Problem 18 - Prealgebra

Question

Write a proportion that could be used to solve for each variable. Then solve. 8 pencils in 2 boxes 20 pencils in \(x\) boxes

Step-by-Step Solution

Answer

The value of \(x\) is 5.

Step 1: Define the Proportion

To set up a proportion, we need two equivalent ratios. We are given two scenarios: one with 8 pencils in 2 boxes and another with 20 pencils in \(x\) boxes. We will set up a proportion comparing these two ratios as follows: \( \frac{8}{2} = \frac{20}{x} \).

Step 2: Cross Multiply

To solve the proportion \( \frac{8}{2} = \frac{20}{x} \), we cross multiply to eliminate the fractions. This gives us \(8x = 20 \times 2\).

Step 3: Simplify the Equation

Next, simplify the equation obtained from cross-multiplication. So, we calculate \(8x = 40\).

Step 4: Solve for \(x\)

To find the value of \(x\), divide both sides of the equation \(8x = 40\) by 8. This results in \(x = \frac{40}{8}\).

Step 5: Simplify the Result

Simplify \(x = \frac{40}{8}\) to obtain \(x = 5\). Therefore, there would be 5 boxes needed to hold 20 pencils at the same rate as 8 pencils in 2 boxes.

Key Concepts

Understanding Ratios

Ratios are essentially a way to compare quantities. They show the relationship between two numbers. Think of it as a way to express how much of one thing there is compared to another. In our exercise, we have two sets of items to compare: pencils and boxes.

• The ratio of pencils to boxes in the first scenario is 8 to 2.
• This can be written as the fraction \(\frac{8}{2}\).
• In the second scenario, the ratio is 20 to \(x\), written as \(\frac{20}{x}\).

Ratios help us set up equations or proportions that let us solve for unknown values. They make it easier to understand relationships and predict outcomes. In our example, the pencil-to-box ratio helps us find out how many boxes we need for a different number of pencils.

Demystifying Cross-Multiplication

Cross-multiplication is a handy tool to solve equations that involve two ratios. It helps eliminate the fractions, making it easier to solve for unknown variables. This technique is especially useful when dealing with proportions, like in our exercise.Let's illustrate how cross-multiplication works:- Start with the proportion \(\frac{8}{2} = \frac{20}{x}\).- Cross-multiply by multiplying each denominator with the opposite numerator: this gives us the equation \(8x = 20 \times 2\).Cross-multiplication simplifies our problem, turning it into a simple algebraic equation. From here, we can solve for \(x\) by isolating it on one side of the equation. Cross-multiplication links the numerators and denominators across the equation, preserving the equality and making problem-solving more straightforward.

Exploring Equivalent Fractions

Equivalent fractions represent the same part of a whole, even if the numerators and denominators differ. They are all about showing that two different fractions can be equal when they represent the same proportion of a whole.In our example:

• The fraction \(\frac{8}{2}\) describes the ratio of pencils to boxes in one scenario.
• \(\frac{20}{x}\) is equal to \(\frac{8}{2}\), representing the same ratio in another scenario.

These fractions are equivalent because the relationship remains constant across both scenarios. Finding equivalent fractions allows us to check our work and ensure our solutions are consistent with the problem's conditions. Simplifying and comparing the fractions helps demonstrate the consistency and establish trust in our solution.

Effective Problem Solving with Proportions

Problem solving with proportions involves using mathematical reasoning to find unknown values when given a set of ratios or proportions. It is a systematic approach that combines logical thinking with mathematical procedures.In our pencil and box scenario:- We start with identifying what we know (8 pencils and 2 boxes) and what we need to find (how many boxes for 20 pencils).- Set up the proportion \(\frac{8}{2} = \frac{20}{x}\) to equate the known and unknown situations.- Use cross-multiplication to form an equation that can be solved algebraically. - Finally, solve the equation step by step, simplify and find the unknown.This approach equips you with a structured method to tackle similar problems. By using proportions, you can cleverly navigate through diverse real-world problems and effortlessly find solutions. Understanding this strategy not only aids in math but can broaden your problem-solving skills in everyday life.