Problem 4 - Prealgebra

Question

A spiral toy that can bounce down a flight of stairs is made from 80 feet of wire. Write and solve an equation to find how many of these toys can be made from a spool of wire that contains 4000 feet.

Step-by-Step Solution

Answer

50 toys can be made from 4000 feet of wire.

Step 1: Understand the Problem

We have a single spiral toy that requires 80 feet of wire. Given a total supply of 4000 feet of wire, we need to find out how many spiral toys can be constructed using this total length.

Step 2: Define the Equation

Let's denote the number of toys that can be made as \( x \). Each toy requires 80 feet of wire, so the total wire required for \( x \) toys is \( 80x \). The total available wire is 4000 feet, leading to the equation \( 80x = 4000 \).

Step 3: Solve the Equation

To find \( x \), we'll solve the equation \( 80x = 4000 \) by dividing each side by 80. \[ x = \frac{4000}{80} \] This calculation simplifies to \( x = 50 \).

Step 4: Verify the Solution

Multiply the number of toys by the wire needed per toy to ensure it matches the total wire. \[ 50 \times 80 = 4000 \] This confirms our solution as the calculated total matches the available wire.

Key Concepts

Equation Solving

Solving equations is like unlocking the answer to a mathematical puzzle. Each step we take helps us get closer to the solution. In the exercise, we had to find out how many spiral toys could be made from a certain length of wire. Here's how we did it: First, we recognized that each toy needed 80 feet of wire. We labeled the number of toys as \( x \). Then, we set up the equation. Writing equations like this helps keep everything organized. The problem becomes a kind of mathematical sentence, with each part telling us something about the situation. Solving the equation \( 80x = 4000 \) involved isolating \( x \). We divided both sides by 80, which is the key operation that revealed our answer. When you solve equations, you're always trying to "undo" operations to find out what the variable is – in this case, how many toys.

Algebraic Expressions

Algebraic expressions are like the language of algebra. They let us represent numbers and relationships in simple ways. In the toy wire problem, the expression \( 80x \) stood for the total amount of wire needed for \( x \) toys. Expressions allow you to turn words into numbers: instead of saying 'an unknown number of toys each using 80 feet of wire,' we just write \( 80x \). Breaking it down:

• "80" is the coefficient. It tells us how much wire is used per toy.
• "x" is the variable. It stands for the number of toys we can make.

To understand algebraic expressions, think of them as a recipe. Each part tells you how much of something to use, so by following the recipe – or the expression – you end up with the final product, like finding out how many toys you can make with the wire.

Mathematical Reasoning

Mathematical reasoning is all about thinking logically and understanding how different math parts fit together. Let's look at how we used it to solve the wire problem.When we set up our equation, we reasoned that if 80 feet make one toy, then 4000 feet can make several toys. This logical step helped us form the equation \( 80x = 4000 \). Mathematical reasoning is not just about doing calculations. It's about making smart decisions about what calculations to do. We verified our answer by using reasoning again. We calculated \( 50 \times 80 = 4000 \) to ensure that it fit the original scenario. This kind of check is crucial in math because it confirms you've solved the problem correctly. Logical thinking in math means you're like a detective, checking all the clues until everything lines up perfectly.