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Chapter 5: Problem 91

Perform the indicated operation. \(\frac{3}{5} \cdot \frac{7}{10}\)

Short Answer

Expert verified
The result of the operation is \( \frac{21}{50} \).

Step by step solution

01

Multiply the Numerators

To multiply two fractions, we begin by multiplying their numerators. Here, multiply the numerators 3 and 7 to get the product:\[ 3 \times 7 = 21 \]
02

Multiply the Denominators

Next, multiply the denominators of the two fractions 5 and 10 to find the product of the denominators:\[ 5 \times 10 = 50 \]
03

Form the Fraction with the Results

Now that we have multiplied the numerators and denominators, we form a new fraction with these results. The numerator is 21 and the denominator is 50, so the new fraction is:\[ \frac{21}{50} \]
04

Simplify the Fraction (if possible)

Check if the fraction \( \frac{21}{50} \) can be simplified by finding the greatest common divisor (GCD) of 21 and 50. Since 21 and 50 have no common factors apart from 1, the fraction \( \frac{21}{50} \) is already in its simplest form.

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Key Concepts

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

Numerators
In a fraction, the numerator is the number located above the fraction bar. It essentially represents how many parts of the fraction's whole are being considered. For instance, in the fraction \( \frac{3}{5} \), the '3' is the numerator. This indicates that 3 parts of a 5-part whole are under consideration. When multiplying fractions, it's important to multiply the numerators of the fractions together to find the new numerator of the resulting product.
By doing this:
  • First, identify the numerators in each fraction.
  • Then, multiply these numbers directly.
In our exercise, multiplying the numerators 3 and 7 gives us 21, forming the top part of the new fraction.
Denominators
The denominator is the number below the fraction bar. It defines how many parts the whole is divided into. In the fraction \( \frac{3}{5} \), the '5' is the denominator, showing that the whole is divided into 5 equal parts. When multiplying fractions, the denominators of each fraction must also be multiplied together to determine the denominator of the resulting product.
To correctly perform this operation:
  • Identify the denominators in each fraction.
  • Multiply these denominators to find the new denominator.
In the given exercise, by multiplying the denominators 5 and 10, we reach 50. This becomes the denominator of the fraction \( \frac{21}{50} \).
Simplifying Fractions
Simplifying fractions, or reducing them, involves dividing the numerator and the denominator by their greatest common divisor (GCD). This makes the fraction as simple as possible鈥攂ut still equivalent鈥攊n value. Simplification is important as it makes fractions easier to work with and understand.
If you encounter a fraction:
  • Identify common factors in the numerator and denominator.
  • Divide both by the greatest of these common factors.
In the case of the fraction \( \frac{21}{50} \), we check for common factors. Since no number other than 1 divides both 21 and 50 evenly, the fraction remains \( \frac{21}{50} \), already in its simplest form.

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