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Chapter 3: Problem 9

Determine if each proportion is true or false. See Example 1 . $$\frac{42}{77}=\frac{6}{11}$$

Short Answer

Expert verified
The proportion is true since \(462 = 462\). Thus, \(\frac{42}{77} = \frac{6}{11}\).

Step by step solution

01

Write down the given proportions

We are given the proportions: \(\frac{42}{77} = \frac{6}{11}\)
02

Cross-multiply

To check if these two proportions are equal, we cross-multiply: (42)(11) ?= (77)(6)
03

Compute the products

Now, we compute the products on both sides: 462 ?= 462
04

Compare the products and determine if the proportion is true or false

Since the products are equal, the proportion is true: \(\frac{42}{77} = \frac{6}{11}\)

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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 helpful technique for verifying if two fractions form a true proportion. A proportion states that two fractions are equal. To use cross-multiplication, you multiply the numerator of one fraction by the denominator of the other fraction. This is done for both sides of the equation.

For example, in the proportion \(\frac{42}{77} = \frac{6}{11}\), you multiply 42 by 11 and 77 by 6.
  • Compute: \(42 \times 11\) and \(77 \times 6\)
If the two products are equal, the proportion is true. If they are not equal, the proportion is false. This method simplifies comparison by transforming it into a straightforward multiplication problem, ensuring easy verification.
Solving Proportions
Solving proportions involves checking if two ratios are equivalent. Ratios compare two numbers, much like a fraction.

To solve a proportion such as \(\frac{42}{77} = \frac{6}{11}\), you can use cross-multiplication as a tool. By doing this, you're confirming whether these two ratios are equal. If the results from cross-multiplying are equal, you've established that the proportion holds true.

Finding the equality of proportions helps in many real-world applications, such as predicting outcomes and calculating quantities. By mastering this concept, you can solve many problems efficiently.
Fractions
Fractions represent a part of a whole and consist of a numerator and a denominator. For the fraction \(\frac{42}{77}\), 42 is the numerator, and 77 is the denominator.

Understanding fractions is crucial when dealing with proportions, as proportions are essentially equations that set two fractions equal to each other.
  • The fraction \(\frac{6}{11}\) can be seen as six parts of a whole that is divided into eleven parts.
  • Similarly, \(\frac{42}{77}\) represents 42 parts of a whole of 77 parts.
The concept of fractions is foundational in understanding and solving proportions by expressing different parts and comparing them using mathematical operations like cross-multiplication.
Equivalence
Equivalence in mathematics means that two expressions represent the same value or amount. With proportions, we're testing the equivalence of two fractions.

For example, if \(\frac{42}{77}\) is equivalent to \(\frac{6}{11}\), it means they represent the same ratio or relationship between numbers. This is checked using cross-multiplication, as shown in the example:
  • Calculate: \(42 \times 11 = 462\)
  • Calculate: \(77 \times 6 = 462\)
Since both calculations equal 462, the fractions are indeed equivalent. Establishing equivalence helps confirm the truth of a proportion and is vital in ensuring accurate interpersonal comparisons in mathematics.

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Most popular questions from this chapter

Solve each equation by first clearing fractions or decimals. $$\frac{a}{8}-1=\frac{a}{3}-\frac{7}{12}$$

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) Three times a number is fifteen more than half the number. Find the number.

Solve each formula for the indicated variable. $$F=m a\( for \)m$ (physics)$$

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Solve each equation and check your answer. $$2(9 z-1)+7=10 z-14+8 z+2$$
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