Problem 61 - Elementary Algebra

Question

The ratio of peanuts to all nuts in a certain brand of packaged mixed nuts is 3 to 5 . If the package contains 475 nuts, then how many peanuts can we expect?

Step-by-Step Solution

Answer

There are 285 peanuts in the package of mixed nuts.

Step 1: Understanding the Ratio

The ratio of peanuts to all nuts is given as 3:5. This means for every 3 peanuts, there are a total of 5 nuts.

Step 2: Calculating Total Ratio Parts

The total parts in the ratio are calculated by adding the numbers in the ratio: 3 (peanuts) + 2 (other nuts) = 5. Therefore, 3 parts of peanuts and 2 parts of other nuts make 5 total parts in the mixed nuts.

Step 3: Setting Up the Proportion

The ratio tells us that peanuts make up 3 parts out of 5. So the proportion of peanuts in the package is \( \frac{3}{5} \).

Step 4: Calculating Number of Peanuts

To find out how many peanuts there are, multiply the total number of nuts by the proportion of peanuts: \( \frac{3}{5} \times 475 \).

Step 5: Performing the Calculation

Calculating this expression gives \( \frac{3}{5} \times 475 = 285 \). So, there are 285 peanuts in the package.

Key Concepts

Understanding Proportions

Proportions are a crucial concept in mathematics, especially when dealing with ratios. When you have a ratio, it compares two quantities, showing how much of one exists relative to the other. In our example, the ratio of peanuts to total nuts is 3 to 5. This tells us that for every 3 peanuts, there are 5 nuts in total. In practical terms, a proportion is a statement that two ratios are equal. For example, if 3 is to 5 as 285 is to 475, a consistent proportional relationship is maintained across both sets. Therefore, understanding and setting up proportions can help solve real-world problems, such as determining quantities in recipes, calculating speeds, or converting currencies. 🔑 Key points to remember:

• A ratio compares two quantities.
• Proportions are equations that show two ratios are equal.
• You can solve proportion problems by cross-multiplying and dividing.

Performing Arithmetic Calculations

Arithmetic calculations form the backbone of solving proportion problems. Once you have your proportion set up, arithmetic is necessary to find the unknown quantity. For the mixed nuts example, we calculated the number of peanuts by multiplying the total number of nuts (475) by the proportion of peanuts, which we deduced was \( \frac{3}{5} \).This operation involved basic arithmetic steps:

• Multiply 475 by 3 to get 1,425.
• Then, divide 1,425 by 5 to extract the quantity of peanuts, yielding 285.

These steps demonstrate how arithmetic operations like multiplication and division are crucial in interpreting proportions. They transform abstract ratios into concrete numbers, allowing us to answer practical questions efficiently.

Applying Elementary Algebra

Elementary algebra provides additional tools to tackle problems involving ratios and proportions. It lets us express the relationships between quantities using variables and equations. In this exercise, the proportion of peanuts can be set up as an equation:\[ \frac{3}{5} = \frac{x}{475} \]Here, \( x \) represents the unknown quantity of peanuts. This equation signifies that the ratio of peanuts to all nuts maintains the same relationship as peanuts alone to the total 475 nuts.By applying cross-multiplication, a fundamental algebraic technique, we solve for \( x \):

• Cross-multiply to get: \( 5x = 3 \times 475 \)
• Simplify to find \( x \): \( x = \frac{3 \times 475}{5} \)
• This computation confirms that \( x = 285 \)

Thus, elementary algebra not only verifies our calculation but also provides a structured approach to solve proportion problems efficiently and accurately.