Problem 98 - Elementary Algebra

Question

For the following problems, find the solution. Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?

Step-by-Step Solution

Answer

Answer: Both pipes can fill the tank together in approximately 2 hours and 13 minutes.

Step 1: Determine the rate at which each pipe fills the tank individually

Pipe A can fill the tank in 4 hours, so its individual filling rate is 1/4 tank per hour. Pipe B can fill the tank in 5 hours, so its individual filling rate is 1/5 tank per hour.

Step 2: Combine the filling rates of both pipes

In order to find how long it will take for both pipes to fill the tank together, we need to add their filling rates together. The combined filling rate R is given by the equation: \( R = (1/4) + (1/5) \).

Step 3: Calculate the combined filling rate

To find the combined filling rate, solve the equation R from step 2: \[ R = (1/4) + (1/5) = 5/20 + 4/20 = 9/20 \]

Step 4: Determine how long it takes for both pipes to fill the tank together

Since the combined filling rate R is 9/20 tank per hour, the time it takes for both pipes to fill the tank together can be found by taking reciprocal of R. Time = 1/R = 1/(9/20) = 20/9 hours Both pipes can fill the tank together in 20/9 hours or approximately 2 hours and 13 minutes.

Key Concepts

Rate Problems

Rate problems often involve finding how fast something happens or how long it takes for something to complete a task. Rate problems include scenarios like filling tanks, painting walls, or traveling at certain speeds. In these problems, rates describe how much of a task is completed in a unit of time. For instance, if a pipe can fill a tank in 4 hours, its rate is one-quarter of the tank per hour, which we express as \( \frac{1}{4} \) tank/hour.

Understanding rate problems involves:

• Identifying the individual rates.
• Understanding how these rates interact when combined.

When multiple rates work together, like two pipes filling a tank, you add their rates to find the overall rate. This is because each pipe contributes to the task simultaneously. For the given problem, combining rates involves adding \( \frac{1}{4} \) and \( \frac{1}{5} \), which are the rates from each pipe working alone.

By handling rate problems step-by-step, you can simplify complex situations into solvable mathematical equations.

Problem Solving

Problem solving is a systematic approach to finding solutions to complex challenges. It often involves breaking down a problem into smaller, manageable parts. When solving word problems, like the one about the pipes filling a tank, it's important to understand what you're asked to find and what information you have. This guides your setup and solution.

In this exercise:

• Understand what each pipe does separately.
• Gather information (like time taken and rate for each pipe).
• Combine this information to solve how long they take together.

Effective problem solving means translating words into mathematical expressions. Once the problem is clear, using algebra to solve it is often straightforward. This exercise involves recognising two parts working together, combining their efforts, and calculating how these joint efforts convert into time taken to fill the tank.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. When solving the rate problem, using algebra is essential to express and solve the situation. Algebra helps form equations from word problems, allowing you to manipulate and solve them easily.

In this tank-filling problem, algebra is used to:

• Express each pipe's rate as an algebraic fraction (\( \frac{1}{4} \) for Pipe A and \( \frac{1}{5} \) for Pipe B).
• Combine these rates algebraically (\( \frac{1}{4} + \frac{1}{5} = \frac{9}{20} \)).
• Determine the time needed by finding the reciprocal of the combined rate (\( \frac{20}{9} \) hours).

Using algebraic expressions to solve rate problems simplifies the process, as it changes word problems into solvable mathematical equations. Once the expressions are set, it's easier to carry out arithmetic operations and arrive at the correct solution.