Problem 30 - Elementary and Intermediate Algebra

Question

Determine whether each equation is a true proportion. $$ \frac{7}{16}=\frac{3}{7} $$

Step-by-Step Solution

Answer

The equation is not a true proportion.

Step 1: Understand the concept of a proportion

A proportion is an equation stating that two ratios are equal. To determine if the given equation is a true proportion, we need to verify that the two ratios are equivalent.

Step 2: Set up the cross-multiplication

To check if \( \frac{7}{16} = \frac{3}{7} \) is a true proportion, use the method of cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the denominator of the first fraction by the numerator of the second fraction.

Step 3: Perform the cross-multiplication

Calculate \( 7 \times 7 \) and \( 16 \times 3 \). This gives us:\( 7 \times 7 = 49 \)\( 16 \times 3 = 48 \)

Step 4: Compare the products

Compare the results from Step 3. Since 49 is not equal to 48, the products of the cross-multiplication are not equal.

Step 5: Conclusion

Since the results of the cross-multiplication are not equal, \( \frac{7}{16} = \frac{3}{7} \) is not a true proportion.

Key Concepts

Cross-Multiplication

Cross-multiplication is an essential tool when working with proportions. It's a straightforward method to determine whether two fractions form a true proportion. The term "cross-multiplication" might sound complex, but it simply involves multiplying across the diagonal of two fractions. Here's how it works.

When you have a proportion, like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves the following steps:

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

If the products of these multiplications are equal, then the original fractions form a true proportion. This technique is both simple and effective for confirming the equality of two ratios.

Ratios

Ratios are a way to compare two quantities by division. They are often expressed in the form \( a:b \) or \( \frac{a}{b} \). Understanding ratios is key to grasping proportions since a proportion is simply an equation showing equal ratios.

Imagine you have two quantities, such as 7 and 16; their ratio can be written as \( \frac{7}{16} \). This means for every 7 of one quantity, you have 16 of the other. In our original exercise, the ratios \( \frac{7}{16} \) and \( \frac{3}{7} \) were compared by determining whether they represent the same relationship.

• A helpful way to think about ratios is in terms of "how much of one thing relative to another?"
• Proportions help us find out if two different ratios express the same relationship between quantities.

Grasping the concept of ratios makes solving real-world problems involving comparisons much easier.

Equivalent Fractions

Equivalent fractions are fractions that represent the same portion of a whole, even if their numerators and denominators differ. It's crucial for understanding proportions, as two fractions in a proportion must be equivalent for it to be true.

To find out if two fractions are equivalent, you can use the idea of cross-multiplication as discussed earlier. However, you can also use simplification or finding a common denominator.

• Simplifying involves dividing the numerator and the denominator by their greatest common divisor.
• Finding a common denominator can also reveal equivalence by comparing numerators once both fractions have the same denominator.

When two fractions are equivalent, it means they slice the whole into a different number of parts, but still cover the same amount. Understanding this concept will help determine accurate proportions and solve exercises like converting recipes or adjusting quantities efficiently.