Problem 55 - Algebra A Combined Function

Question

A manufacturer of cans of salted mixed nuts states that the ratio of peanuts to other nuts is 3 to 2 . If 324 peanuts are in a can, find how many other nuts should also be in the can.

Step-by-Step Solution

Answer

There should be 216 other nuts in the can.

Step 1: Identify the Ratio

The ratio of peanuts to other nuts is given as 3:2. This means for every 3 peanuts, there are 2 other nuts.

Step 2: Express the Ratio Algebraically

Let the number of other nuts be represented by x. Since the ratio is 3:2, we can express this relationship as 3/2 = 324/x.

Step 3: Cross-Multiply to Solve for x

Cross-multiply to find x: 3x = 2 × 324.

Step 4: Calculate the Number of Other Nuts

Simplify the equation: 3x = 648. Divide both sides by 3 to find x: x = 648 / 3.

Step 5: Final Calculation

Perform the division: x = 216. So, there should be 216 other nuts in the can.

Key Concepts

Ratios and Proportions

Ratios are a way to compare two quantities by showing the relative size of one quantity to another. In this exercise, the manufacturer states that for every 3 peanuts, there are 2 other nuts. This relationship can be described as a ratio of 3:2. Ratios can be scaled up or down to determine quantities based on a specific situation. For example, if you know the number of peanuts in a can, you can use the ratio to find out how many of the other nuts should be there as well.
Proportions are equations that state that two ratios are equivalent. When dealing with ratios like 3:2, a proportion helps you solve for an unknown quantity when you have one complete pair and a partial portion of the other. This makes calculations simpler and straightforward.

Cross-Multiplication

Cross-multiplication is a useful technique for solving proportions. When you have a proportion set up, such as the one in this exercise, which states that 3/2 = 324/x, cross-multiplication allows you to get rid of the fractions by multiplying across the equation.
Here's how it works: you multiply the numerator of one fraction by the denominator of the other fraction and set it equal to the product of the other pair. In this case, you end up with the equation 3x = 2 × 324.
This method is effective because it simplifies the equation to just a simple multiplication, making it much easier to solve.

Algebraic Equations

An algebraic equation involves variables, numbers, and operations. In the context of this exercise, an algebraic equation was used to express the proportion: \[3x = 2 \times 324\]This equation represents the relationship given in the problem after using cross-multiplication. Solving an algebraic equation typically involves isolating the variable you are solving for, which here is \(x\).
By organizing the relationship in equation form, it becomes much easier to apply mathematical operations and solve for unknown quantities efficiently.

Basic Algebraic Operations

Basic algebraic operations include addition, subtraction, multiplication, and division. These operations are foundational for solving equations and manipulating expressions. In this particular exercise, once the equation \(3x = 648\) was established, the next step was to isolate \(x\) by performing division.
To isolate \(x\), divide both sides of the equation by 3 resulting in:\[x = \frac{648}{3}\]By performing division, the solution for \(x\) is discovered to be 216. This shows how basic arithmetic operations are crucial for solving algebraic equations and finding values in real-world problems.