Problem 4 - Prealgebra

Question

For each of the following proportions, name the means, name the extremes, and show that the product of the means is equal to the product of the extremes. $$\frac{5}{3}=\frac{10}{16}$$

Step-by-Step Solution

Answer

The means are 3 and 10; the extremes are 5 and 16; the products are not equal, so it's not a true proportion.

Step 1: Identify the Proportion

The given proportion is \( \frac{5}{3} = \frac{10}{16} \).

Step 2: Name the Means

In a proportion \( \frac{a}{b} = \frac{c}{d} \), the means are the inner terms. So for \( \frac{5}{3} = \frac{10}{16} \), the means are 3 and 10.

Step 3: Name the Extremes

In the same proportion \( \frac{5}{3} = \frac{10}{16} \), the extremes are the outer terms. Thus, the extremes are 5 and 16.

Step 4: Calculate the Product of the Means

Multiply the means: \( 3 \times 10 = 30 \).

Step 5: Calculate the Product of the Extremes

Multiply the extremes: \( 5 \times 16 = 80 \).

Step 6: Verify the Proportion Condition

The product of the means should equal the product of the extremes. Since \( 30 eq 80 \), the original equation does not hold as a true proportion.

Key Concepts

Means

In any proportion, we observe a set of numbers that form a relationship of equality between two fractions. The concept of "means" refers to the numbers found in the middle of this proportional setup. It’s like looking at the center points in a face-off.

• If you have a general proportion represented as \( \frac{a}{b} = \frac{c}{d} \), the means are \( b \) and \( c \).
• Thus, in our example \( \frac{5}{3} = \frac{10}{16} \), the means are the inner numbers 3 and 10.

Understanding means helps us confirm whether an equation is genuinely a proportion by using the products of these numbers.

Extremes

The concept of "extremes" in a proportion is complementary to that of "means." Here, the extremes are the numbers positioned on the outer ends of the proportional equation.

• In a proportion \( \frac{a}{b} = \frac{c}{d} \), the extremes are \( a \) and \( d \).
• In the proportion \( \frac{5}{3} = \frac{10}{16} \), these extremes are the numbers 5 and 16.

Understanding extremes helps you verify the proportionality by focusing on the outer position of the numbers, emphasizing the check between products.

Cross-Multiplication

Cross-multiplication is a straightforward technique to check the validity of a proportion. By multiplying numbers in a cross pattern, it confirms if two ratios form a true proportion.

• In any proportion \( \frac{a}{b} = \frac{c}{d} \), you multiply falsely diagonally: \( a \times d \) and \( b \times c \).
• Using \( \frac{5}{3} = \frac{10}{16} \), cross-multiplied, we calculate \( 5 \times 16 \) and \( 3 \times 10 \).
• If these products are equal, the proportion holds true. Otherwise, as in our exercise, where 30 is not equal to 80, the proportion does not hold.

Cross-multiplication is effective for validating or negating proportions.

Fraction Equality

Understanding the equality between two fractions is crucial in evaluating if a proportion is balanced. Fraction equality signifies that the two sides share identical ratios.

• To confirm fraction equality, we rely on products of both means and extremes after cross-multiplication.
• If \( \frac{a}{b} = \frac{c}{d} \) holds true, then both \( a \times d = b \times c \) confirm the equality.
• In the example, even though \( \frac{5}{3} \) doesn’t equal \( \frac{10}{16} \) due to mismatched products of 30 and 80, investigating through this method showcases fraction inequality.

Getting a grasp of fraction equality aids significantly in algebraic manipulation and understanding ratios.