Problem 67 - Algebra 2

Question

PREREQUISITE SKILL Solve each proportion. $$ \frac{16}{v}=\frac{32}{9} $$

Step-by-Step Solution

Answer

The value of \( v \) is \( \frac{9}{2} \).

Step 1: Understand the Proportion

A proportion states that two ratios are equal. Here, we have the proportion \( \frac{16}{v} = \frac{32}{9} \). Our goal is to solve for \( v \).

Step 2: Cross-Multiply the Ratios

Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, equating the two products. For this equation, cross-multiplying gives us:\[ 16 \times 9 = 32 \times v \]

Step 3: Simplify the Product on the Left

Calculate \( 16 \times 9 \) to simplify the left side of the equation:\[ 16 \times 9 = 144 \]

Step 4: Formulate the Equation

Substitute the simplified product back into the equation from Step 2:\[ 144 = 32v \]

Step 5: Solve for v

To isolate \( v \), we need to divide both sides of the equation by 32:\[ v = \frac{144}{32} \]

Step 6: Simplify the Fraction

Simplify \( \frac{144}{32} \) by finding the greatest common divisor (GCD) of 144 and 32. The GCD is 16.\[ \frac{144}{32} = \frac{144 \div 16}{32 \div 16} = \frac{9}{2} \]

Step 7: Final Solution

Thus, the value of \( v \) that satisfies the proportion is:\[ v = \frac{9}{2} \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve proportions where two ratios are equated. It's like tying a knot between the opposite parts of fractions. This method helps you derive an equation that's easier to handle.
For example, if you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication transforms it into a simple equation \( a \times d = b \times c \).

• This technique works because multiplying both sides of an equation by the same number keeps them equal.
• By setting these two products equal, we effectively eliminate the fractions, moving directly to multiplication.
• In our exercise, by cross-multiplying \(16 \times 9\) and \(32 \times v\), we shift from a complex fraction form to a standard equation, \(144 = 32v\).

Cross-multiplication is particularly handy when dealing with equations that involve fractions, offering a direct route to solving for unknown variables.

Ratios

Ratios are a way to compare two quantities. They're expressed as fractions, like \( \frac{a}{b} \), meaning 'a' parts compared to 'b' parts. Ratios help us understand proportionate relationships in various contexts.

• A ratio remains the same when both parts are multiplied or divided by the same number. This is why you can scale up or down without altering the essence.
• In the exercise, the ratio \( \frac{16}{v} \) and \( \frac{32}{9} \) imply two quantities differ in the same manner.

Recognizing and setting up these ratios is key to solving equations where different values are compared, as shown when solving proportion problems. Understanding that two ratios form a proportion allows you to see the relationships between quantities and solve for unknowns effectively.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its lowest terms, which means making the numerator and denominator as small as possible without changing its value. This is done by dividing both by their greatest common divisor (GCD).

• Finding the GCD ensures the fraction retains the same ratio in its simplest form. For instance, to simplify \( \frac{144}{32} \), you determine that their GCD is 16.
• Divide both terms by the GCD: \( \frac{144 \div 16}{32 \div 16} = \frac{9}{2} \).

In the exercise, simplifying \( \frac{144}{32} \) was the last step to find \( v \) in its simplest form, \( \frac{9}{2} \). By simplifying, we not only make fractions easier to interpret but also easier to work with in future calculations. Simplification is crucial for clarity and accuracy when dealing with mathematical operations.