Problem 60 - Prealgebra

Question

Pieces of Pipe. How many pieces of pipe that are \(\frac{2}{3}\) foot long must be laid together to make a pipe 16 feet long?

Step-by-Step Solution

Answer

24 pieces of pipe are needed.

Step 1: Understanding the Problem

We need to find how many pieces of pipe, each \(\frac{2}{3}\) foot in length, are required to make a total pipe length of 16 feet. This involves determining how many \(\frac{2}{3}\) foot sections fit into 16 feet.

Step 2: Set Up the Equation

Let \(x\) represent the number of \(\frac{2}{3}\) foot pipes. The total length of these pieces is represented by the equation: \((x \times \frac{2}{3}) = 16\).

Step 3: Solve for x

To find \(x\), divide both sides by \(\frac{2}{3}\). This gives: \[x = \frac{16}{\left(\frac{2}{3}\right)}. \]

Step 4: Simplify the Division

Simplify the division of a fraction by multiplying by its reciprocal. Therefore, \[x = 16 \times \frac{3}{2}. \]

Step 5: Calculate the Result

Multiply 16 by \(\frac{3}{2}\) to find \(x\). First, multiply 16 and 3 to get 48, and then divide by 2 to get 24. So, \(x = 24\).

Key Concepts

Division with Fractions

Division with fractions might seem tricky at first, but it's just a different way of thinking about numbers. When you divide by a fraction, you're asking how many times the fraction fits into the number you're dividing. For instance, in our problem, we want to know how many \( \frac{2}{3} \) foot pieces fit into a 16-foot pipe.

Here's a simple way to do it:

• First, remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is made by flipping its numerator and denominator. So, the reciprocal of \(\frac{2}{3} \) is \(\frac{3}{2} \).


• Now, replace the division with multiplication by the reciprocal: \[ x = 16 \times \frac{3}{2}. \]

Understanding this concept means you can handle any fraction division confidently.

Multiplication with Fractions

Multiplying fractions is more straightforward than it sounds. It's about increasing or decreasing a number by the fraction you're multiplying with.

Here's how it works:

• When you multiply two fractions, multiply the numerators together to get the new numerator.


• Multiply the denominators together to get the new denominator.

In our problem, once we convert the division to a multiplication: \(16 \times \frac{3}{2} \), we treat 16 as \(\frac{16}{1} \). Therefore, multiply 16 (numerator) by 3, giving 48, and 1 (denominator) by 2, giving 2.

• The resulting fraction \(\frac{48}{2} \) simplifies to 24.

This method shows how we can multiply entire numbers by fractions and maintain balance.

Problem Solving in Math

Problem solving in math requires a systematic approach to unpacking and addressing the problem. Here, understanding the situation is key. When facing a problem like laying pipe pieces together, you should:

• Understand the Problem: Recognize what you're asked to find. With our pipe problem, we're looking for the number of pieces needed to reach a certain length.


• Set Up the Equation: Use known values to form an equation. Start with variables to represent unknown quantities.


• Solve the Equation: Use mathematical operations such as division and multiplication with fractions to find the answer.

Once the equation \(x \times \frac{2}{3} = 16\) was established, each step logically followed until we found the answer. Through practice, breaking problems into simple steps makes solving math challenges easier and more intuitive.