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Chapter 2: Problem 34

Determine whether each proportion is true or false. $$ \frac{27}{160}=\frac{18}{110} $$

Short Answer

Expert verified
The proportion is false.

Step by step solution

01

Cross-Multiplication

To determine whether the given proportion \( \frac{27}{160}=\frac{18}{110} \) is true, use the cross-multiplication method. Multiply the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second fraction.
02

Perform the Multiplications

Multiply 27 by 110 and then multiply 160 by 18: \[ 27 \times 110 = 2970 \] \[ 160 \times 18 = 2880 \]
03

Compare the Products

Compare the products from Step 2. Since 2970 is not equal to 2880: \[ 2970 eq 2880 \]
04

Determine the Truth Value

Since the cross-products are not equal, the given 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 quick way to determine if two fractions are equivalent. To use this method, you multiply the numerator of the first fraction by the denominator of the second fraction. Then you multiply the denominator of the first fraction by the numerator of the second fraction. If both products are the same, the proportions are equal. In the example \( \frac{27}{160} = \frac{18}{110} \), we perform the following multiplications: \[ 27 \times 110 = 2970 \] and \[ 160 \times 18 = 2880 \]. Because 2970 is not equal to 2880, the proportion is false.
Numerator
In a fraction, the numerator is the top number. It represents how many parts of a whole are being considered. For example, in the fraction \( \frac{27}{160} \), the numerator is 27. When performing cross-multiplication, the numerator of the first fraction is multiplied by the denominator of the second fraction. In our example, the numerator 27 is multiplied by the denominator 110 of the second fraction, giving: \[ 27 \times 110 = 2970 \].
Denominator
The denominator is the bottom number in a fraction. It shows the total number of equal parts the whole is divided into. For instance, in the fraction \( \frac{18}{110} \), the denominator is 110. During cross-multiplication, the denominator of the first fraction is multiplied by the numerator of the second fraction. Here, the denominators are 160 and 110. In our solution, multiplying 160 (denominator of the first fraction) by 18 (numerator of the second fraction) we get: \[ 160 \times 18 = 2880 \].
Truth Value
The truth value tells us if a statement or proportion is true or false. By comparing the products obtained from cross-multiplication, we can determine the truth value of a proportion. In the exercise, we've calculated \[ 27 \times 110 = 2970 \] and \[ 160 \times 18 = 2880 \]. Since 2970 is not equal to 2880, the proportion \( \frac{27}{160} = \frac{18}{110} \) is false. Hence, the truth value of this proportion is false.

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