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Chapter 7: Problem 53

Write each sentence as a proportion. 17 is to 12 as \(x\) is to 20

Short Answer

Expert verified
\( \frac{17}{12} = \frac{x}{20} \)

Step by step solution

01

Understanding Proportional Sentences

To express '17 is to 12 as \( x \) is to 20' as a proportion, we need to understand that this statement is describing a relationship comparing two ratios. A proportion is an equation stating that two ratios are equal.
02

Setting Up the Ratios

Identify the numbers in the sentence: '17 is to 12' can be written as the ratio \( \frac{17}{12} \) and 'x is to 20' can be written as the ratio \( \frac{x}{20} \).
03

Expressing as a Proportion

Now we write the equation where these two ratios are equal: \( \frac{17}{12} = \frac{x}{20} \). This is the proportional statement derived from the sentence.

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Key Concepts

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

Ratios
A ratio is a way to compare two quantities. It's like saying how much of one thing there is compared to another. For example, the statement '17 is to 12' is a ratio. You see ratios in everyday life鈥攍ike when noticing that there are twice as many apples as oranges in a fruit basket. A ratio can be expressed in different forms such as '17:12', '17 to 12', or as a fraction \( \frac{17}{12} \). Each form communicates the same relationship.

Understanding ratios is crucial because they help make comparisons clear and straightforward. By looking at the numbers involved, you can quickly gauge the proportion of one quantity to another.
  • Ratios can compare similar types of quantities, like length to length or time to time.
  • They help in solving problems where quantities need to be compared.
Equations
Equations are mathematical statements that show the equality between two expressions. In the realm of proportions, they play a vital role because a proportion is essentially an equation stating that two ratios are equivalent.

For instance, the equation \( \frac{17}{12} = \frac{x}{20} \) signifies that the two ratios \( \frac{17}{12} \) and \( \frac{x}{20} \) are equal. This equation is formed by setting two ratios against each other and finding the missing value that makes them equivalent. Here, an unknown variable, \( x \), is used in the equation, signifying a missing value to solve for.
  • Equations in proportions often involve unknowns that need to be solved.
  • They are a key tool in mathematics to express and solve real-world problems.
Proportional Relationships
Proportional relationships describe how two quantities increase or decrease at the same rate. When two ratios are equal, they form a proportion, which is the basis of solving proportional relationships. In the example '17 is to 12 as \( x \) is to 20', the proportional relationship is expressed as \( \frac{17}{12} = \frac{x}{20} \).

Recognizing proportional relationships is crucial in daily life鈥攆rom cooking recipes to scaling up models. They are helpful because they simplify complex comparative relationships into manageable equations.
  • When quantities are proportional, they will always have a constant ratio.
  • Understanding this concept allows for predictions of one quantity when given the other.

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Most popular questions from this chapter

Write each percent as a decimal and a fraction. Spending for advertisements on the Internet in 2016 increased by \(14.2 \%\) over spending in 2015.

If a number is increased by \(100 \%,\) how does the increased number compare with the original number? Explain your answer.

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

Solve. What percent of 800 is \(4 ?\)
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