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Chapter 7: Problem 55

Write each sentence as a proportion. 8 is to 9 as 14 is to \(x\).

Short Answer

Expert verified
The proportion is \(\frac{8}{9} = \frac{14}{x}\), with \(x = 15.75\).

Step by step solution

01

Understanding Ratios and Proportions

A proportion states that two ratios are equal. In the problem, "8 is to 9 as 14 is to \(x\)", means the ratios \(\frac{8}{9}\) and \(\frac{14}{x}\) are equal. We can write this as a proportion equation: \(\frac{8}{9} = \frac{14}{x}\).
02

Setting Up the Equation

Using the information given, set up the equation of the two ratios being equal. This translates the sentence into a mathematical expression as follows: \(\frac{8}{9} = \frac{14}{x}\).
03

Cross-Multiplying to Solve for \(x\)

To solve the proportion, use cross-multiplication. This involves multiplying diagonally across the equal sign. Multiply 8 by \(x\) and 9 by 14, to give the equation: \(8x = 126\).
04

Solving for \(x\)

Now, solve for \(x\) by dividing both sides of the equation by 8. This gives: \(x = \frac{126}{8}\). When simplified, divide 126 by 8 to find that \(x = 15.75\).

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Key Concepts

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

Understanding Ratios
Ratios are a way to compare two quantities or numbers, showing the relative size of one quantity to another. They are often written in the form of a fraction, like in the example given: \(\frac{8}{9}\). This means that for every 8 of one quantity, we have 9 of another.
  • Ratios can be written in different forms: \(a:b\), \(\frac{a}{b}\), or "a to b".
  • It's important to maintain the order when comparing the numbers, as it affects the relationship.
In a proportion like "8 is to 9 as 14 is to \(x\)", you're essentially seeing two ratios that have been set equal to each other. This equality allows us to solve for the unknown, given that the relationships are consistent.
Cross-Multiplication Explained
Cross-multiplication is a technique often used to solve proportions - it helps to simplify the identification of the unknown variable. When you have a proportion, such as \(\frac{8}{9} = \frac{14}{x}\), cross-multiplication allows you to eliminate the fractions and create a simple equation to solve.How does cross-multiplication work?
  • Take the numerator of one fraction and multiply it by the denominator of the other: \(8 \times x\).
  • Do the same for the other pair: \(9 \times 14\).
  • Set these two products equal to each other: \(8x = 126\).
Cross-multiplication is effective because it uses the property of equality鈥攖he equal sign鈥攙alidating that both sides of the equation are balanced when multiplied this way.
Solving Equations
Solving equations involves finding the value of the unknown that makes the equation true. In our example, cross-multiplication gave us the equation \(8x = 126\). The goal is to isolate \(x\) on one side of the equation.Here's how to solve for \(x\):
  • Start by dividing both sides of the equation by 8, the coefficient of \(x\).
  • This simplifies the equation to \(x = \frac{126}{8}\).
  • Perform the division to find \(x = 15.75\).
By following these steps, you solve for the unknown, making sure each operation is mathematically sound. This process demonstrates how equations can be manipulated to uncover the needed values for unknowns, maintaining consistency and accuracy.

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Most popular questions from this chapter

Check each answer to see if each makes the proportion a true proportion. If any proportion is not true, solve it to find the correct solution. $$\begin{aligned}&\frac{p}{100}=\frac{13}{52}\\\&\text { Is the percent equal to } 25(25 \%) \end{aligned}$$

Solve. \(75 \%\) of 14 is what number?

Solve. 9310 is what percent of \(3800 ?\)

Solve. 550.4 is what percent of \(172 ?\)

Write each fraction as a decimal and then write each decimal as a percent. Round the decimal to three decimal places (nearest thousandth) and the percent to the nearest tenth percent. $$ \frac{21}{79} $$
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