Problem 31 - Elementary and Intermediate Algebra

Question

Determine whether each equation is a true proportion. $$ \frac{5}{8}=\frac{12}{19.4} $$

Step-by-Step Solution

Answer

The equation is not a true proportion.

Step 1: Understanding Proportions

To decide if the given equation \( \frac{5}{8} = \frac{12}{19.4} \) is a true proportion, we must check if these two fractions are equivalent. A proportion means both sides of the equation have the same value.

Step 2: Cross-Multiply the Fractions

To verify if two fractions form a true proportion, cross-multiply the numerators and denominators: multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. Calculate: \( 5 \times 19.4 = 97 \) and \( 12 \times 8 = 96 \).

Step 3: Compare the Products

Now, compare the results from the cross-multiplication. If the two products are equal, the fractions form a true proportion; otherwise, they do not. Here, \( 97 eq 96 \).

Step 4: Conclusion of Proportion Verification

Since 97 is not equal to 96, the cross-products are not the same, indicating that the given equation \( \frac{5}{8} = \frac{12}{19.4} \) is not a true proportion.

Key Concepts

Cross-Multiplication

Cross-multiplication is a simple yet powerful technique used to determine if two fractions form a true proportion.
When you have an equation that looks like this: \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves two basic steps:

• Multiply the numerator of the first fraction (\(a\)) by the denominator of the second fraction (\(d\)).
• Then, multiply the numerator of the second fraction (\(c\)) by the denominator of the first fraction (\(b\)).

After conducting these calculations, compare the two products you obtained. If both products are identical, then the fractions are indeed in proportion. Otherwise, they are not. In our example, we multiplied \(5\) by \(19.4\) resulting in \(97\), and \(12\) by \(8\) resulting in \(96\). Since \(97e96\), these fractions do not form a true proportion.

Fractions

Fractions represent parts of a whole, expressed as a ratio of two values: the numerator and the denominator. The numerator is the top number and tells you how many parts you have, while the denominator is the bottom number and tells you into how many parts the whole is divided.
For example, \(\frac{5}{8}\) is a fraction where 5 is the numerator and 8 is the denominator.
Fractions are crucial in mathematics because they allow us to deal with numbers that aren't whole numbers.

• Improper fractions: where the numerator is bigger than the denominator, like \(\frac{9}{4}\).
• Mixed numbers: a combination of a whole number and a fraction, like 2\(\frac{1}{2}\).

Understanding fractions is essential for learning more advanced math concepts like proportions. They are important in comparing quantities, determining equivalency, and performing arithmetic operations in various mathematical problems.

Equivalent Fractions

Equivalent fractions describe the same part of a whole using different numerators and denominators. They might look different, but they represent the same value.
An easy way to think about equivalent fractions is to visualize them as ways to slice the same pie. You might cut the pie into more slices, or fewer slices, and still have the same portion of the pie.
For instance, \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{4}{8}\) are all equivalent fractions because they all represent the same quantity of the whole. To check if two fractions are equivalent, you can use cross-multiplication

• If the cross-products are equal, the fractions are equivalent.
• If not, they are not equivalent fractions.

Understanding equivalent fractions is a fundamental concept in solving problems involving proportions and helps in simplifying calculations.