Question

The population of City A is three times the population of City B. Together, Cities A and B have twice the population of City C. What is the ratio of the population of City \(C\) to the population of City \(B ?\)

Short answer

The ratio is 2:1.

Step-by-step solution

- Define Variables

Let the population of City B be denoted by \(B\). Then, the population of City A is 3 times that of City B, which we can represent as \(A = 3B\). Let the population of City C be denoted by \(C\).

- Set Up Equation

Based on the problem's information, the total population of Cities A and B together is twice that of City C. We can write this relationship as an equation: \[ A + B = 2C \]

- Substitute Values

Substitute \(A\) in the equation with \(3B\): \[ 3B + B = 2C \]

- Simplify the Equation

Combine like terms to simplify: \[ 4B = 2C \]

- Solve for C in Terms of B

Divide both sides of the equation by 2 to isolate C: \[ 2B = C \] This shows that the population of City C is 2 times the population of City B.

- Find the Ratio

The ratio of the population of City C to the population of City B is: \[ \frac{C}{B} = \frac{2B}{B} = 2 \]

Key concepts

ratio and proportion

Ratios and proportions are fundamental in solving many problems, including this one involving city populations. A ratio compares two quantities by division, like 3:1, which means for every 3 units of one quantity, there is 1 unit of another. Here, we are working with ratios to understand the relationship between populations. City A's population is three times that of City B. This can be written as the ratio 3:1. Also, Cities A and B together have twice the population of City C. Being comfortable with ratios helps simplify these comparisons and set up equations.

algebraic equations

Algebraic equations allow us to use variables to solve problems. In this case, we use algebra to find unknown population values. We start by defining our variables clearly: let B represent the population of City B. Then, the population of City A, being three times B, is represented as 3B. We also introduce C for the population of City C. We use these variables to create an equation based on the problem's information: \( A + B = 2C \). Substituting values and simplifying this equation helps us find the relationships needed.

population comparison

Comparing populations quickly becomes straightforward once familiar with the ratios and equations. Initially, we know City A’s population is three times City B's. To include City C, we use the fact that Cities A and B together equal twice City C's population. This gives us the equation \( A + B = 2C \). With A represented as 3B (from the ratio), substituting this into the equation lets us find how populations compare in a manageable way, leading us logically to the solution.

variables in algebra

Using variables in algebra helps to generalize the problem and find the solution flexibly. Let’s define the variables: B for City B's population, A as 3B, and C for City C. Substituting and manipulating these in equations lets us solve for unknowns. Our equation \( A + B = 2C \) becomes \( 3B + B = 2C \), simplifying to \( 4B = 2C \), and further \( 2B = C \). This shows City C's population is double that of City B. Thus, by substituting and solving for variables, algebra simplifies complex relationships.