Maria has some blueberries and some cups. If she puts 9 blueberries into each cup, she will have two cups left empty. If she puts 6 blueberries into each cup, she will have three blueberries left over. How many cups and how many blueberries does Maria have in total?

Introduction / Context:
This puzzle is a nice example of how everyday situations can be translated into simple algebra. Maria has an unknown number of cups and an unknown number of blueberries. Two different filling rules are described, and in each scenario something is left over: either extra cups or extra blueberries. The question asks you to determine both quantities. This type of problem trains your ability to form equations from word statements, an essential skill for quantitative aptitude tests and school mathematics.

Given Data / Assumptions:

• Maria has a certain number of cups; call this number c.
• She has a certain number of blueberries; call this number b.
• If she puts 9 blueberries into each cup, exactly two cups remain empty.
• If she puts 6 blueberries into each cup, she has 3 blueberries left over after filling all cups.
• We assume all blueberries and cups are identical and that she fills cups fully following the rule in each scenario.

Concept / Approach:
To solve the puzzle, we convert each description into an equation. When Maria puts 9 blueberries into each cup and leaves two cups empty, she fills (c - 2) cups, each with 9 blueberries. That means b = 9 * (c - 2). In the second scenario, she puts 6 blueberries into each of the c cups, using 6 * c blueberries, and still has 3 blueberries left over, so b = 6 * c + 3. We now have two expressions for b in terms of c, which lets us create and solve a simple linear equation for c, and then compute b.

Step-by-Step Solution:
Step 1: Let c be the number of cups and b be the number of blueberries. Step 2: From the first scenario, write b = 9 * (c - 2), because only c - 2 cups are filled with 9 blueberries each. Step 3: From the second scenario, write b = 6 * c + 3, because every cup gets 6 blueberries and 3 remain unused. Step 4: Set the two expressions for b equal to each other: 9 * (c - 2) = 6 * c + 3. Step 5: Expand and simplify: 9c - 18 = 6c + 3, so 9c - 6c = 3 + 18, giving 3c = 21. Step 6: Solve for c: c = 21 / 3 = 7 cups. Step 7: Substitute c = 7 into b = 6 * c + 3: b = 6 * 7 + 3 = 42 + 3 = 45 blueberries. Step 8: Check with the first formula: b = 9 * (c - 2) = 9 * (7 - 2) = 9 * 5 = 45, which matches, confirming the solution.

Verification / Alternative check:
We can verify the answer using a direct reasoning check instead of algebra. Assume Maria has 7 cups and 45 blueberries. If she tries to put 9 blueberries in each cup, she needs 9 * 7 = 63 blueberries to fill all 7 cups. She only has 45, so she can fill 5 cups fully, using 45 blueberries, and 2 cups stay empty, matching the first condition. If she puts 6 blueberries in each of 7 cups, she uses 6 * 7 = 42 blueberries and has 45 - 42 = 3 blueberries left, which matches the second condition. Since both conditions are satisfied, the solution is consistent and unique.

Why Other Options Are Wrong:
5 cups and 45 blueberries: With 5 cups, leaving 2 cups empty would not make sense, and the second scenario calculation fails.
7 cups and 39 blueberries: 39 cannot satisfy both the 9-per-cup and 6-per-cup descriptions simultaneously.
9 cups and 45 blueberries: The counts do not fit both conditions; too many cups compared to berries.
9 cups and 57 blueberries: This also fails when you test each scenario step by step; it does not satisfy both conditions at the same time.

Common Pitfalls:
A common error is to misinterpret "two cups left" as "two blueberries left" or to forget that two different scenarios must be true for the same values of cups and blueberries. Another frequent mistake is to create incorrect equations, such as b = 9 * c + 2, which ignores the wording about empty cups. Carefully translating each sentence into a mathematical expression is crucial. Students should also remember to verify proposed answers with both scenarios instead of trusting algebraic manipulation alone, as a miswritten equation can lead to consistent but wrong numbers.

Final Answer:
Maria has 7 cups and 45 blueberries in total.