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Chapter 6: Problem 403

Determine whether each equation is a proportion. $$ \frac{16}{7}=\frac{48}{23} $$

Short Answer

Expert verified
\( \frac{16}{7} eq \frac{48}{23} \) because the cross-products are not equal, so the equation is not a proportion.

Step by step solution

01

- Understand Proportions

A proportion means that two ratios or fractions are equal. To determine if \( \frac{16}{7} = \frac{48}{23} \) is a proportion, test if the cross-products are equal.
02

- Calculate Cross-Products

Set up the cross-products for the given fractions: \( 16 \times 23 \) and \( 7 \times 48 \). Calculate these: \( 16 \times 23 = 368 \) and \( 7 \times 48 = 336 \).
03

- Compare Cross-Products

Compare the cross-products calculated in step 2: \( 368 \) and \( 336 \). Since 368 is not equal to 336, the given fractions are not in proportion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Products
To determine if two fractions form a proportion, we use the method of cross-products. This involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. If these products are equal, then the fractions are proportional. For example in our exercise, you need to compare:
\(16 \times 23 = 368\) and \(7 \times 48 = 336\). Since \(368 e 336\), the fractions \( \frac{16}{7}\) and \( \frac{48}{23} \) do not form a proportion.
Equal Fractions
Proportions are essentially equations declaring that two ratios are equal. To check whether two ratios are equal, you can compare their fractions. If we simplify the two ratios and they turn out to be the same, they are in proportion.
Another way to test for equal fractions is to cross-multiply, as mentioned before:
  • If the cross-products are equal, the fractions are proportionate.
  • If the cross-products are not equal, the fractions are not proportionate.
This method provides a simple way to test for proportionality without the need to simplify the fractions.
Ratios
Ratios compare two quantities by division. For instance, the ratio 16:7 compares 16 units to 7 units. When ratios are expressed as fractions, they can be compared to determine if they form a proportion.
A proportion is an equation of the form \( \frac{a}{b} = \frac{c}{d} \), where \(a, b, c, \) and \(d\) are numbers.
To decide if two ratios are proportional, their cross-products must be equal, which shows that one ratio can be scaled by a constant to get the other. For example, to check if \( \frac{16}{7} \) and \( \frac{48}{23} \) are proportional, compare their cross-products. Since \( 368 e 336 \), they are not proportional.

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Most popular questions from this chapter

In the following exercises, solve the percent problem. Kiyoshi bought a television set on sale for \(\$ 899\) . The original price was \(\$ 1,200\) . Find: (a) the amount of discount (b) the discount rate (round to the nearest tenth of a percent)

Dimple gets paid \(\$ 3,200\) per month. She pays \(\$ 960\) a month for rent. What percent of her monthly pay goes to rent?

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In the following exercises, write each sentence as a proportion. 3 is to 8 as 12 is to 32

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