Problem 77 - Algebra A Combined Function

Question

Which of the following are incorrect and why? $$ \frac{1+2}{1+3} \text { simplifies to } \frac{2}{3} $$

Step-by-Step Solution

Answer

The simplification is incorrect; it should be \( \frac{3}{4} \), not \( \frac{2}{3} \).

Step 1: Understanding the Expression

The given expression is \( \frac{1+2}{1+3} \). We must verify whether this simplifies correctly to \( \frac{2}{3} \).

Step 2: Simplifying the Numerator and Denominator

First, simplify the numerator, \(1 + 2\), which equals \(3\). Next, simplify the denominator, \(1 + 3\), which equals \(4\).

Step 3: Form the Simplified Fraction from Calculations

After calculating, the actual expression simplifies to \( \frac{3}{4} \).

Step 4: Compare Simplified Expression with Given Result

We compare the calculated value \( \frac{3}{4} \) with the given claim \( \frac{2}{3} \). They are not equal, meaning the given simplification is incorrect.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. To do this, you need to find common factors in the numerator and the denominator and cancel them out. This ensures that the fraction is in its lowest terms. However, when simplifying, certain operations like addition or subtraction inside fractions need careful attention.
Remember to fully simplify each part of the fraction before division. This means separately simplifying both the numerator and denominator. Once each part is simplified, check for any common factors and simplify further if possible. In the provided exercise, simplifying individual parts of an expression first helps see that \[ \frac{1+2}{1+3} \] simplifies not to \( \frac{2}{3} \) but rather to \( \frac{3}{4} \).

• Perform operations on numerators and denominators first.
• Complete all additions or subtractions before simplification.
• Find and cancel common factors afterward.

Numerator and Denominator

A fraction has two main components: the numerator and the denominator. The numerator is the top number that shows how many parts of a whole you have, and the denominator is the bottom number that shows how many parts the whole is divided into. Understanding these components is crucial when simplifying fractions because each needs to be correctly interpreted and simplified separately.
For example, in the expression \( \frac{1+2}{1+3} \), the numerator is initially \(1+2\) and the denominator is \(1+3\). Each simplifies to \(3\) and \(4\), respectively, leading to the fraction \( \frac{3}{4} \). Many mistakes occur if the entire fraction is not carefully broken down part by part. Simplification fails if you overlook any part of the form \[ \frac{{\text{numerator}}}{{\text{denominator}}} \] by skipping basic arithmetic operations first.

• Clarify the operation needed for each part.
• Simplify the numerator and denominator separately.
• Double-check your calculations.

Problem-Solving Steps

Breaking down a problem into manageable steps is essential to avoid mistakes. Having a structured approach helps in maintaining clarity and correctness in solutions. The given exercise can be accurately tackled by following clear problem-solving steps.

• Identify the expression you're working with. Here, \( \frac{1+2}{1+3} \).
• Simplify both the numerator and the denominator separately and perform any basic arithmetic needed.
• Write out the simplified expression— in our exercise, it's \( \frac{3}{4} \).
• Compare your results with any provided solutions to check consistency.

By following these steps, errors are minimized, ensuring a correct understanding of whether a simplification is accurate or not. Each of these steps highlights the importance of clarity and patience in problem-solving to achieve correct mathematical solutions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators representing a value. They require proper interpretation and computation to unveil their true simplified form. Simplifying a mathematical expression involves sequential and correct application of arithmetic operations.
Working with expressions like \( \frac{1+2}{1+3} \) requires tackling one component at a time: calculate meaningful parts before attempting to simplify the fraction as a whole. This ensures accuracy and identifies any flaws or false simplifications provided initially.

• Break down expressions into smaller, calculable segments.
• Use correct order of operations: parentheses, exponents, multiplication/division, addition/subtraction (PEMDAS).
• Review expressions thoroughly to prevent errors.

Understanding these expressions and correctly solving them is crucial in presenting the correct answers in mathematical exercises and beyond. Following these steps ensures you compute expressions swiftly and confidently.