Cistern Capacity: Calculate Total Volume Easily
Hey guys! Ever wondered how to figure out the total capacity of a cistern when you only know a fraction of it? It's a common problem, and today, we're going to break it down step-by-step. We'll tackle the classic problem: "If 3/7 of a cistern's capacity is 8,136 gallons, what's the total capacity?" This isn't just about math; it's about understanding proportions and how they apply in the real world. So, let's dive in and unlock the secrets of cistern volumes!
Understanding the Problem: Visualizing the Cistern
Before we jump into calculations, let's visualize what we're dealing with. Imagine a big water tank, our cistern. We know that if we divide this tank into 7 equal parts, 3 of those parts hold 8,136 gallons. Our mission is to find out how much water all 7 parts can hold. This is a classic problem involving fractions and proportions, and it's a great example of how math can help us solve everyday puzzles.
Key Concepts:
- Fraction: A fraction represents a part of a whole. In our case, 3/7 represents three parts out of seven total parts.
- Proportion: A proportion is a statement that two ratios are equal. We'll use proportions to relate the known fraction (3/7) and its corresponding volume (8,136 gallons) to the unknown total capacity.
- Capacity: The capacity of the cistern is the total volume of water it can hold.
To really grasp this, think about it like a pizza. If you have a pizza cut into 7 slices and you know 3 slices weigh a certain amount, you can figure out the weight of the whole pizza using proportions. The cistern problem is the same idea, just with gallons of water instead of pizza slices.
Step-by-Step Solution: Cracking the Capacity Code
Now, let's get down to the nitty-gritty and solve this problem. We'll break it down into simple steps so you can follow along easily.
Step 1: Find the Volume of One Part
We know that 3/7 of the cistern holds 8,136 gallons. To find the volume of just 1/7, we need to divide the known volume (8,136 gallons) by 3. This will tell us how much water each individual part represents.
8,136 gallons / 3 = 2,712 gallons
So, each 1/7 of the cistern holds 2,712 gallons. We're one step closer to finding the total capacity!
Step 2: Calculate the Total Capacity
Now that we know the volume of 1/7 of the cistern, we can easily find the total capacity. Since the cistern is divided into 7 parts, we simply multiply the volume of one part (2,712 gallons) by 7. This will give us the total number of gallons the cistern can hold.
2,712 gallons/part * 7 parts = 18,984 gallons
Eureka! We've found the answer. The total capacity of the cistern is 18,984 gallons.
Step 3: Verify Your Answer (Because Mistakes Happen!)
It's always a good idea to double-check your work. To verify our answer, we can calculate 3/7 of the total capacity we found (18,984 gallons) and see if it matches the original given volume (8,136 gallons). If it does, we know we're on the right track.
(3/7) * 18,984 gallons = 8,136 gallons
Success! Our calculation is correct. We've successfully determined the cistern's total capacity.
Alternative Approach: The Power of Proportions
There's another way to solve this problem using proportions. Remember, a proportion is a statement that two ratios are equal. We can set up a proportion to relate the known fraction (3/7) and its corresponding volume (8,136 gallons) to the unknown total capacity (let's call it 'x').
Setting up the Proportion:
We can write the proportion as follows:
3/7 = 8,136 gallons / x
This reads as
