Problem 31 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-\frac{3}{5} \div-\frac{5}{9}$$

Step-by-Step Solution

Answer

The quotient is \(\frac{27}{25}\).

Step 1: Understand Division of Fractions

When you divide by a fraction, it is equivalent to multiplying by its reciprocal. Therefore, \[-\frac{3}{5} \div -\frac{5}{9}\] can be rewritten as \[-\frac{3}{5} \times -\frac{9}{5}.\]

Step 2: Multiply the Fractions

Multiply the numerators together and the denominators together. For \[-\frac{3}{5} \times -\frac{9}{5},\] we multiply the numerators: \[-3 \times -9 = 27\] and the denominators: \[5 \times 5 = 25.\] So, the product is \[\frac{27}{25}.\]

Step 3: Simplify the Fraction

The fraction \(\frac{27}{25}\) is already in its simplest form since 27 and 25 have no common factors other than 1. Hence, \[\frac{27}{25}\] is the simplified form of the quotient.

Key Concepts

Reciprocal of a Fraction

When we're dividing fractions, understanding the concept of a reciprocal is essential. The reciprocal of a fraction is what you get when you flip the numerator and the denominator. For example, the reciprocal of \(-\frac{5}{9}\) is \(-\frac{9}{5}\).

This flipping operation turns the fraction upside down.

• If your original fraction has a negative sign, your reciprocal will maintain that sign.
• It's key to remember that multiplying any number by its reciprocal equals 1. This property helps us when we switch from division to multiplication.

So, if you have a fraction \(-\frac{a}{b}\), its reciprocal will be \(-\frac{b}{a}\).

Whenever you divide by a fraction, you multiply by its reciprocal. This simplification makes the operation easier.

Multiplication of Fractions

Once you've changed the division to multiplication using the reciprocal, the next step is straightforward: multiply the fractions.

• Multiply the numerators—these are the top numbers—for example, \(-3\) and \(-9\) become \(27\).
• Multiply the denominators, the bottom numbers, so \(5 \times 5\) results in \(25\).

Thus, the product of \(-\frac{3}{5}\) and \(-\frac{9}{5}\) is \(\frac{27}{25}\).

Multiplying fractions is a simple and rule-governed step because you do not need to find a common denominator. Just multiply across and you're done!

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its smallest form. This means ensuring that the numerator and the denominator have no common factors other than 1.

• In our exercise, \(27\) and \(25\) have no common factors except 1, meaning \(\frac{27}{25}\) is already in its simplest form.
• A quick way to check is to perform a divisibility test: see if both numbers are divisible by any prime numbers like 2, 3, or 5.

If a fraction is non-simplified, continue dividing by common factors until no more exist. Simplifying helps us better understand and work with the fraction in problem-solving.