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Chapter 4: Problem 28

Multiply the fractions, and simplify your result. $$\frac{21}{20} \cdot \frac{-4}{5}$$

Short Answer

Expert verified
The simplified product is \(-\frac{21}{25}\).

Step by step solution

01

Multiply the Numerators

To multiply the fractions, start by multiplying the numerators. The numerators are 21 and -4. So calculate: \[ 21 \times (-4) = -84 \]
02

Multiply the Denominators

Next, multiply the denominators. The denominators are 20 and 5. Thus, we have: \[ 20 \times 5 = 100 \]
03

Write the Fraction Result

Combine the results from Step 1 and Step 2 to form the fraction: \[ \frac{-84}{100} \]
04

Simplify the Fraction

To simplify \( \frac{-84}{100} \), find the greatest common divisor (GCD) of 84 and 100, which is 4. Divide both the numerator and the denominator by their GCD: \[ \frac{-84 \div 4}{100 \div 4} = \frac{-21}{25} \]

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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 top number that tells us how many parts of the whole or group we are considering. In our exercise, the numerators are 21 and -4 from the fractions \( \frac{21}{20} \) and \( \frac{-4}{5} \), respectively. When multiplying fractions, we multiply the numerators together to find the numerator of the product.
So, in our example, we multiply 21 by -4 to get -84. This means the fraction of the product has -84 as its numerator, showcasing how many parts are being considered in this calculation.
Denominators
The denominator is the bottom number in a fraction and indicates how many equal parts the whole is divided into. In the fractions \( \frac{21}{20} \) and \( \frac{-4}{5} \), the denominators are 20 and 5, respectively. When multiplying fractions, multiplying the denominators gives you the denominator of the resulting fraction.
This action combines the divisions of the whole into a larger number of parts. For our exercise, multiplying the denominators 20 and 5 gives 100, meaning the whole has been divided into 100 parts for this calculation.
Simplifying Fractions
Simplifying a fraction involves making it as simple or as small as possible by ensuring the numerator and the denominator share no common factors other than 1. After obtaining the new fraction from multiplication, we often need to simplify it to make it easier to interpret.
Our resulting fraction from the multiplication was \( \frac{-84}{100} \). At this stage, it is clear both numbers can be reduced by identifying a common factor. Simplifying is a crucial skill in mathematics as it helps in recognizing equivalent fractions and making complex numbers manageable.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is a critical step in simplifying fractions. To simplify \( \frac{-84}{100} \), we need to find the GCD of 84 and 100.
To find this, we note the factors of each number, and determine 4 is the largest number that evenly divides both. Dividing both the numerator and denominator by this GCD, we arrive at \( \frac{-21}{25} \), which is the simplified form of our fraction. Understanding the concept of GCD helps in breaking down numbers, ensuring calculations are straightforward.

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