Problem 29 - Elementary and Intermediate Algebra

Question

Determine whether each equation is a true proportion. $$ \frac{7}{3}=\frac{14}{6} $$

Step-by-Step Solution

Answer

Yes, \( \frac{7}{3} = \frac{14}{6} \) is a true proportion because both sides equal 42 when cross-multiplied.

Step 1: Understand the concept of proportion

A proportion states that two ratios are equal. For example, if \( \frac{a}{b} = \frac{c}{d} \), then \( a, b, c, \) and \( d \) are in proportion.

Step 2: Cross-multiply the ratios

To check if the equation \( \frac{7}{3} = \frac{14}{6} \) is a true proportion, cross-multiply: 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. This gives us: \( 7 \times 6 = 3 \times 14 \).

Step 3: Calculate both products

Calculate each product from the cross-multiplication: \( 7 \times 6 = 42 \) and \( 3 \times 14 = 42 \).

Step 4: Compare the products

Compare the results from Step 3: Since both products are equal (42 = 42), the equation is a true proportion.

Key Concepts

ratios

Ratios are a way to compare two quantities by using division. They tell us how much of one thing there is compared to another. For example, if you have 7 apples and 3 oranges, you might say the ratio of apples to oranges is 7 to 3. This can be written as \( \frac{7}{3} \). Ratios are useful because they provide a way to scale up or scale down quantities while keeping the proportion consistent.

• Ratios are often simplified, much like fractions. If both numbers in a ratio can be divided by the same amount, they should be simplified to see if they are truly comparable to another ratio.
• Understanding ratios is key when determining if two ratios form a true proportion.

In essence, when dealing with problems involving proportions, it's all about comparing these ratios and checking their equivalence.

cross-multiplication

Cross-multiplication is a method used to determine if two ratios form a proportion. When you have a proportion such as \( \frac{7}{3} = \frac{14}{6} \), cross-multiplying involves the following steps:

• Multiply the numerator of the first ratio (7) by the denominator of the second ratio (6).
• Multiply the denominator of the first ratio (3) by the numerator of the second ratio (14).

The process will look like this: \( 7 \times 6 \) and \( 3 \times 14 \).
By calculating these, you help verify if the two ratios are indeed equivalent.

• Cross-multiplication is invaluable in demonstrating the equality of two ratios.
• It avoids potential errors that can occur with decimal division or simplification.

true proportion

To determine if an equation represents a true proportion, you need to see if the two ratios are indeed equal after cross-multiplying their respective terms.
A true proportion means the two products obtained from cross-multiplying should be equal. From the example \( \frac{7}{3} = \frac{14}{6} \), cross-multiplication gives us both products as 42.

• If \( 7 \times 6 = 42 \) and \( 3 \times 14 = 42 \), then the two sides of the equation are balanced, confirming they are in true proportion.
• Thus, a true proportion signifies that the relationship between quantities is maintained consistently.

Recognizing true proportions helps in solving many real-world problems where scaling or adjusting quantities is needed while maintaining original relationships.