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Chapter 2: Problem 28

Decide whether each proportion is true or false. $$ \frac{\frac{1}{3}}{6}=\frac{1}{18} $$

Short Answer

Expert verified
True. The proportion is \(\frac{1}{18} = \frac{1}{18}\).

Step by step solution

01

Simplify the left-hand side

Simplify the left-hand side of the proportion which is \(\frac{\frac{1}{3}}{6}\). To divide by a fraction, multiply by its reciprocal. This can be rewritten as \(\frac{1}{3} \times \frac{1}{6}\).
02

Multiply the fractions

Multiply the fractions \(\frac{1}{3} \times \frac{1}{6}\). To do this, multiply the numerators together and the denominators together: \(\frac{1 \times 1}{3 \times 6} = \frac{1}{18}\).
03

Compare both sides

Now that the left-hand side is simplified to \(\frac{1}{18}\), compare it to the right-hand side which is also \(\frac{1}{18}\). Since both sides are equal, the proportion is true.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form. A fraction is simplified when the numerator (top number) and the denominator (bottom number) have no common factors other than 1. To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD).

For example, to simplify \(\frac{4}{8}\), identify the GCD of 4 and 8, which is 4. Divide both by 4 to get \(\frac{1}{2}\). This simplified fraction represents the same value using smaller numbers. Simplifying fractions makes calculations easier and more understandable.
Multiplying Fractions
Multiplying fractions might seem tricky, but it's straightforward if you follow the steps. To multiply two fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together.

For example, if we multiply \(\frac{1}{3} \times \frac{1}{6}\), we do \(\frac{1 \times 1}{3 \times 6} = \frac{1}{18}\). Multiply the top numbers: 1 and 1 which equals 1. Then, multiply the bottom numbers: 3 and 6 which equals 18.

Using this method makes complex fraction calculations manageable and ensures accurate results.
Comparing Fractions
Comparing fractions involves determining if one fraction is greater than, less than, or equal to another fraction. The key is to have a common denominator or convert fractions to decimals.

For instance, to compare \(\frac{1}{3}\) and \(\frac{1}{4}\), convert both fractions to have a common denominator. Multiply the denominators: 3 and 4 to get 12. Convert \(\frac{1}{3}\) to \(\frac{4}{12}\) and \(\frac{1}{4}\) to \(\frac{3}{12}\). Now, compare the numerators: 4 is greater than 3, so \(\frac{1}{3}\) is greater than \(\frac{1}{4}\).

Understanding this process helps in solving proportion problems and ensures accuracy in mathematical comparisons.

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