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Chapter 10: Problem 30

Divide. $$ -\frac{3}{5} \div\left(-\frac{5}{8}\right) $$

Short Answer

Expert verified
The result is \frac{24}{25}.

Step by step solution

01

Identify the Division and Convert to Multiplication

Dividing fractions can be simplified by converting the division into multiplication. For the given problem \(-\frac{3}{5} \div\left(-\frac{5}{8}\right)\), this becomes \(-\frac{3}{5}\ \times\ \left(-\frac{8}{5}\right)\). The second fraction is flipped from \(-\frac{5}{8}\) to \(-\frac{8}{5}\).
02

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together. This gives: \[ -\frac{3}{5} \times \left( -\frac{8}{5}\right) \ = \ \frac{(-3) \times (-8)}{(5) \times (5)} \ = \ \frac{24}{25} \]
03

Simplify the Expression

Notice that the negative signs will cancel each other out and result in a positive fraction. Therefore, \(-\frac{3}{5} \div \left( -\frac{5}{8} \right) \= \frac{24}{25}\).

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Key Concepts

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

Fraction Division
Fraction division is an essential math skill. When you divide one fraction by another, you actually multiply by the reciprocal of the second fraction. For example, in the given problem, \(-\frac{3}{5} \div \left(-\frac{5}{8}\right)\), you turn this into a multiplication problem: \(-\frac{3}{5} \times \left(-\frac{8}{5}\right)\). The reciprocal of \(-\frac{5}{8}\) is \(-\frac{8}{5}\). This step simplifies the operation to something familiar鈥攎ultiplication.
Always remember these steps:
  • Flip the second fraction, so the numerator becomes the denominator and vice versa.
  • Change the division sign to a multiplication sign.
Mastering this conversion will make fraction division much more manageable.
Multiplying Fractions
Multiplying fractions is much simpler than it sounds. Once you've converted a division problem into a multiplication problem, as we did with \(-\frac{3}{5} \times \left(-\frac{8}{5}\right)\), the next step is to multiply the numerators and then the denominators.
In this case, you multiply the numerators: \(-3)\ \times \left(-8\right)\) which equals \(24\). Then multiply the denominators: \(5 \times 5\) which equals \(25\). This gives us the fraction \(\frac{24}{25}\).
Key tips for multiplying fractions:
  • Multiply the top numbers (numerators) together.
  • Multiply the bottom numbers (denominators) together.
Multiplication of fractions is straightforward once this method is understood.
Simplifying Fractions
Simplifying fractions is crucial for making your final answer as clear and simple as possible. After you鈥檝e done the multiplication, you might end up with a fraction that can be simplified.
In our example, \(\frac{24}{25}\), there are no common factors between 24 and 25, so the fraction is already in its simplest form. But, in other cases, you may have to divide both the numerator and the denominator by their greatest common divisor (GCD) to simplify the fraction.
  • Check if the numerator and denominator have any common factors.
  • Divide both by the greatest common divisor.
By practicing this regularly, you'll get better at spotting which fractions can be simplified quickly.

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Most popular questions from this chapter

Divide. $$ \frac{48.6}{-3} $$

Simplify. $$ 256 \div(-32) \div(-4) $$

Simplify. $$ \begin{array}{l} 4^{3}+10 \cdot 20+ \\ 8^{2}-23 \end{array} $$

Multiply. \(-\frac{3}{5}\left(-\frac{2}{7}\right)\)

Multiply. \(4 \cdot(-4) \cdot(-5) \cdot(-12)\)
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