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Chapter 2: Problem 33

Determine whether each proportion is true or false. $$ \frac{120}{82}=\frac{7}{10} $$

Short Answer

Expert verified
False

Step by step solution

01

Cross-Multiply the Ratios

In order to determine if the proportion is true, cross-multiply the numbers in the ratios. This means multiplying the numerator of one ratio by the denominator of the other ratio. For our problem, multiply 120 by 10 and 82 by 7.
02

Perform the Multiplications

Calculate the products from the cross-multiplication: \(120 \times 10 = 1200\) and \(82 \times 7 = 574\).
03

Compare the Products

If the two products from Step 2 are equal, then the proportion is true. Otherwise, it is false. Here, compare 1200 and 574.
04

Determine the Result

Since 1200 is not equal to 574, the given proportion \( \frac{120}{82} = \frac{7}{10} \) is false.

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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 powerful technique used to determine if two fractions are equivalent. It involves multiplying the numerator of one fraction by the denominator of the other fraction and comparing the results. If these products are equal, the fractions are in proportion.

Here's how it works step-by-step:
  • Identify the numerators and denominators in the proportion.
  • Multiply the numerator of the first fraction by the denominator of the second fraction.
  • Multiply the numerator of the second fraction by the denominator of the first fraction.

For the given problem:
\(\frac{120}{82} = \frac{7}{10}\)

We cross-multiply to get:
\(120 \times 10 = 1200\)
and
\(82 \times 7 = 574\)
. Since 1200 does not equal 574, these fractions are not equivalent, and the proportion is false.
numerator and denominator
In a fraction, the number on top is called the numerator, and the number at the bottom is called the denominator.

Understanding these terms is vital for working with proportions.

The numerator represents how many parts of a whole are being considered, while the denominator indicates the total number of equal parts that make up the whole.

Consider the fraction \( \frac{7}{10} \). Here, 7 is the numerator and 10 is the denominator. This fraction tells us that we have 7 out of 10 equal parts.

When comparing fractions using cross-multiplication, the numerator and denominator play key roles. Their relationship helps us understand whether two fractions represent the same proportion.
true or false proportions
Determining whether a proportion is true or false involves comparing two fractions by cross-multiplying their numerators and denominators. A proportion is true if and only if the cross-multiplication of the numerators and denominators gives equal results.

For example, to check if \(\frac{120}{82} = \frac{7}{10}\) is a true proportion:
  • Cross-multiply: 120 multiplied by 10 equals 1200 and 82 multiplied by 7 equals 574.
  • Compare the results: 1200 is not equal to 574.

Since the products are not equal, we conclude that the given proportion is false.

Understanding this process helps simplify many algebra problems and confirms whether two ratios are truly equivalent.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set, and write it using interval notation. \(9<-3 x<15\)

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The sum of three times a number and 7 more than the number is the same as the difference of -11 and twice the number. What is the number?

Find the measure of an angle whose supplement measures \(39^{\circ}\) more than twice its complement.

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