Question

Explain why \(\frac{4+3}{2+5}\) is not equal to \(\frac{4}{2}+\frac{3}{5}\).

Short answer

The simplified form of \( \frac{4+3}{2+5} \) is 1, while \( \frac{4}{2}+\frac{3}{5} \) simplifies to \ \frac{13}{5} \, which are not equal.

Step-by-step solution

Write down the initial expression

Consider the initial expression: \ \ \( \frac{4+3}{2+5} \)

Simplify the numerator and denominator

Add the numbers in the numerator and the denominator: \ \ The numerator is \(4+3=7\) \ The denominator is \(2+5=7\) \ Thus, \( \frac{4+3}{2+5} = \frac{7}{7} = 1 \)

Analyze the separate fractions

Now consider the separate fractions: \ \ \( \frac{4}{2} \) and \( \frac{3}{5} \) \ Simplify each fraction: \ \( \frac{4}{2} = 2 \) and \( \frac{3}{5} \) remains as is. \ Add these results together: \( 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} \)

Compare the results

Compare the simplified expressions: \( \frac{4+3}{2+5} = 1 \) which is not equal to \ \ \( \frac{4}{2} + \frac{3}{5} = \frac{13}{5} \)

Key concepts

fraction addition

When adding fractions, the denominators must be the same. This can be confusing, so let's break it down. For example, if you want to add \(\frac{1}{4}\) and \(\frac{1}{2}\), you can't just add the numerators and denominators directly. You first need to find a common denominator.
A common denominator is a shared multiple of the denominators. For \(\frac{1}{4}\) and \(\frac{1}{2}\), the common denominator is 4.
So, you convert \(\frac{1}{2}\) to \(\frac{2}{4}\). Now the fractions look like this: \(\frac{1}{4} + \frac{2}{4}\). With a common denominator, you can add the numerators: (1+2), then put this over the common denominator: \(\frac{3}{4}\).
This is why \(\frac{4+3}{2+5} ≠ \frac{4}{2}+\frac{3}{5}\). The denominators were not the same.

simplifying fractions

Simplifying fractions makes them easier to understand and work with. You simplify a fraction by dividing the numerator and the denominator by the greatest common divisor (GCD), which is the largest number that can divide both.
For instance, take the fraction \(\frac{8}{12}\). Both 8 and 12 are divisible by 4, which is their GCD. When you divide both by 4, you get \(\frac{2}{3}\). Thus, \(\frac{8}{12} = \frac{2}{3}\).
This step is crucial because smaller numbers are easier to handle and understand. Also, no calculation errors arise from working with reduced fractions.

numerator and denominator

Fractions consist of two main parts: the numerator and the denominator. The numerator is the top number, which shows how many parts you have. The denominator is the bottom number, which shows how many parts make up the whole.
For example, in \(\frac{3}{5}\), 3 is the numerator, and 5 is the denominator. This fraction means you have 3 parts of something that is divided into 5 equal parts.
Understanding the roles of the numerator and the denominator helps you to easily grasp why \(\frac{4+3}{2+5} eq \frac{4}{2} + \frac{3}{5}\). Adding numerators and denominators separately changes the fraction's value, which is not the intended calculation for fraction addition.