Solving Math Puzzles: Operations And Equations

Welcome to a fun dive into the world of mathematics, where we'll tackle some intriguing problems involving various operations and equations. These types of questions are great for sharpening your logical thinking and problem-solving skills. We'll break down each problem step-by-step, making sure you understand the logic behind each solution. So, grab your thinking cap, and let's get started!

Problem 3: Understanding the Difference of Squares

Our first mathematical puzzle introduces a special notation: $6 \% y = (x+y)(x-y)$. The question asks us to find the value of $8 \% 5$. The notation $x \% y$ here represents a specific operation defined by the formula $(x+y)(x-y)$. This formula is famously known as the difference of squares identity, which states that the product of the sum and difference of two numbers is equal to the difference of their squares: $(x+y)(x-y) = x^2 - y^2$. Therefore, when we see $x \% y$, we can directly interpret it as $x^2 - y^2$. Now, let's apply this to find $8 \% 5$. Here, $x=8$ and $y=5$. Using our derived formula, we substitute these values: $8^2 - 5^2$. First, calculate the squares: $8^2 = 64$ and $5^2 = 25$. Finally, subtract the second square from the first: $64 - 25 = 39$. So, $8 \% 5 = 39$. This problem highlights how recognizing a known mathematical identity can simplify complex-looking expressions. It's a common strategy in mathematics to look for familiar patterns, as they often provide a shortcut to the solution.

Problem 4: A Unique Division Operation

Next, we encounter a different kind of operation defined as $m \& n = \frac{m+n}{n-m}$. We need to find the value of $8 \& 7$. In this definition, the symbol {{content}}amp; $ signifies a particular way of combining two numbers, $m$ and $n$. The rule is to add the two numbers ($m+n$) and then divide the sum by the result of subtracting the second number from the first ($n-m$). Let's apply this rule to $8 \& 7$. Here, $m=8$ and $n=7$. Following the defined operation, we first calculate the sum of $m$ and $n$: $8 + 7 = 15$. Next, we calculate the difference $n-m$: $7 - 8 = -1$. Now, we divide the sum by the difference: $\frac{15}{-1} = -15$. Therefore, $8 \& 7 = -15$. It's important to pay close attention to the order of operations and the specific formula provided. In this case, the denominator ($n-m$) is crucial, and a mistake in its calculation, especially with the signs, would lead to an incorrect answer. This problem emphasizes the importance of precise substitution and adherence to the given definition.

Problem 5: A Simple Subtraction and Division Mix

Let's move on to problem 5, where the operation is defined as $a * b = \frac{b-a}{2}$. We are asked to find the value of $6+8$. The symbol $*$ in this context represents a distinct mathematical operation. The rule states that to find $a * b$, you should subtract the first number ($a$) from the second number ($b$) and then divide the result by 2. Now, let's apply this to $6+8$. It seems there might be a slight typo in the problem description, as the operation is defined using $a * b$, but the question asks to find $6+8$. Assuming the question meant to ask for $6 * 8$ using the defined operation, we proceed as follows. Here, $a=6$ and $b=8$. According to the rule, we first calculate the difference $b-a$: $8 - 6 = 2$. Then, we divide this difference by 2: $\frac{2}{2} = 1$. So, if the question intended to be $6 * 8$, the answer would be 1. If the question literally meant to find $6+8$ using this operation definition, it's not directly applicable as the operation is defined for $*$, not $+$. However, given the context of the other problems, it's highly probable that $6+8$ was meant to be $6 * 8$. This scenario highlights the need for careful checking of problem statements to ensure all symbols and operations are correctly interpreted.

Problem 6: A Custom Operator with Exponents

Problem 6 introduces another new symbol: $o < i < p = o^2 + (p-o)$. We need to find the value of $5 < 9$. The symbol $<$ here is not the usual less-than sign but a custom operator that takes three inputs, though the definition only explicitly uses two of them in a way that implies the middle input might not be directly used in the calculation itself, or perhaps the notation $o < i < p$ should be interpreted as applying the operation sequentially or in a specific grouping. However, the formula provided is $o^2 + (p-o)$, which only involves the first and the last variables ($o$ and $p$). Let's assume the definition implies that for $o < i < p$, the calculation is $o^2 + (p-o)$, where $o$ is the first number and $p$ is the last. In our case, we need to find $5 < 9$. This notation is slightly ambiguous as it's written with only two numbers when the definition seems to imply three ($o < i < p$). If we interpret $5 < 9$ as meaning $o=5$ and $p=9$, and the middle variable $i$ is either irrelevant or implicitly handled, then we substitute these values into the formula $o^2 + (p-o)$. So, $o=5$ and $p=9$. The calculation becomes $5^2 + (9-5)$. First, calculate the square: $5^2 = 25$. Next, calculate the term in the parentheses: $9-5 = 4$. Finally, add the results: $25 + 4 = 29$. Thus, $5 < 9 = 29$. This problem tests our ability to adapt to new definitions and handle potentially ambiguous notation, focusing on the explicit mathematical operations provided.

Problem 7: A Three-Variable Operation

Now we face problem 7 with the definition x # y # z = x^3 + 5y - z. We are asked to find 3 # 5 # 2. This is a clear definition involving three variables. The operator is represented by the symbol $\#$. The rule states that for any three numbers $x, y, z$, the operation $x \# y \# z$ is calculated by cubing the first number ($x^3$), adding five times the second number ($5y$), and then subtracting the third number ($z$). Let's apply this to the specific values given: $x=3$, $y=5$, and $z=2$. First, we calculate $x^3$: $3^3 = 3 \times 3 \times 3 = 27$. Next, we calculate $5y$: $5 \times 5 = 25$. Finally, we subtract $z$: $2$. Now, we combine these results according to the formula: $x^3 + 5y - z = 27 + 25 - 2$. Performing the addition first: $27 + 25 = 52$. Then, performing the subtraction: $52 - 2 = 50$. Therefore, $3 \# 5 \# 2 = 50$. This problem is straightforward once the definition is understood. It involves basic arithmetic operations: exponentiation, multiplication, addition, and subtraction, applied in a specific order based on the given formula.

Problem 8: An Extended Three-Variable Operation

Finally, we have problem 8, defined as $a$. The question asks us to find the value of $10$. Similar to the previous problem, this definition involves three variables: $a, b,$ and $c$, combined using the operator $% $. The formula indicates that we should first multiply the first two numbers ($a \cdot b$) and then subtract the third number ($c$) from the product. It's important to note that the symbol $\%$ is used here in a completely different context than in Problem 3. Always rely on the specific definition provided for each problem. For $10 \% 2 \% 3$, we have $a=10$, $b=2$, and $c=3$. Following the rule, we first compute the product of $a$ and $b$: $10 \cdot 2 = 20$. Then, we subtract $c$ from this product: $20 - 3 = 17$. So, $10 \% 2 \% 3 = 17$. This problem reinforces the concept that mathematical symbols can have different meanings depending on the context and the definitions provided within a specific problem set. Careful reading and application of the given rules are key to solving these types of mathematical puzzles.

Conclusion

We've successfully navigated through a series of mathematical puzzles, each presenting unique definitions and operations. These exercises are invaluable for developing analytical skills and reinforcing fundamental mathematical concepts. Remember, the key to solving such problems lies in carefully reading the definitions, accurately substituting the given values, and meticulously following the order of operations. Practice is essential, and the more you engage with these types of problems, the quicker and more confident you will become in deciphering and solving them.

For further exploration into mathematical operations and problem-solving techniques, you can visit Khan Academy or MathWorld.