Fuel Tank Capacity Calculation Determining Fuel Mass In A Gasoline-Ethanol Mixture

Introduction

Hey guys! Today, let's dive into a cool math problem about a fuel tank filled with a mix of gasoline and ethanol. We'll figure out how many kilograms of fuel are in the tank, considering the different densities of gasoline and ethanol. It’s a practical problem that shows how math can be used in real-world situations. This article aims to break down the problem step-by-step, making it easy to understand even if you're not a math whiz. So, grab your thinking caps, and let’s get started!

Problem Statement

Okay, so here's the deal: We have a fuel tank with a total capacity of 70 liters. This tank is filled with a mixture of gasoline and ethanol. Specifically, 60% of the tank's volume is gasoline, and the remaining 40% is ethanol. We're given the densities of both fuels: gasoline has a density of 0.77 kg/L, and ethanol has a density of 0.80 kg/L. Our mission, should we choose to accept it, is to determine the total mass of the fuel mixture in the tank, measured in kilograms. To solve this, we'll need to calculate the individual volumes of gasoline and ethanol, then use their densities to find their masses, and finally, add those masses together. Let's break it down further in the subsequent sections.

Breaking Down the Problem

To tackle this fuel tank conundrum, let's break it down into smaller, more manageable steps. First, we need to figure out the volume of gasoline in the tank. Since gasoline occupies 60% of the 70-liter capacity, we'll calculate this percentage to find the exact volume. Next, we'll do the same for ethanol, which makes up the remaining 40% of the tank's volume. Once we know the volumes of both gasoline and ethanol, we can move on to calculating their masses. To find the mass of gasoline, we'll multiply its volume by its density (0.77 kg/L). Similarly, we'll calculate the mass of ethanol by multiplying its volume by its density (0.80 kg/L). Finally, we'll add the mass of gasoline and the mass of ethanol to find the total mass of the fuel mixture in the tank. This step-by-step approach will help us keep things organized and ensure we don't miss any crucial details. Stay tuned as we dive into the calculations!

Calculating Gasoline Volume

Alright, let's get those calculators out and start crunching some numbers! Our first task is to determine the volume of gasoline in the tank. Remember, the tank has a total capacity of 70 liters, and 60% of that is filled with gasoline. To find this volume, we simply need to calculate 60% of 70 liters. In mathematical terms, this means we'll multiply 70 liters by 0.60 (since 60% is equivalent to 0.60 as a decimal). So, the calculation looks like this: Gasoline Volume = 0.60 * 70 liters. Doing the math, we get a gasoline volume of 42 liters. This is a crucial piece of information because it tells us exactly how much gasoline we're dealing with. With this volume in hand, we can now move on to the next step: figuring out the volume of ethanol in the tank. Let's keep the momentum going!

Understanding Percentage Calculations

Before we move on, it's worth taking a quick detour to make sure we're all on the same page when it comes to percentage calculations. Understanding percentages is super important, not just for this problem, but for all sorts of real-life situations, like calculating discounts, figuring out tips, or even understanding statistics. A percentage is simply a way of expressing a number as a fraction of 100. So, when we say 60%, we mean 60 out of 100, or 60/100. To convert a percentage to a decimal, we divide it by 100. That's why 60% becomes 0.60. In our gasoline volume calculation, we used this decimal form to multiply by the total tank capacity. This method works for any percentage problem – just convert the percentage to a decimal and multiply it by the total amount. Now that we've refreshed our percentage skills, let's jump back into the problem and calculate the ethanol volume!

Calculating Ethanol Volume

Now that we've nailed down the gasoline volume, let's turn our attention to the ethanol. The problem states that ethanol occupies the remaining 40% of the tank's 70-liter capacity. Just like we did with gasoline, we need to calculate this percentage to find the volume of ethanol. So, we'll multiply the total capacity (70 liters) by 0.40 (since 40% is equivalent to 0.40 as a decimal). The calculation is: Ethanol Volume = 0.40 * 70 liters. If you do the math, you'll find that the ethanol volume is 28 liters. Awesome! We now know the volumes of both gasoline (42 liters) and ethanol (28 liters) in the tank. This is a major step forward because it allows us to calculate the mass of each fuel type. Remember, we need the volumes to use the densities provided in the problem. So, let's move on to the next stage: calculating the mass of gasoline.

Verifying Volume Calculations

Before we proceed, it's always a good idea to double-check our calculations and make sure everything adds up. In this case, we can verify our volume calculations by adding the volume of gasoline (42 liters) and the volume of ethanol (28 liters). If the sum equals the total tank capacity (70 liters), we know we're on the right track. Let's do it: 42 liters (gasoline) + 28 liters (ethanol) = 70 liters. Hooray! The volumes add up perfectly, which confirms that our calculations are accurate. This little verification step is a great habit to get into, especially when dealing with multi-step problems. It helps catch any potential errors early on and prevents them from snowballing into bigger issues later. Now that we've given our volumes the thumbs-up, let's confidently move on to calculating the mass of gasoline using its density.

Calculating Gasoline Mass

Alright, with the gasoline volume safely in our grasp, it's time to calculate its mass. Remember, we're given the density of gasoline as 0.77 kg/L. Density is a measure of how much mass is contained in a given volume, and it's the key to converting volume to mass. The formula we'll use is: Mass = Density * Volume. In our case, the density of gasoline is 0.77 kg/L, and we've calculated the volume to be 42 liters. So, we plug those numbers into the formula: Gasoline Mass = 0.77 kg/L * 42 liters. Performing the multiplication, we find that the mass of gasoline in the tank is 32.34 kg. Fantastic! We've successfully calculated the mass of the gasoline. Now, let's do the same for ethanol. We'll use the same formula, but with the density of ethanol and the volume we calculated earlier.

The Importance of Units

Before we move on to ethanol, let's take a moment to appreciate the importance of units in our calculations. Units are our friends, and they help us keep track of what we're measuring and ensure our answers make sense. In the gasoline mass calculation, we multiplied density (kg/L) by volume (liters). Notice how the