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Mathematical problem-solving involves understanding the issue at hand, devising a plan to tackle it, carrying out that plan, and then examining the results to ensure accuracy. In the context of our example, Becky's problem is a practical one: calculating fuel consumption for her trip. She needs to determine how many gallons of gasoline her car will consume over a 2100-mile journey. To solve this, she must identify the relevant data (total distance and car's fuel efficiency) and select the appropriate mathematical operation to use—in this case, division.

Breaking Down the Problem

Initially, Becky discerns that her car's fuel efficiency is the key to solving the problem. Understanding the efficiency as a rate allows for the utilization of mathematical operations to find the solution. By recognizing that the total number of gallons needed is the unknown variable, Becky sets the stage for an arithmetic division problem.

The unit rate is a comparative measure that expresses the quantity of one thing in relation to a single unit of another thing. It is particularly useful when dealing with ratios and comparisons. In our car trip scenario, the car's fuel efficiency is presented as a unit rate of 30 miles per gallon. This tells us how many miles the car can travel for every gallon of fuel.

Applying Unit Rates

To apply unit rate calculations to our problem, we look at the rate of 30 miles per gallon and the total distance of 2100 miles. By understanding that the car's efficiency rate provides the necessary link between distance and fuel consumption, we can calculate the total amount of fuel needed for the trip simply by dividing the distance by the rate.

Division of rational numbers is a fundamental concept in mathematics, used to split a number into specified portions. Rational numbers include all integers, fractions, and decimal numbers that can be expressed as the quotient of two integers. In Becky's problem, both the total distance (2100 miles) and the car's fuel efficiency (30 miles per gallon) are rational numbers.

Executing the Division

When performing the division, the total distance becomes the dividend, and the car's fuel efficiency is the divisor. The quotient is the number of gallons of fuel needed. This operation simplifies the process of determining fuel consumption for a given distance when we have an efficiency rate. In the example given, dividing 2100 miles by 30 miles per gallon precisely results in the number of gallons required.

The distance to gallon ratio is a specific expression of fuel consumption, indicating the distance that can be traveled per unit of fuel. This offers a clear and immediate understanding of a vehicle's fuel efficiency. In Becky's case, the ratio is essential for planning her trip, as it directly affects the calculation of her travel costs.

Utilizing the Ratio

With the knowledge that her car runs 30 miles for every gallon of gas, Becky uses this distance to gallon ratio to forecast the fuel needs for a 2100-mile journey. This is a straightforward ratio application, indicating that for every 30 miles traveled, one gallon of fuel is used. The ratio simplifies the complex relationship between distance and fuel into a manageable, one-step calculation. By dividing the total trip distance by the distance per gallon, Becky easily determines that 70 gallons of fuel will be required for the trip.