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Chapter 7: Problem 59

Solve each proportion. $$\frac{b}{7.8}=\frac{2}{3}$$

Short Answer

Expert verified
The solution is \(b = 5.2\).

Step by step solution

01

Cross-Multiply

The first step in solving a proportion is to perform cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives us: \ \(b \times 3 = 7.8 \times 2\)
02

Calculate Products

Calculate the products from Step 1: \ \(3b = 15.6\)
03

Solve for b

To isolate \(b\), divide both sides of the equation by 3: \ \(b = \frac{15.6}{3}\) \ Simplifying gives \(b = 5.2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
Cross-multiplication is a reliable method to solve proportions, especially when you have an equation involving two fractions set equal to each other. A proportion means that two fractions represent the same value or ratio. To cross-multiply:
  • Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
  • Do the same with the numerator of the second fraction and the denominator of the first.
This process transforms the proportion into a straightforward equation that can easily be solved. For instance, given the proportion \( \frac{b}{7.8} = \frac{2}{3}\), cross-multiplication results in the equation \(b \times 3 = 7.8 \times 2\). This step helps eliminate the fractions and simplifies the solving process greatly.
By understanding this core concept, you can solve almost any proportional equation with confidence.
Isolating Variables
Once you've used cross-multiplication, you're left with a simple equation where solving involves finding the value of a variable, such as \(b\). Isolating the variable means you perform operations to get the variable alone on one side of the equation. Here鈥檚 how you do it:
  • First, perform any multiplication or division required to maintain balance in the equation.
  • In the example \(3b = 15.6\), you divide both sides by 3 to isolate \(b\).
This operation gives you \(b = \frac{15.6}{3}\).
Understanding this essential part helps you solve equations efficiently. It is essential because isolating the variable allows you to directly find its value, which is necessary for obtaining your final solution.
Simplification in Algebra
Simplification means reducing an equation or an expression to its simplest form. This is crucial for verifying your solutions and understanding algebraic relationships clearly. In the step-by-step solution:
  • Once you have \(b = \frac{15.6}{3}\), simply perform the division to get \(b = 5.2\).
  • This result is more accessible and easier to use, especially in further calculations, if needed, as it is in the most basic form.
Simplification helps confirm that your calculations are correct and that your solution is appropriately reduced. By practicing this concept, you not only improve your algebraic skills but also cultivate a keen sense of clarity in mathematical problem-solving.

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