Question

\(\frac{?}{\mathrm{MN}}=\frac{G H}{G N}\)

Short answer

Use cross-multiplication to find ? as \(? = \frac{GH \times MN}{GN}\).

Step-by-step solution

Understand the Problem

We are given the proportion \( \frac{?}{\mathrm{MN}} = \frac{GH}{GN} \) and we need to determine the value that should replace '?' to make the proportion true.

Identify the Variables

In the given proportion, \(?\) is the unknown we need to solve for, and MN, GH, GN are segments or values given in the problem.

Use Cross-Multiplication

To solve the proportion \( \frac{?}{\mathrm{MN}} = \frac{GH}{GN} \), apply cross-multiplication to eliminate the fractions. This gives the equation: \( ? \times GN = GH \times MN \).

Solve for the Unknown

To find \(?\), simply isolate it on one side of the equation. Divide both sides by GN to solve for \(?\), resulting in: \( ? = \frac{GH \times MN}{GN} \).

Substitute and Simplify

If numerical values for GH, MN, and GN are provided, substitute them into the equation from Step 4 and simplify to find the exact value of \(?\). Otherwise, the expression \( ? = \frac{GH \times MN}{GN} \) is the simplest form of the solution.

Key concepts

Understanding Proportions

A proportion is an equation that states two ratios are equivalent. In simple terms, it shows that two fractions or ratios are equal to each other. For instance, if we have a proportion like \(\frac{a}{b} = \frac{c}{d}\), it means that the ratio \(a:b\) is equivalent to \(c:d\). Proportions are used to solve various mathematical problems because they represent relationships in a balanced equation.

• Numerator: The top number of a fraction.
• Denominator: The bottom number of a fraction.

In the context of the original problem, we are trying to establish equality between the fraction involving the unknown and the one with known values, such as \(\frac{?}{MN} = \frac{GH}{GN}\). This is the starting point of understanding how proportions work.

Algebraic Manipulation Techniques

Algebraic manipulation involves rearranging equations and expressions to solve for a desired variable. In this exercise, we are manipulating the equation using a technique known as cross-multiplication. This allows us to eliminate the fractions and work with a simpler, linear equation. Cross-multiplying involves multiplying the numerator of one ratio by the denominator of the other:

• \(? \times GN = GH \times MN\)

This converts the proportion into a more manageable equation without fractions. Such manipulation helps unlock a straightforward pathway to isolating and identifying the unknown variable, which is essential for solving the problem.

Solving Equations for an Unknown

The process of solving equations involves finding the value of the unknown variable that makes the equation true. Once we have manipulated the equation into a format like \(? \times GN = GH \times MN\), we can proceed to solve for the unknown. To find the value of \(?\), it needs to be isolated on one side of the equation.

• Divide both sides by \(GN\) to isolate \(?\).
• This gives you \(? = \frac{GH \times MN}{GN}\).

At this step, if numerical values are provided for \(GH\), \(MN\), and \(GN\), substitute these into the equation to find the precise numerical value of the unknown. Otherwise, this expression shows the relationship between these values and maintains the equation's balance, effectively solving the problem.