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Chapter 6: Problem 26

Determine whether each statement is a proportion. $$\frac{13}{-19}=\frac{-65}{95}$$

Short Answer

Expert verified
The cross product of the first and the second ratio is \(1235\). Therefore, the statement represents a proportion because the cross products are equal.

Step by step solution

01

Identify the ratios and the cross products

In our case, the ratios are \(\frac{13}{-19}\) and \(\frac{-65}{95}\). Let's find the cross products. According to the proportion rule, we equate the product of the extremes (13 and 95) with the product of the means (-19 and -65).
02

Calculate the cross products

Calculate both the products: \(13 * 95\) and \(-19 * -65\).
03

Compare the cross products

If the products in Step 2 are equal, then the two ratios form a proportion. Otherwise, they do not.

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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 often used to solve problems involving proportions. It allows us to compare two ratios by multiplying across the equals sign. When you have a proportion, like \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication involves multiplying \(a\) by \(d\) and \(b\) by \(c\), and then comparing these products.
  • Imagine having two fractions equal to each other.
  • Multiply the numerator of the first fraction by the denominator of the second fraction.
  • Do the same for the other numerator and denominator combination.
  • If these two products are equal, the fractions form a valid proportion.
This method is particularly handy as it simplifies the process of checking if two given fractions or ratios are truly proportionate, eliminating the need to reduce them to a common denominator.
Ratios
A ratio represents a relationship between two numbers showing how many times the first number contains the second. Ratios can be expressed in various forms: fractions, 1:2, or even by using the word "to" as in "1 to 2".
  • They are useful for comparing sizes, quantities, or amounts.
  • Can be simplified just like fractions.
  • Offer a straightforward way to represent relative amounts.
When working with ratios in algebra, understanding how they differ from fractions is essential. Ratios are inherently a comparison, whereas fractions represent parts of a whole. This concept becomes important when dealing with proportions, as proportions are simply statements of equality between two ratios.
Proportion Rule
The proportion rule is central to solving many algebra problems. It states that two ratios are equal if the cross products are equal. This rule can help in various scenarios such as scaling recipes or altering map scales.
  • If two fractions are equal, they form a proportion.
  • It helps in verifying proportions or solving for missing values.
  • The rule is foundational in subjects like geometry and physics, where proportions are common.
To apply this rule effectively, always set up your ratios correctly and perform cross-multiplication with care. When a proportion is confirmed, it means that the two ratios are equitably related, maintaining consistency across whatever units or quantities are being compared.

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Most popular questions from this chapter

A recipe for spaghetti sauce requires four 16 -ounce bottles of catsup to make two gallons of sauce. How many bottles of catsup are needed to make 10 gallons of sauce?

Comparative shopping A 30-pound bag of fertilizer costs \(\$ 12.25,\) and an 80-pound bag costs \(\$ 30.25 .\) Which is the better buy?

Set up and solve a proportion. An airplane ascends 100 feet as it flies a horizontal distance of 1,000 feet. The two triangles in the illustration are similar. How much altitude will it gain as it flies a horizontal distance of 1 mile? (Hint: 5,280feet = 1 mile.)

Let \(x\) equal a number of your choosing. Without simplifying first, use a calculator to evaluate $$\frac{x^{2}+x-6}{x^{2}+3 x} \cdot \frac{x^{2}}{x-2}$$ Try again, with a different value of \(x\). If you were to simplify the expression, what do you think you would get?

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{45}{9 a}$$
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