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Chapter 7: Problem 58

In the following exercises, simplify. $$ \frac{7}{12} \cdot \frac{9}{17} \cdot \frac{24}{7} $$

Short Answer

Expert verified
The simplified form is \( \frac{18}{17} \).

Step by step solution

01

- Multiply the numerators

Multiply the numerators together. \[ 7 \times 9 \times 24 = 1512 \]
02

- Multiply the denominators

Multiply the denominators together. \[ 12 \times 17 \times 7 = 1428 \]
03

- Form the new fraction

Form the new fraction using the products of the numerators and denominators. \[ \frac{1512}{1428} \]
04

- Simplify the fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 1512 and 1428, which is 84. Then, divide both the numerator and the denominator by their GCD.\[ \frac{1512 \text{ (numerator)} \text{ divided by 84}}{1428 \text{ (denominator)} \text{ divided by 84}} = \frac{18}{17} \]

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Key Concepts

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

fraction multiplication
When multiplying fractions, the operation involves just a few simple steps. You multiply the numerators together and then multiply the denominators together. For example, if you have \(\frac{7}{12}\) and \(\frac{9}{17}\), you'd multiply the numerators: 7 and 9 to get 63. Then, you'd multiply the denominators: 12 and 17 to get 204. This process is essential because each fraction represents a part of a whole. By multiplying fractions, you're determining the product of these parts.
numerator and denominator multiplication
In our exercise, we need to multiply the numerators and denominators of the given fractions. The fractions provided are \(\frac{7}{12} \cdot \frac{9}{17} \cdot \frac{24}{7}\). Hence, we use the numerators (7, 9, and 24) and multiply them: 7 \times 9 \times 24 = 1512. Then, we take the denominators (12, 17, and 7) and multiply them: 12 \times 17 \times 7 = 1428. This gives us a new fraction \(\frac{1512}{1428}\). Multiplying these parts separately ensures that the structure of the fraction is maintained.
greatest common divisor
To simplify the fraction \(\frac{1512}{1428}\), we need to use the greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For our fraction, the GCD of 1512 and 1428 is 84. Finding the GCD can be done via methods like the Euclidean algorithm. Once identified, you divide both the numerator and the denominator by this GCD: 1512 divided by 84 is 18, and 1428 divided by 84 is 17. Thus, \(\frac{1512}{1428}\) simplifies to \(\frac{18}{17}\).
fraction simplification
The final goal in fraction multiplication problems is to simplify the fraction. Simplification makes the fraction easier to understand and use. After you've formed the new fraction by multiplying the numerators and denominators, you check if it can be simplified. That's when you find the GCD and divide both parts of the fraction. For instance, in \(\frac{1512}{1428}\), we found the GCD to be 84 and divided both numerator and denominator by 84. This reduced the fraction to \(\frac{18}{17}\). Simplifying fractions is crucial because it reduces the fraction to its simplest form, ensuring that it is as clear and concise as possible.

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