Problem 81 - Elementary Algebra Within Reach

Question

Creating a Problem Create a proportion problem. Exchange problems with another student and solve the problem you receive.

Step-by-Step Solution

Answer

The number of red marbles in the bag is 18.

Step 1: Understand Proportions

Firstly, the concept of proportions needs to be understood. A proportion is a statement that two ratios are equal. Therfore, if you have four quantities, \(a\), \(b\), \(c\), and \(d\), you could say \(\frac{a}{b} = \frac{c}{d}\) if \(a\), \(b\), \(c\), and \(d\) form a proportion.

Step 2: Create a Proportion Problem

Create a proportion problem using the understanding of proportions. For instance: 'The ratio of girls to boys in a class is 5 to 4. If there are 20 boys in the class, how many girls are there?'.

Step 3: Exchange and Understand New Problem

Exchange your problem with another student and understand the new problem received. For sake of example, let's assume the received problem is: 'The ratio of red to blue marbles in a bag is 3 to 2. If there are 12 blue marbles, how many red marbles are there?'.

Step 4: Solve the Problem

Solve the received problem using the concept of proportions. Here, the ratio of red to blue marbles is 3 to 2, meaning \( \frac{red}{blue} = \frac{3}{2} \). Now if blue marbles are 12, we can set up the proportion as \( \frac{red}{12} = \frac{3}{2} \) to solve for the number of red marbles. Cross multiplying gives \( red = 18 \).

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by using division. They tell us how many times one quantity contains another. For example, if we say the ratio of girls to boys in a class is 5 to 4, we mean:

• For every 5 girls, there are 4 boys.
• This can be expressed as 5:4 or in fraction form as \(\frac{5}{4}\).

Ratios can be scaled up or down depending on the situation. For example, if there are 20 boys, we could set up a proportion to find out how many girls there are. Using the ratio 5:4:

• Scale both numbers by the same factor.
• If we multiply 4 by 5 to get 20, we must also multiply 5 by 5, resulting in 25 girls.

Solving Algebra Problems Involving Ratios

Algebra problems that involve ratios require setting up equations based on the given ratios. Consider the example: "The ratio of red to blue marbles in a bag is 3 to 2. If there are 12 blue marbles, how many red marbles are there?"To solve it, you set up a proportion:

• The ratio given is \( \frac{3}{2} \).
• Let the number of red marbles be \( x \).
• Set up the equation: \( \frac{x}{12} = \frac{3}{2} \).

From here, we would use algebraic techniques to solve for \( x \), finding the number of red marbles by isolating \( x \) through operations like cross-multiplication.

Using Cross-Multiplication

Cross-multiplication is a powerful tool for solving proportion problems. When you have an equation where two fractions are set equal, like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication helps find unknowns.Here's how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.
• Set the results equal to each other: \( ad = bc \).

This method quickly simplifies solving for unknown variables. In our example problem, using cross-multiplication, \( \frac{x}{12} = \frac{3}{2} \), results in \( 2x = 36 \), helping us easily find \( x = 18 \).

Developing Mathematical Reasoning

Mathematical reasoning is the ability to logically think about and solve mathematical problems. When dealing with proportions, you're not just using numbers, but also engaging in critical thinking. Here's how to strengthen this skill:

• Break down problems into smaller parts.
• Analyze each part and how it connects with others.
• Use logic to deduce relations between numbers, like understanding how changing one part of a ratio affects the whole.
• Practice with different problems to get used to seeing patterns.

Through practice, solving ratio problems becomes a more intuitive process, leading to better problem-solving strategies and understanding.