Question

Insert the missing quantity. $$\frac{x+2}{5 x}=\frac{?}{5}$$

Short answer

The missing quantity is \(x + 2\).

Step-by-step solution

Identify the Cross-Multiplication

To find the missing quantity in a proportion where the denominators are the same, you can use the concept of cross-multiplication or simply equate the numerators since the denominators are equal.

Equate the Numerators

Since the denominators are the same (both are 5), you can set the numerators equal to each other: \(x + 2 = ?\)

Solve for the Missing Quantity

The numerator on the left-hand side of the equation is the missing quantity on the right-hand side. Thus, the missing quantity is: \(x + 2\)

Key concepts

Proportions

A proportion is an equation that expresses the equality of two ratios or fractions. It is a cornerstone concept in algebra and is often used to solve problems involving relationships between quantities. When you see a proportion like \( \frac{a}{b} = \frac{c}{d} \), it means that the ratio of \(a\) to \(b\) is the same as the ratio of \(c\) to \(d\).

In our exercise, \( \frac{x+2}{5x} = \frac{?}{5} \) is a simple form of proportion where the denominators already match, simplifying our task to finding the missing numerator. Understanding how to manipulate proportions is essential as they can model and solve real-world problems, including scaling, density calculations, and map readings.

Proportions also offer a way to find missing values when some quantities are known, and they ensure that the relationships between quantities remain consistent when some values change.

Equating Numerators

Equating numerators comes into play when you are dealing with fractions that have the same denominator. This method simplifies solving proportions by taking advantage of equality. It saves time and calculations when compared to cross-multiplying.

Imagine two fractions side by side in equality, \( \frac{N_1}{D} = \frac{N_2}{D} \), where \(N_1\) and \(N_2\) are numerators, and \(D\) is the common denominator. Since both parts of the fraction share \(D\), it logically follows that \(N_1\) must equal \(N_2\).

In the given exercise, setting \(x+2\) equal to the missing quantity helps us deduce the answer quickly. This technique is a valuable shortcut when you encounter fractions with equal denominators, as it allows you to concentrate solely on the numerators to find your solution.

Algebraic Fractions

Algebraic fractions are simply fractions where the numerator, the denominator, or both contain algebraic expressions. Just like numerical fractions, algebraic fractions can be added, subtracted, multiplied, and divided using the same principles.

An essential skill in dealing with algebraic fractions is simplifying them, which includes factoring and cancelling common factors when possible. In the context of our exercise, we don't need to perform these steps because the fractions involved are already in a simplified form.

Understanding how algebraic fractions work is necessary when moving on to more advanced topics in algebra, including solving equations and understanding functions. Algebraic fractions encapsulate ideas fundamental in algebra; thus, mastering them opens up avenues to a broader understanding of mathematical concepts.