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Chapter 4: Problem 19

Multiply the fractions, and simplify your result. $$\frac{-6}{23} \cdot \frac{9}{10}$$

Short Answer

Expert verified
The simplified product is \(\frac{-27}{115}\).

Step by step solution

01

Multiply the Numerators

To multiply the fractions, start by multiplying the numerators together: \(-6 \times 9.\) Thus, we have the numerator: \[-6 \cdot 9 = -54.\]
02

Multiply the Denominators

Next, multiply the denominators together: \(23 \times 10.\) This results in the denominator: \[23 \cdot 10 = 230.\]
03

Write the Fraction Product

Combine the results from Steps 1 and 2 into a single fraction: \[\frac{-54}{230}.\]
04

Simplify the Fraction

To simplify \(\frac{-54}{230}\), first find the greatest common divisor (GCD) of 54 and 230. The GCD is 2. Divide both the numerator and the denominator by their GCD:\[\frac{-54}{230} = \frac{-54 \div 2}{230 \div 2} = \frac{-27}{115}.\]

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Key Concepts

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

Fractions
Fractions represent a way of expressing numbers that are not whole. They contain two parts: the numerator and the denominator. The numerator is the top number, indicating how many parts we have, and the denominator is the bottom number, specifying how many equal parts something is divided into.
Understanding fractions can be greatly beneficial, especially when tackling tasks such as multiplying or simplifying them.
  • They allow us to express numbers that lie between whole numbers.
  • Fractions are crucial in helping represent values in different formats.
Learning to work with fractions effectively forms a strong foundation for more advanced mathematics.
Numerators and Denominators
When it comes to multiplying fractions, focusing on the numerators and denominators is essential.
  • **Numerator:** This is the top part of the fraction, representing the number of parts you have.
  • **Denominator:** This is the bottom part of the fraction, representing the number of total equal parts the whole is divided into.
In the example of multiplying \(\frac{-6}{23} \cdot \frac{9}{10},\) we simply focus on two tasks:- Multiplying the numerators: \(-6 \times 9 = -54\)- Multiplying the denominators: \(23 \times 10 = 230\)Combining these gives us the new fraction: \(\frac{-54}{230}.\)This intuitive approach makes managing fractions much more straightforward.
Simplifying Fractions
Simplifying fractions helps express them in their simplest form, making calculations easier and results more understandable. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.
In our example \(\frac{-54}{230},\) the GCD is 2.
  • Divide the numerator \(-54 \div 2 = -27\)
  • Divide the denominator \(230 \div 2 = 115\)
Putting these together gives us the simplified fraction: \(\frac{-27}{115}\).Recognizing when and how to simplify fractions is valuable because it helps with comparing fractions and performing further mathematical operations efficiently.

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Most popular questions from this chapter

Simplify the complex rational expression. $$\frac{-\frac{3}{7}-\frac{1}{3}}{\frac{1}{3}-\frac{6}{7}}$$

Solve the equation and simplify your answer. $$\frac{2}{5} x=-\frac{1}{6}$$

Solve the equation and simplify your answer. $$-\frac{7}{2} x-\frac{5}{4}=\frac{4}{5}$$

Solve the equation and simplify your answer. $$\frac{1}{2} x-\frac{8}{3}=-\frac{2}{5}$$

Solve the equation and simplify your answer. $$\frac{1}{2} x+\frac{1}{3}=-\frac{5}{2} x-\frac{1}{4}$$
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