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Chapter 6: Problem 28

Determine whether the ratios form a proportion. $$\frac{4.7}{-7} \triangleq \frac{23.5}{-35}$$

Short Answer

Expert verified
Yes, the ratios form a proportion.

Step by step solution

01

- Simplify the First Ratio

Simplify the first ratio \(\frac{4.7}{-7}\). Both numerator and denominator can be divided by 0.1, simplifying it to \(\frac{47}{-70}\) which simplifies further to \(\frac{47 \div 23.5}{-70 \div 35} = \frac{2}{-3}\).
02

- Simplify the Second Ratio

Now simplify the second ratio \(\frac{23.5}{-35}\). Both numerator and denominator can be divided by 0.5, simplifying it to \(\frac{47}{-70}\) which simplifies further to \(\frac{47 \div 23.5}{-70 \div 35} = \frac{2}{-3}\).
03

- Compare the Simplified Ratios

After simplifying both ratios, we find that \(\frac{2}{-3}\) is the same for both the first ratio and the second ratio.
04

Conclusion

Since both simplified forms \(\frac{2}{-3}\) are equal, the ratios \(\frac{4.7}{-7}\) and \(\frac{23.5}{-35}\) form a proportion.

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Key Concepts

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

Simplifying Ratios
Simplifying ratios is the process of reducing them to their simplest form, similar to simplifying fractions. For instance, consider the ratio \(\frac{4.7}{-7}\). This means we are comparing 4.7 parts to -7 parts. By dividing both the numerator and the denominator by the same number, we reduce the ratio. In this case, both parts can be divided by 0.1. Doing so, we get: \(\frac{47}{-70}\). This simplification makes it easier to understand and compare the ratios.
Comparing Fractions
Comparing fractions involves determining if two fractions represent the same value. In the given exercise, we simplified \(\frac{4.7}{-7}\) and \(\frac{23.5}{-35}\) to their simplest forms: \(\frac{2}{-3}\). Since both fractions simplify to the same value, they are equivalent. To compare fractions, it鈥檚 essential to simplify them first and then see if they match. When fractions (or ratios) reduce to the same simplest form, they are considered equal.
Proportional Relationships
A proportional relationship exists when two ratios are equivalent. In our problem, we found that the ratios \(\frac{4.7}{-7}\) and \(\frac{23.5}{-35}\) are proportional because they both simplify to \(\frac{2}{-3}\). In a proportional relationship, if you multiply or divide the numerator and denominator of one ratio by the same number, you get the other ratio. This principle helps in identifying whether two ratios or fractions form a proportion.
Numerator and Denominator
The numerator is the top part of a fraction, and the denominator is the bottom part. In the ratio \(\frac{4.7}{-7}\), 4.7 is the numerator, and -7 is the denominator. Simplifying a ratio involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{23.5}{-35}\), the GCD is 0.5, reducing it to \(\frac{47}{-70}\) and finally to \(\frac{2}{-3}\). Understanding numerators and denominators helps in performing operations such as simplifying and comparing values.

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Most popular questions from this chapter

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