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Chapter 4: Problem 14

In Exercises \(11-18,\) solve each proportion. $$\frac{a}{8}=\frac{9}{2}$$

Short Answer

Expert verified
a=36

Step by step solution

01

Cross-Multiply the Proportion

To solve the proportion, use cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the numerator of the second fraction and the denominator of the first fraction. This gives the equation: .
02

Set Up the Equation

From cross-multiplication, Set the two products equal to each other: .
03

Solve for

Isolate Multiply both sides of the equation by .

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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 powerful technique for solving proportions. When you have an equation in the form of \(\frac{a}{b} = \frac{c}{d}\), cross-multiplying means you multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction. This transforms the proportion into an algebraic equation: \a \times d = b \times c\.

Here's an easy way to think about it: you 'cross' the fractions by diagonally connecting the terms with multiplication, creating a single equation that's often easier to solve.
Algebraic Equations
An algebraic equation is a mathematical statement that shows the equality of two expressions. In the context of solving proportions, after you cross-multiply, you turn the proportion into a straightforward algebraic equation.

For example, if the proportion is \(\frac{a}{8} = \frac{9}{2}\), cross-multiplication gives you the equation 2a = 72. This equation is simpler and can be solved using basic algebra techniques.
Isolation of Variables
Isolation of variables means manipulating an equation to get the unknown variable by itself on one side of the equation. This makes it straightforward to find the value of the variable.

When we have an equation like 2a = 72, our goal is to get 'a' by itself. To do that, we divide both sides by 2, the coefficient of 'a'. This results in:
  • 2a/2 = 72/2
  • a = 36
By isolating the variable, we discover that a = 36.

Remember: the key to isolation is performing the same operation on both sides of the equation.

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Most popular questions from this chapter

Solve the equations and inequalities. $$0.155 x+0.075(80-x)=8.4$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. A collection of nickels and dimes is worth \(\$ 1.30 .\) If there are 5 fewer dimes than nickels, how many of each are there?

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{3 y}{x}-\frac{x}{4 y}+\frac{2 y}{5 x}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Takima has a semester average of \(94 .\) She has four exam scores of \(88,85,93\) and \(95 .\) If her final exam counted for 3 exams, find her score on the final.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Lam takes five exams and has a semester average of \(92 .\) Four of his exam scores were \(79,89,94,\) and \(98 .\) Find his fifth exam score.
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