = 20 doubled + 2 doubled = 40 + 4 = 44. She sold 44 glasses of lemonade on Wednesday. To find the total number of glasses of lemonade sold, add the number sold on each day. 11 + 22 + 44 = 77. Gabriella sold a total of 77 glasses of lemonade.
Upper Elementary:
Question: A book store is having a buy two, get one free sale. If you got a total of 45 books, how many of those books were free?
Answer: 15 books
Solution: Fractional Parts Method:
For every 2 books you buy, 1 book is free. This means that for every 3 books, 2 are bought. So 2/3 of the books are bought. Similarly, for every 3 books, 1 is free. So 1/3 of the books are free. Find 1/3 of the total books received to find out how many books were free. 1/3 of 45 is 15. You received 15 books for free.
Counting Method:
For every 2 books you buy, 1 book is free. 2 + 1 = 3. This is a total of 3 books. If we buy 4 books, we get 2 for free. 4 + 2 = 6. Let’s continue the pattern:
2 buy + 1 free = 3 books
4 buy + 2 free = 6 books
6 buy + 3 free = 9 books
8 buy + 4 free = 12 books
Notice that the buy books go up by 2, the free books go up by 1, and the total books go up by 3 each time. Continuing this pattern, you will get to 30 books bought, 15 books free, and 45 books total. Thus, if you get 45 books you received 15 books for free.
Middle School:
Question: The sum of 3 consecutive even integers is 72. What are the 3 digits?
Answer: 22, 24, 26
Solution: One way to solve this problem is to set up an equation. Let n be the first even integer. Since every even integer is 2 numbers apart, that means the next even integer is n + 2. The next even integer after n + 2 would be n + 4. So if we add these 3 values together, set it equal to 72, and solve for n, we will find the first even integer.
n + (n + 2) + (n + 4) = 72
Combine the like terms.
3n + 6 = 72
Subtract 6 from both sides.
3n = 66
Divide by 3.
n = 22
The first even integer is 22. That means the next even integer is 24, and the next even integer is 26. So the 3 consecutive even integers that add to 72 are 22, 24, and 26.
Algebra and Up:
Question: You roll two 6-sided dice and take the product of the numbers. What is the probability that the product is a prime number?
Answer: 1/6
Solution: There are 6 outcomes for the first die and 6 outcomes for the second die. Therefore, there are 6 x 6 = 36 total combinations of rolls. A prime number is a number where its only factors are 1 and itself. Since the product of any number other than 1 would make a composite number, the only values out of the 36 possible combinations we need to consider are the rolls where at least one die has a value of 1: 1×1, 1×2, 1×3, 1×4, 1×5, 1×6, 2×1, 3×1, 4×1, 5×1, and 6×1. The values of these products are 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, and 6, respectively. Of this list, 2, 3, and 5 are prime and there are 2 of each. Thus, we have 6 total combinations where the product is a prime number: 1×2, 1×3, 1×5, 2×1, 3×1, and 5×1. Out of the 36 combinations, 6 of them yield prime numbers. Thus, the probability of rolling 2 dice and having the product be a prime number is 6/36 = 1/6.
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