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Chapter 6: Problem 21

Determine whether the ratios form a proportion. $$\frac{5}{18}=\frac{4}{16}$$

Short Answer

Expert verified
The ratios \( \frac{5}{18} \) and \( \frac{4}{16} \) do not form a proportion.

Step by step solution

01

- Understand the Problem

To determine if the ratios form a proportion, compare the two ratios to see if they are equivalent. Equivalent ratios have the same value when simplified.
02

- Simplify each Ratio

Simplify both ratios. Start with the ratio on the left: \( \frac{5}{18} \) cannot be simplified further as 5 and 18 have no common factors other than 1. Now, simplify the ratio on the right: \( \frac{4}{16} \). Divide the numerator and the denominator by their greatest common factor, which is 4: \(\frac{4 \div 4}{16 \div 4} = \frac{1}{4} \).
03

- Compare the Simplified Ratios

After simplifying, the ratios are \( \frac{5}{18} \) and \( \frac{1}{4} \). Since \( \frac{5}{18} \) is not equal to \( \frac{1}{4} \), the original ratios \( \frac{5}{18} \) and \( \frac{4}{16} \) do not form a proportion.

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Key Concepts

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

equivalent ratios
When working with ratios, it's essential to determine if two ratios are equivalent. Equivalent ratios are ratios that express the same relationship between numbers.
For example, the ratios \ \( \frac{3}{6} \) and \ \( \frac{1}{2} \) are equivalent because if you simplify \ \( \frac{3}{6} \), you get \ \( \frac{1}{2} \).
A quick way to check for equivalent ratios is to cross-multiply and compare results. For example, for the ratios \ \( \frac{a}{b} \) and \ \( \frac{c}{d} \), check if \ \( a \times d = b \times c \).
If the products are equal, the ratios are equivalent.
simplifying fractions
Simplifying fractions makes it easier to identify equivalent ratios.
To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common factor (GCF).
Consider the fraction \ \( \frac{20}{30} \). The GCF of 20 and 30 is 10. So, you divide both by 10:
\[ \frac{20 \div 10}{30 \div 10} = \frac{2}{3} \]
The simplified fraction \ \( \frac{2}{3} \) is easier to work with and compare.
For complicated fractions, you may need to factor each number by its prime factors to find the GCF.
ratio comparison
To determine if two ratios form a proportion, you need to compare them directly.
You can do this by simplifying both ratios, as we did in the exercise. Once simplified, if the ratios are identical, they form a proportion.
Let鈥檚 look at the exercise: \ \( \frac{5}{18} \) and \ \( \frac{4}{16} \). First, simplify \ \( \frac{4}{16} \) to \ \( \frac{1}{4} \).
Next, compare \ \( \frac{5}{18} \) and \ \( \frac{1}{4} \). They are not the same, so these ratios do not form a proportion.
Cross-multiplying is another method. For \ \( \frac{5}{18} \) and \ \( \frac{1}{4} \), check if \ \( 5 \times 4 = 18 \times 1 \). Since \ \( 20 eq 18 \), they don't form a proportion.

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Most popular questions from this chapter

The ratio of a person's height to the length of the person's lower arm (from elbow to wrist) is approximately 6.5 to \(1 .\) Measure your own height and lower arm length. Is the ratio you get close to the average of 6.5 to \(1 ?\)

Refer to the table that shows the average spending per person for reading (books, newspapers, magazines, etc.) by age group. Write each ratio in lowest terms. (PICTURE NOT COPY) \begin{array}{|l|c|} \hline \text { Age Group } & \text { Annual Average (\$) } \\ \hline \text { Under 25 years } & 60 \\ \hline 25 \text { to } 34 \text { years } & 111 \\ \hline 35 \text { to } 44 \text { years } & 136 \\ \hline 45 \text { to } 54 \text { years } & 172 \\ \hline 55 \text { to } 64 \text { years } & 183 \\ \hline 65 \text { to } 74 \text { years } & 159 \\ \hline 75 \text { years and over } & 128 \\ \hline \end{array} Find the ratio of spending for the group under 25 years old to the spending for the group of 55 to 64 years old.

Determine whether the pairs of numbers are proportional. Are the numbers 48 and 18 proportional to the numbers 24 and \(9 ?\)

In a recent year, the rate of prostate disease in U.S. men was 118 cases per 1000 men. At that time, the state of Massachusetts had approximately \(2,500,000\) men. How many men in Massachusetts would be expected to have prostate disease?

Solve the proportion. Be sure to check your answers. $$\frac{12}{16}=\frac{3}{x}$$
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