Problem 12 - Thinking Mathematically

Question

At the beginning of a year, the odometer on a car read 25,124 miles. At the end of the year, it read 37,364 miles. If the car averaged 24 miles per gallon, how many gallons of gasoline did it use during the year?

Step-by-Step Solution

Answer

The car used \( 510 \) gallons of gasoline during the year.

Step 1: Find the total miles driven

Subtract the initial reading on the odometer from the final reading. So, 37,364 miles - 25,124 miles = 12,240 miles.

Step 2: Calculate the gallons of gasoline used

Divide the total distance travelled by the fuel efficiency. Therefore, 12,240 miles / 24 miles/gallon gives the total gallons used.

Step 3: Final Answer

Perform the division from step 2 to get the final answer.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in equations and formulas. It's useful for solving word problems because it allows us to translate a problem's words into mathematical symbols. In our exercise, the idea of algebra helps us understand how to transform the scenario into a mathematical operation.

For example, if you know the total miles driven and the fuel efficiency, you can set up the equation: \( \text{Total Miles} = \text{Miles per Gallon} \times \text{Gallons of Gas} \). This equation is a simple algebraic expression that allows us to solve for the "Gallons of Gas" once we substitute the known values.

Arithmetic

Arithmetic is the process of working with numbers through basic operations such as addition, subtraction, multiplication, and division. It's the foundation of mathematics that everyone uses in daily life.

In this exercise, arithmetic is used to perform calculations like subtracting the initial odometer reading from the final reading to find the total miles driven. It also involves dividing the total miles by the car's fuel efficiency to find out how many gallons of gasoline were used. These operations might seem basic, but they are crucial for solving many real-world problems.

Problem Solving

Problem-solving is an essential skill in mathematics. It involves understanding the problem at hand, planning a strategy to solve it, carrying out the computations correctly, and reflecting on whether the solution makes sense.

For this exercise, the process starts with understanding what information is given (odometer readings and fuel efficiency) and what is being asked (the total gallons of gasoline used). The strategy involves using arithmetic to compute the total miles traveled and then dividing by the miles per gallon to find the gallons used. It’s about breaking down the word problem into these manageable parts and ensuring each step logically follows from the last.

Odometer Reading

An odometer is a tool in a vehicle that measures the total miles that the car has traveled. Reading the odometer is like looking at a "distance counter" on your dashboard.

In our problem, we start with an initial odometer reading and end with a final reading after a year. The difference between these two readings gives us the total distance driven throughout the year. Knowing how to interpret odometer readings is key to figuring out the distance when solving related word problems.

Fuel Efficiency

Fuel efficiency refers to how effectively a vehicle uses fuel, typically measured in miles per gallon (mpg). This metric tells us how many miles the car can travel on a single gallon of gasoline.

In problems like this one, understanding fuel efficiency is crucial. When you know the total miles driven and the vehicle's efficiency, you can determine how many gallons of gas were consumed over a certain distance. By dividing the total miles by the vehicle's miles per gallon, you find out the number of gallons needed for that journey. Understanding this concept helps us make calculations for the cost of travel and environmental impact.