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Chapter 2: Problem 25

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

Short Answer

Expert verified
The proportion is false because 1200 鈮� 574.

Step by step solution

01

Write the proportion as an equation

The given proportion is \(\frac{120}{82} = \frac{7}{10}\).
02

Cross multiply the fractions

To determine if the proportion is true or false, cross multiply the fractions: \(120 \times 10\) and \(82 \times 7\).
03

Perform the multiplications

Calculate each side of the cross multiplication: \(120 \times 10 = 1200\) and \(82 \times 7 = 574\).
04

Compare the results

Compare the two products obtained from the multiplication: \(1200\) and \(574\). Since \(1200 eq 574\), the proportion 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 method 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 products.

For example, given the proportion \(\frac{a}{b} = \frac{c}{d}\), cross multiplication involves calculating \(a \times d\) and \(b \times c\).

Here are the steps to follow:
  • Multiply the numerator of the first fraction by the denominator of the second fraction.
  • Multiply the denominator of the first fraction by the numerator of the second fraction.
  • Compare the two products obtained.

If the products are equal, the two fractions are equivalent. Otherwise, they are not. This method simplifies the comparison of fractions without needing to find a common denominator.
Fraction Comparison
Comparing fractions can be tricky, but cross multiplication simplifies the process significantly. It avoids the need for converting fractions to have the same denominator or converting them to decimals.

When given two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), and asked to determine if they are equal, follow these steps:
  • Use cross multiplication as described above.
  • Compare the resulting products.

If the cross products are equal, the fractions represent the same value. If not, they represent different values. This method is especially useful in solving problems quickly and efficiently without the need for more complex arithmetic.
Equation Solving
Equation solving involves finding the value of an unknown that makes the equation true. In proportion problems, you often work with equations that involve fractions.

Here's how it relates to the example given:
  • First, express the proportion as an equation: \(\frac{120}{82} = \frac{7}{10}\).
  • Next, use cross multiplication to rewrite the equation in a form that鈥檚 easier to compare: \(120 \times 10 = 82 \times 7\).
  • Solve each multiplication separately: \(1200\) and \(574\).
  • Finally, compare the results to see if the equation holds true.
In this problem, the results were not equal, leading to the conclusion that the proportion is false. Mastering equation solving is essential as it forms the basis for understanding more complex algebraic concepts later on.

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Most popular questions from this chapter

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Find the value of \(y\) when \((a) x=-2\) and \((b) x=4 .\) Sections 1.3 and 2.3 $$ y+3 x=8 $$
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