Problem 30 - Prealgebra

Question

Solve each proportion. $$\frac{3}{2.2}=\frac{7.5}{y}$$

Step-by-Step Solution

Answer

The value of \(y\) is 5.5.

Step 1: Cross Multiply

To solve the proportion \(\frac{3}{2.2} = \frac{7.5}{y}\), we begin by cross-multiplying. This means we multiply the numerator of one fraction by the denominator of the other and set the products equal. So, \(3 \times y = 7.5 \times 2.2\).

Step 2: Simplify the Right Side

Now we calculate the multiplication on the right side: \(7.5 \times 2.2\). This gives us \(16.5\). Thus, we have the equation \(3y = 16.5\).

Step 3: Solve for \(y\)

To solve for \(y\), we divide both sides of the equation by 3: \(y = \frac{16.5}{3}\).

Step 4: Complete the Division

Finally, we perform the division to find the value of \(y\): \(y = 5.5\).

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve proportions, which are equations that show two fractions are equivalent. When we have a proportion like \( \frac{3}{2.2} = \frac{7.5}{y} \), cross multiplication helps to eliminate the fractions and make the equation easier to solve.

To perform cross multiplication, you follow these steps:

• Multiply the numerator of the first fraction (3 in this case) by the denominator of the second fraction (\(y\)).
• Multiply the numerator of the second fraction (7.5) by the denominator of the first fraction (2.2).
• Set the two products equal to each other, resulting in the equation \(3 \times y = 7.5 \times 2.2\).

This process allows you to remove the fractions, providing a straightforward equation to solve. The beauty of cross multiplication is its simplicity and its ability to transform a proportion into a more familiar equation form quickly.

Solving Equations

When solving proportions, like the one provided, we often end up with equations that need to be solved for a specific variable, which in this case is \(y\).

After cross multiplying our original proportion, we simplified it to the equation \(3y = 16.5\). The next step is to isolate \(y\) on one side of the equation. To do this, we use the following steps:

• Recognize this as a simple linear equation in the form \(ax = b\).
• To solve for \(y\), divide both sides of the equation by 3: \(y = \frac{16.5}{3}\).

These steps allow us to isolate the variable, providing the solution for \(y\) once the division is completed. Solving equations is a foundational skill in algebra, helping to unravel unknowns and find precise values.

Fractions

Fractions represent a part of a whole and consist of two parts: a numerator and a denominator. When working with proportions, understanding fractions is essential since proportions are essentially comparisons between two fractions.

In our given problem, we began with \( \frac{3}{2.2} = \frac{7.5}{y} \). To solve this, we relied on our understanding of fractions:

• The numerator (top number) indicates how many parts we have.
• The denominator (bottom number) shows into how many parts the whole is divided.

By manipulating fractions through techniques like cross multiplication, we can find unknown values like \(y\). Working with fractions often requires additional steps like adopting common denominators or converting them into decimals, depending on the context and operation needed. Mastering these methods allows for seamless transition between fractions and other numerical expressions.