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Chapter 4: Problem 13

In Exercises \(11-18,\) solve each proportion. $$\frac{a}{6}=\frac{5}{3}$$

Short Answer

Expert verified
The value of \(a\) is 10.

Step by step solution

01

Set up the Proportion

Given the proportion \(\frac{a}{6} = \frac{5}{3}\), identify the two fractions that are equal.
02

Cross Multiply

To solve for \(a\), cross multiply the fractions. This means multiplying the numerator of each fraction by the denominator of the other fraction. Here鈥檚 the setup: \(a \times 3 = 6 \times 5\).
03

Simplify the Equation

Now simplify the equation from the cross multiplication: \(3a = 30\).
04

Solve for the Variable

Divide both sides of the equation by 3 to isolate \(a\): \(a = \frac{30}{3}\).
05

Simplify the Result

Simplify \(\frac{30}{3}\) to get the value of \(a\): \(a = 10\).

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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 technique used to solve proportions, which are equations that show two ratios are equal. For example, in the proportion \(\frac{a}{6}=\frac{5}{3}\), cross multiplication involves multiplying the numerator of each fraction by the denominator of the other. It transforms the equation into a simple linear equation. Cross multiplication is like this: \(a \times 3 = 6 \times 5\). This step makes solving proportions straightforward, as it eliminates the fractions.
Fraction Equations
Fraction equations involve expressions with fractions. When solving these, it's important to eliminate the fractions to make it easier to isolate the variable. This is where cross multiplication comes in handy. In the given exercise, \(\frac{a}{6} = \frac{5}{3}\), cross multiplication helps get rid of the fractions by creating a single equation: \(3a = 30\). Once the fractions are gone, it鈥檚 much simpler to solve for the variable.
Solving for a Variable
Solving for a variable means finding the value of the unknown in the equation. After cross multiplying in our example, we end up with \(3a = 30\). To solve for \(a\), we need to isolate it. We do this by dividing both sides of the equation by the coefficient of \(a\), which is 3: \(a = \frac{30}{3}\). Finally, simplifying \(\frac{30}{3}\), we get \(a = 10\). Now we know the value of \(a\)!
Proportion Equations
Proportion equations show that two ratios are equal. They can be written as fractions and involve variables. For example, \(\frac{a}{6}=\frac{5}{3}\) shows that the fraction \(\frac{a}{6}\) is equal to the fraction \(\frac{5}{3}\). Solving these types of equations typically involves cross multiplication to eliminate the fractions, leading to a simpler equation where you can easily solve for the unknown variable.

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Most popular questions from this chapter

Solve the equations and inequalities. $$\frac{x+1}{2}+x=6$$

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. The length of a rectangle is 1 less than twice the width. If the width is 1 more than one seventh of the perimeter, find the dimensions of the rectangle.

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{x}{3}+\frac{x}{2}+\frac{x}{5}$$

Solve the equations and inequalities. $$\frac{2 y-3}{2}-\frac{y-5}{3}=\frac{1}{6}$$

Make up a verbal problem involving a purchase of some items that would give rise to the following equation (be sure to indicate what the variable would represent): $$ 30 x+50(80-x)=3,600 $$
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