Problem 11 - Algebra 1: Concepts and Skills

Question

The average mileage for your old truck is 10.5 miles per gallon. Estimate the number of miles you can travel on a full 22 gallon tank of diesel fuel.

Step-by-Step Solution

Answer

The estimated number of miles that can be covered on a 22-gallon tank of diesel fuel is 231 miles.

Step 1: Identify the Given Information

The truck's average mileage is 10.5 miles per gallon. And, the capacity of the diesel tank is 22 gallons.

Step 2: Setup the Calculation

To estimate the number of miles travelled on a full tank, multiply the average miles per gallon by the number of gallons. So, the formula will be: estimated miles = average mileage per gallon * total gallons.

Step 3: Solve the Calculation

Substitute the given values into the equation: estimated miles = 10.5 miles per gallon * 22 gallons = 231 miles.

Key Concepts

Solving Algebraic Equations

In every algebraic journey, understanding how to solve equations is your secret map to treasure. Think of algebraic equations as puzzles where you're trying to find the missing piece, often represented by a variable. In the case of our example with the truck's mileage, the equation is straightforward and doesn't even involve variables.

However, to solve more complex algebraic equations, you would normally isolate the variable on one side of the equation by performing inverse operations. These operations may include addition, subtraction, multiplication, and division, applied to both sides of the equation to maintain balance. Let's say the equation was a bit more complex, and we had to solve for miles per gallon: \( x = \frac{total \ miles}{total \ gallons} \). In this scenario, the equation would require division to isolate 'x'.

Rate Problems in Algebra

Rate problems are all about relationships – they tell us how one quantity changes in relation to another. Our example is a classic rate problem where we calculate the distance traveled based on the rate of fuel consumption.

In algebra, the rate is typically represented by the formula \( rate = \frac{distance}{time} \), or in our truck example, \( rate = \frac{miles}{gallon} \), where 'rate' is the average mileage. Working with rate problems involves setting up proportions and cross-multiplication to find the unknown variable when given the other two quantities. It's like a dance where each step precisely mirrors the other – balance and proportion are key.

Unit Conversion in Mathematics

Traveling through the world of math often requires a passport for unit conversion, as various countries (or subjects) speak in different mathematical 'languages' or units. Different problems necessitate different units of measurement, and it’s critical to be fluent in converting between them to understand the true distance of your mathematical travels.

For example, if we were asked to convert our truck's mileage into kilometers per liter, we'd need to convert miles to kilometers (1 mile ≈ 1.60934 kilometers) and gallons to liters (1 gallon ≈ 3.78541 liters). Mastering unit conversion is like being able to communicate in multiple languages seamlessly. It takes practice, but once you get the hang of it, it opens up a whole new world of possibilities and allows you to effortlessly navigate various mathematical scenarios.