Question

Evaluate each expression. \(\frac{7+6-2 \cdot 6}{11-5 \cdot 2}\)

Short answer

1

Step-by-step solution

Simplify the Numerator

Start by simplifying the numerator, which is \( 7 + 6 - 2 \cdot 6 \). Follow the order of operations (PEMDAS/BODMAS). First, perform the multiplication: \(-2 \cdot 6 = -12 \). Then, you get: \[ 7 + 6 - 12 \]. Next, add and subtract from left to right: \( 7 + 6 = 13 \) and then \( 13 - 12 = 1 \). The numerator simplifies to 1.

Simplify the Denominator

Next, simplify the denominator, which is \( 11 - 5 \cdot 2 \). Again, follow the order of operations. Perform the multiplication first: \( 5 \cdot 2 = 10 \). Then, subtract: \( 11 - 10 = 1 \). The denominator simplifies to 1.

Divide the Simplified Numerator by the Simplified Denominator

Finally, divide the simplified numerator by the simplified denominator: \[ \frac{1}{1} = 1 \]. Therefore, the given expression evaluates to 1.

Key concepts

PEMDAS

To correctly solve any mathematical expression, we must follow the order of operations, often memorized by the acronym PEMDAS. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Without following this specific order, we could end up with incorrect results. For example, in the numerator of our exercise: \(7 + 6 - 2 \times 6\), we start by performing the multiplication first: \(-2 \times 6 = -12\). We then continue with the addition and subtraction from left to right: \(7 + 6 = 13\) and \(13 - 12 = 1\). This sequence ensures accurate calculations.

Numerator and Denominator Simplification

Simplifying fractions involves breaking down both the numerator (the top part of the fraction) and the denominator (the bottom part of the fraction). In our exercise, for the numerator \(7 + 6 - 2 \times 6\), we've already simplified it to 1. For the denominator \(11 - 5 \times 2\), we begin by performing the multiplication first: \5 \times 2 = 10\. Then, we subtract to get: \11 - 10 = 1\. Simplification like this lets us work with smaller, more manageable numbers before we perform any division.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. Understanding and accurately performing these operations is essential for solving mathematical problems. In our original problem, we applied these operations step-by-step to simplify \(7 + 6 - 2 \times 6\) to 1 for the numerator and \(11 - 5 \times 2\) to 1 for the denominator. Finally, we performed the division: \frac{1}{1} = 1\. Mastering arithmetic operations ensures precision and correctness in mathematical problem solving.