Questions: Determine whether the statement makes sense or does not make sense, and explain your reasoning. I can solve x/9 = 4/6 by using the cross-products principle or by multiplying both sides by 18, the least common denominator. Choose the correct answer below. A. The statement does not make sense. Multiplying the terms on both sides of the equation by the least common denominator missing quantity in a proportion cannot be found by using the cross-products principle if only three of the numbers are B. The statement makes sense. If three of the numbers in a proportion are known, the value of the missing quantity can be terms on both sides of the equation by the least common denominator will eliminate the fractions in the equation. C. The statement does not make sense. If three of the numbers in a proportion are known, the value of the missing quantity both sides of the equation would need to be multiplied by 6 * 4=24, and not 18, to eliminate the fractions in the equation D. The statement makes sense. If two of the numbers in a proportion are known, the value of the missing quantities can be terms on both sides of the equation by the least common denominator will eliminate the fractions in the equation.

Solution Steps

To determine whether the statement makes sense, we need to verify if the given methods (cross-products principle and multiplying both sides by the least common denominator) can solve the equation \(\frac{x}{9} = \frac{4}{6}\).

1. Cross-Products Principle: This principle states that for the proportion \(\frac{a}{b} = \frac{c}{d}\), the cross-products \(a \cdot d\) and \(b \cdot c\) are equal. Applying this to \(\frac{x}{9} = \frac{4}{6}\), we get \(x \cdot 6 = 9 \cdot 4\).

2. Multiplying by the Least Common Denominator (LCD): The LCD of 9 and 6 is 18. Multiplying both sides of the equation \(\frac{x}{9} = \frac{4}{6}\) by 18 should eliminate the fractions.

For the equation \(\frac{x}{9} = \frac{4}{6}\), we apply the cross-products principle: \[ x \cdot 6 = 9 \cdot 4 \] This simplifies to: \[ 6x = 36 \] Solving for \(x\): \[ x = \frac{36}{6} = 6 \] Thus, the cross-products principle gives \(x = 6\).

The least common denominator (LCD) of 9 and 6 is 18. Multiplying both sides of the equation \(\frac{x}{9} = \frac{4}{6}\) by 18: \[ 18 \cdot \frac{x}{9} = 18 \cdot \frac{4}{6} \] This simplifies to: \[ 2x = 12 \] Solving for \(x\): \[ x = \frac{12}{2} = 6 \] Thus, multiplying by the LCD also gives \(x = 6\).

The statement claims that the equation \(\frac{x}{9} = \frac{4}{6}\) can be solved using either the cross-products principle or by multiplying both sides by the least common denominator. Both methods yield \(x = 6\), confirming the statement is correct.

Final Answer