Problem 70 - Elementary Algebra Within Reach

Question

Map Scale On a map, \(1 \frac{1}{2}\) inches represents 40 miles. Estimate the distance between two cities that are 4 inches apart on the map.

Step-by-Step Solution

Answer

The distance between the two cities that are 4 inches apart on the map is approximately 106.67 miles.

Step 1: Analyze the scale of the map

Here, it is given that 1.5 inches on the map represent 40 miles in real life. This gives us a scale 'X' to which we can compare the 4 inches. This ratio, in a proportional format, can be expressed as \(\frac{1.5}{40} = X\).

Step 2: Define the equation to solve

Since the provided scale relates inches to miles, it can be used to find the real distance corresponding to 4 inches on the map. Thus we can write the equation as follows: \(X = \frac{4}{Y}\), where 'Y' represents the unknown distance equivalent to 4 inches.

Step 3: Simplify the equation

We already have the equation \(\frac{1.5}{40} = \frac{4}{Y}\), and we can now cross-multiply and solve for 'Y'. 1.5 * Y = 4 * 40 -> Y = \(\frac{4 * 40}{1.5}\).

Step 4: Compute the final answer

Perform the calculation to compute the value of 'Y'. This gives us the real distance between the two cities that corresponds to 4 inches on the map.

Key Concepts

Elementary Algebra

Elementary algebra forms the basis for solving problems like the one presented in our map scale scenario. It involves understanding how letters and symbols represent numbers and operations. For map scale problems, algebra allows us to set up an equation that represents the relationship between distances on a map and real-life distances.

In the provided exercise, we establish a proportional equation using algebraic symbols where the unknown real-life distance, represented by variable 'Y', corresponds to the 4 inches on the map. Through algebra, we equate this to the known ratio obtained from the scale of the map.

Ratios and Proportions

Ratios and proportions are vital when interpreting map scales. A ratio compares two quantities, while a proportion shows that two ratios are equal. In our map scale problem, the ratio is given as 1.5 inches to 40 miles.

This ratio helps us understand the proportion of the map's scale; for every 1.5 inches on the map, it represents 40 miles of real distance. When faced with a 4-inch measurement, we need to set up a proportion to find the unknown distance, forming the equation \(\frac{1.5}{40} = \frac{4}{Y}\).

Equation Solving

Equation solving is the process of finding the value of a variable that makes an equation true. It is a critical step in map scale problems to find the unknown real distance. From our setup proportion, \(\frac{1.5}{40} = \frac{4}{Y}\), we are tasked to solve for 'Y'.

To do this, we must isolate the variable on one side of the equation, which is often achieved through methods like cross-multiplication. Once 'Y', the unknown real distance, is by itself on one side, we get the solution that represents the actual distance between two points on a map.

Cross Multiplication

Cross multiplication is a technique used to solve proportions. It involves multiplying across the equal sign diagonally and then dividing to solve for the unknown variable. It's extremely useful in map scale problems where we need to find equivalence between different units of measurement.

In step 3 of our solution, \(1.5 * Y = 4 * 40\) is an example of cross multiplication. By multiplying the cross-sections of our proportion, we set up an equation that allows us to easily compute 'Y'. Dividing 4 * 40 by 1.5 gives us the distance between the two cities in miles.