Problem 11 - Prealgebra

Question

Solve each proportion. $$\frac{51}{z}=\frac{17}{7}$$

Step-by-Step Solution

Answer

z = 21.

Step 1: Understand Proportions

A proportion is an equation that states that two ratios are equivalent. In this problem, we have \( \frac{51}{z} = \frac{17}{7} \). Our goal is to solve for \( z \), making the two ratios equivalent.

Step 2: Cross Multiply

To solve the proportion, cross multiply the terms. Multiply the numerator of the first ratio by the denominator of the second ratio and set it equal to the numerator of the second ratio multiplied by the denominator of the first ratio. This gives us:\[ 51 \times 7 = 17 \times z \] Simplifying the left side gives us \( 357 = 17z \).

Step 3: Solve for z

Now, solve for \( z \) by dividing both sides of the equation from Step 2 by 17:\[ z = \frac{357}{17} \]Perform the division to find \( z = 21 \).

Step 4: Verify the Solution

Substitute \( z = 21 \) back into the original proportion to verify it:\( \frac{51}{21} = \frac{17}{7} \)Simplify \( \frac{51}{21} \) by dividing both the numerator and denominator by 3 to check equivalence:\( \frac{51}{21} = \frac{17}{7} \), confirming that \( z = 21 \) is correct.

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method to solve equations that involve proportions. A proportion consists of two ratios set equal to each other. It tells us that the products of the inner terms (means) are equal to the products of the outer terms (extremes). For example, in the proportion \( \frac{51}{z} = \frac{17}{7} \), the cross multiplication method comes into play by multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. We set these two products equal to each other:

• First product: \( 51 \times 7 \)
• Second product: \( 17 \times z \)

This gives us the equation \( 51 \times 7 = 17 \times z \). Simplifying this results in \( 357 = 17z \). Cross multiplication is especially handy as it removes the fractions, allowing us to deal with a straightforward linear equation.

Solving Equations

Once you've cross-multiplied and come up with a new equation, the next step is solving for the unknown variable, which in this case is \( z \). From our previous step, we arrived at the equation \( 357 = 17z \). To isolate \( z \), you perform division, a key operation in the process of solving equations.
We divide both sides of the equation by 17:

• Left side: \( \frac{357}{17} \)
• Right side: \( \frac{17z}{17} \), which simplifies to \( z \)

After performing the division, we find that \( z = 21 \). This means our initial assumption about the proportion was correct, which implies that both sets of numbers maintain their ratio when \( z \) equals 21. This stage is crucial as it directly pinpoints the value we are solving for.

Verifying Solutions

After calculating a solution, verification is a critical step to ensure accuracy. This helps confirm that our solution satisfies the original equation or proportion. In this instance, we substitute \( z = 21 \) back into the original proportion \( \frac{51}{z} = \frac{17}{7} \) to check for equivalency.
Replace \( z \) by 21:

• We have: \( \frac{51}{21} \)
• and need to compare it to: \( \frac{17}{7} \)

Simplifying \( \frac{51}{21} \) by finding the greatest common divisor for 51 and 21, which is 3, gives us \( \frac{17}{7} \). Since both values simplify to the same ratio of \( \frac{17}{7} \), it verifies our solution is indeed correct. Verification reassures the accuracy of our solution and demonstrates the validity of using cross multiplication in solving proportions effectively.