Question

If x = 5, y = 6, z = 4, and w = 3.5, evaluate each of the following statements, if possible. If it is not possible, state the reason. a. (x + z) % y b. (x + y) % w c. (y + w) % x d. (x + y) *w e. (x % y) % z f. (y % z) % x g. (x *z) % y h. ((x *y) *w) *z

Short answer

a) 3, b) Not possible, c) Not possible, d) 38.5, e) 1, f) 2, g) 2, h) 420

Step-by-step solution

Evaluate (x + z) % y

First, calculate the sum of \( x \) and \( z \), which is \( 5 + 4 = 9 \). Then, find the remainder when 9 is divided by \( y = 6 \). The remainder is \( 9 \, \text{mod} \, 6 = 3 \).

Evaluate (x + y) % w

Calculate the sum of \( x \) and \( y \), which is \( 5 + 6 = 11 \). Then, find the remainder when 11 is divided by \( w = 3.5 \). As remainder operations usually require integer divisors, this operation is not possible using standard remainder calculations with non-integer values.

Evaluate (y + w) % x

Calculate the sum of \( y \) and \( w \), which is \( 6 + 3.5 = 9.5 \). Then, find the remainder when 9.5 is divided by \( x = 5 \). This calculation is usually not possible in standard integer remainder operations with floating-point arithmetic.

Evaluate (x + y) * w

Calculate the sum of \( x \) and \( y \), which is \( 5 + 6 = 11 \). Then multiply by \( w = 3.5 \), giving \( 11 \times 3.5 = 38.5 \).

Evaluate (x % y) % z

First, find the remainder when \( x \) is divided by \( y \): \( 5 \, \text{mod} \, 6 = 5 \). Then find the remainder when 5 is divided by \( z = 4 \), resulting in \( 5 \, \text{mod} \, 4 = 1 \).

Evaluate (y % z) % x

First, find the remainder when \( y \) is divided by \( z \): \( 6 \, \text{mod} \, 4 = 2 \). Then find the remainder when 2 is divided by \( x = 5 \), resulting in \( 2 \, \text{mod} \, 5 = 2 \).

Evaluate (x * z) % y

Calculate the product of \( x \) and \( z \), which is \( 5 \times 4 = 20 \). Then find the remainder when 20 is divided by \( y = 6 \), resulting in \( 20 \, \text{mod} \, 6 = 2 \).

Evaluate ((x * y) * w) * z

Calculate the product of \( x \) and \( y \), which is \( 5 \times 6 = 30 \). Multiply the result by \( w \): \( 30 \times 3.5 = 105 \). Finally, multiply by \( z \): \( 105 \times 4 = 420 \).

Key concepts

Integer Arithmetic

Integer arithmetic consists of operations such as addition, subtraction, multiplication, and modulus that are performed on whole numbers. These operations are essential in programming for manipulating and analyzing data.
When we conduct integer arithmetic in C++, the operations follow standard mathematical principles:

• **Addition:** Adds two integer values.
• **Subtraction:** Subtracts one integer from another.
• **Multiplication:** Multiplies two integers.
• **Modulus (%):** Returns the remainder of a division of one number by another. This operation is useful for cyclic or repeating patterns, like clock arithmetic.

For example, in the expression \[(x + z) \% y\]if \(x = 5\) and \(z = 4\), we calculate:\[5 + 4 = 9\]and find \[9 \% 6 = 3\].Thus, 3 is the remainder. Each part of the operation deals solely with integers, making it straightforward and predictable.

Floating-Point Arithmetic

Floating-point arithmetic allows for operations on real numbers, which can include decimals. This is crucial for calculations that require precision beyond integers.
In C++, floating-point arithmetic is conducted with data types like `float` and `double`. However, modulus operations typically don't accommodate floating-point numbers as they rely on whole number divisors.
For instance, in the C++ expression\[(x + y) \% w\]where \(x = 5\), \(y = 6\), and \(w = 3.5\), finding the modulus of floating-point values isn't straightforward and often isn't supported natively. This is because remainder operations require the divisor to be an integer for a precise result. Always be cautious of potential pitfalls when using these operations in programming scenarios.

Step-by-Step Evaluation

A step-by-step evaluation process is essential for breaking down complex expressions into manageable parts. This approach helps in understanding exactly how results are derived, particularly in programming and algebraic expressions.
Let's consider how a step-by-step breakdown is applied in the expression \((x + y) \times w\),where \(x = 5\), \(y = 6\), and \(w = 3.5\).

• **Step 1:** Compute the sum \(x + y = 5 + 6 = 11\).
• **Step 2:** Multiply the result by \(w\):
  \[11 \times 3.5 = 38.5\].

By tackling each operation individually, you ensure clarity and accuracy. This method is vital for both learning and debugging complex codes.

Programming Concepts

Several programming concepts can be understood through these arithmetic operations. Fundamental concepts like data types, operator precedence, and error handling play crucial roles in programming execution.

• **Data Types:** Understanding the choice between integers (`int`) and floating-point numbers (`float`, `double`) affects how operations are performed and what results are expected.
• **Operator Precedence:** Knowing the hierarchy of operations (which are executed first) helps in predicting outcomes correctly. For instance, multiplication and division are performed before addition and subtraction unless parentheses dictate otherwise.
• **Error Handling:** Being aware of potential errors, such as attempting invalid modulus operations with floating-point divisors, is crucial in developing robust programs.

Given these concepts, effective programming requires comprehension of how a language like C++ interprets and executes arithmetic operations. Mastering these skills equips you to write efficient and accurate code.