Problem 44 - Algebra 1: Concepts and Skills

Question

There are a total of 28 marbles in a bag. Six of the marbles are red and the rest are blue. What is the ratio of red marbles to blue marbles? A. \(\frac{2}{11}\) B. \(\frac{6}{28}\) C. \(\frac{3}{14}\) D. \(\frac{3}{11}\)

Step-by-Step Solution

Answer

The ratio of red to blue marbles is \( \frac{3}{11} \). Therefore, the correct answer is D. \( \frac{3}{11} \)

Step 1: Find the number of blue marbles

Subtract the given number of red marbles from the total marbles: \(28 - 6 = 22\). So, there are 22 blue marbles.

Step 2: Find the ratio of red to blue marbles

The ratio of red to blue marbles is calculated by dividing the number of red marbles by the number of blue marbles, which is \( \frac{6}{22} \)

Step 3: Simplify the ratio

The ratio \( \frac{6}{22} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, the simplified ratio is \( \frac{3}{11} \)

Key Concepts

Simplifying Ratios

When you see a ratio like \( \frac{6}{22} \), it can often be simplified. Simplifying a ratio means writing it in its simplest form, using the smallest possible numbers that give you the same relation. To do this, you need to find a number that can be evenly divided into both the numerator and the denominator. This number is called the greatest common divisor (GCD). After finding the GCD, divide both parts of the ratio by this number.

For example, in the ratio of red to blue marbles, \( \frac{6}{22} \) becomes \( \frac{3}{11} \) after simplification. We divided both 6 and 22 by their GCD, which is 2.
To simplify ratios, always ensure that:

• Both numbers in the ratio can be divided by the greatest common divisor.
• The result gives you smaller numbers that still represent the original comparison accurately.

By simplifying ratios, we make them easier to understand and compare.

Greatest Common Divisor

The greatest common divisor, or GCD, is a key concept for simplifying fractions and ratios. It is the largest number that can exactly divide two or more numbers without leaving a remainder. Finding the GCD helps us simplify ratios by reducing them to their simplest form.
Here's how you can find the GCD of two numbers:

• List the factors of each number. Factors are numbers that divide evenly into another number.
• Find the common factors from both lists. These are the numbers that appear in both lists.
• Select the largest number from the common factors. This is your GCD.

For the ratio \( \frac{6}{22} \), the factors of 6 are 1, 2, 3, and 6, while the factors of 22 are 1, 2, 11, and 22. The common factor here is 2, making it the GCD. Using the GCD allows us to simplify \( \frac{6}{22} \) to \( \frac{3}{11} \). With practice, identifying the GCD quickly becomes easier and makes dealing with fractions and ratios a breeze.

Subtraction in Word Problems

Subtraction is often a crucial step in solving word problems, particularly when quantities need to be determined separately. To handle such problems effectively, start by clearly identifying what needs to be subtracted and from which value.

As in the original exercise, we needed to find out how many blue marbles were in the bag. We knew the total number of marbles was 28 and that 6 were red. By subtracting the number of red marbles from the total marbles, we found the number of blue marbles: \(28 - 6 = 22\).
When tackling word problems involving subtraction:

• Understand what each quantity in the problem represents.
• Identify which value you need to subtract to find the unknown quantity.
• Write down the subtraction equation and solve for the unknown.

Approaching word problems step-by-step allows you to organize your thoughts and ensures you've interpreted the information correctly. This clear, methodical process reduces the likelihood of mistakes and helps you arrive at the correct solution.