Problem 202 - Elementary Algebra

Question

June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?

Step-by-Step Solution

Answer

The smaller cooler holds 8 gallons, and the bigger cooler holds 40 gallons.

Step 1 - Define Variables

Let the capacity of the smaller cooler be denoted as \( x \) gallons. The capacity of the bigger cooler, being five times as large, is then \( 5x \) gallons.

Step 2 - Set Up the Equation

The total capacity of both coolers together must equal 48 gallons. Therefore, set up the equation: \( x + 5x = 48 \).

Step 3 - Combine Like Terms

Combine like terms in the equation to simplify it: \( 6x = 48 \).

Step 4 - Solve for \( x \)

Solve the simplified equation for \( x \): \( x = \frac{48}{6} = 8 \).

Step 5 - Calculate the Capacities

Using the value of \( x \), find the capacities of the coolers. The smaller cooler holds \( 8 \) gallons, and the bigger cooler holds \( 5 \times 8 = 40 \) gallons.

Key Concepts

Variable Definition

In algebra word problems, starting with defining your variables can make solving the problem much easier. A variable is a symbol, often a letter like 'x', that represents a quantity we need to find.

In this exercise, we are tasked with finding out how many gallons each cooler can hold. Let's call the capacity of the smaller cooler 'x'. This makes it simple to reference this unknown quantity.

Since the problem states that the bigger cooler is five times the size of the smaller one, we can express the capacity of the bigger cooler as '5x'. This step of defining variables is crucial because it translates the word problem into a mathematical form we can work with.

Equation Setup

Setting up the equation correctly is the next important step. An equation shows the relationship between the different parts of the problem.

In our problem, the total capacity that the coolers must hold together is 48 gallons. We have already defined the capacities as 'x' for the smaller cooler and '5x' for the bigger cooler.

To set up the equation, we add these two expressions together because the capacities combine to total 48 gallons. This gives us the equation:

\(x + 5x = 48\).

Setting up the correct equation will guide you to the right solution.

Combining Like Terms

Once the equation is set up, it's time to simplify it by combining like terms. Combining like terms makes the equation easier to solve.

In our equation, \(x + 5x = 48\),
'x' and '5x' are like terms because they both contain the variable 'x'. Adding them together, we get:

\(x + 5x = 6x\).
So the equation simplifies to:

\(6x = 48\).
Combining like terms simplifies the equation and brings us one step closer to finding the solution.

Solving Equations

The final step is solving the equation. This involves finding the value of the variable.

We have the equation:

\(6x = 48\).
To solve for 'x', we need to isolate it. We do this by dividing both sides of the equation by 6, like this:

\(x = \frac{48}{6}\).

When we divide, we find:

\(x = 8\).
So, the smaller cooler can hold 8 gallons. To find the capacity of the bigger cooler, we multiply the value of 'x' by 5, getting:

\(5x = 5 \times 8 = 40\).
So, the bigger cooler can hold 40 gallons. Solving the equation reveals the capacities of both coolers.