Question

Insert the missing quantity. $$\frac{x}{3}=\frac{?}{9}$$

Short answer

\(3x\)

Step-by-step solution

Understand the Problem

The exercise presents a proportion where the numerator x is divided by 3 on one side, and an unknown number (represented by ?) is divided by 9 on the other. The goal is to find the missing number that would make the two ratios equal.

Set up the Proportion

To find the missing number (?), we can set up the equation \( \frac{x}{3} = \frac{?}{9} \) so that both sides have equal value.

Solve for the Missing Number

We solve for '?' by finding what number needs to be placed in the numerator on the right side of the equation so that the proportion is correct. We do this by cross-multiplying: \( x \cdot 9 = 3 \cdot ? \) which simplifies to \( 9x = 3? \)

Calculate the Missing Quantity

To isolate the missing number '?', we divide both sides by 3: \( \frac{9x}{3}=\frac{3?}{3} \) which simplifies to \( 3x=? \) So, the missing quantity is 3 times x.

Key concepts

Proportional Relationships

A proportional relationship is a fundamental concept in mathematics where two quantities have a constant ratio. In simpler terms, one quantity is always a fixed multiple of the other. This relationship is crucial in a variety of real-life situations, including recipe adjustments, map reading, and converting currencies.

In our exercise, we look for a missing quantity that maintains the proportion between two ratios. The original ratio given is \( \frac{x}{3} \) and we aim to find a number that forms the same type of ratio with 9. The concept of proportional relationships emphasizes the importance of the 'constant of proportionality' - the fixed number that can be multiplied with one quantity to obtain the other. In our case, it's the number that relates x to the missing number in the ratio \( \frac{?}{9} \).

Cross Multiplication

Cross multiplication is a technique used to solve equations involving two ratios that are set equal to each other, known as proportions. This method involves multiplying across the equal sign diagonally, hence the name. It simplifies the problem by eliminating the fractions and making it easier to find the unknown value.

When applying cross multiplication to our problem, we multiply x by 9 and 3 by the missing number \( ? \), giving us the equation \( x \cdot 9 = 3 \cdot ? \), which highlights that the products of the cross-multiplied pairs are equal. This is an invaluable tool for verifying if two ratios form a proportional relationship or for finding a missing term in a ratio.

Ratios and Proportions

Ratios and proportions are all about the comparison of numbers or measurements. A ratio represents the relative size of two quantities, while a proportion shows that two ratios are equivalent. They're like the mathematical version of saying 'apples to apples'.

In terms of our exercise, the ratio of x to 3 is set to be equivalent to the ratio of the missing number to 9. We express this relationship using a proportion: \( \frac{x}{3} = \frac{?}{9} \). Solving proportions involves finding a missing term, which, when plugged into the ratio, would make the two ratios equivalent. Ratios and proportions are particularly useful in fields such as science, finance, and engineering where comparing quantities is essential.

Algebraic Equations

Algebraic equations are like puzzles where you solve for an unknown – in our case, the mysterious '?'. These equations involve variables, numbers, and operational symbols, and they follow the basic principles of algebra to maintain equality when manipulated.

For the exercise at hand, the equation is set up as a proportion, which we then transform into the algebraic equation \( 9x = 3? \). By isolating the variable '?', we divide both sides by 3, leading us to \( 3x = ? \). Algebraic equations are essentially a balance scale; whatever operation is done to one side must be done to the other to maintain balance. This is a cornerstone concept of algebra that's applicable in countless mathematical scenarios.