Problem 20 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{28}{125} \div \frac{5}{2}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{56}{625} \).

Step 1: Identify the Problem

We have the expression \( \frac{28}{125} \div \frac{5}{2} \). To solve this division problem using multiplication, we need to replace the divisor by its reciprocal and then multiply.

Step 2: Find the Reciprocal of the Divisor

The divisor in this expression is \( \frac{5}{2} \). The reciprocal of \( \frac{5}{2} \) is \( \frac{2}{5} \). The reciprocal is found by swapping the numerator and the denominator.

Step 3: Replace Division with Multiplication

Rewrite the original problem by replacing the division with multiplication by the reciprocal of the divisor:\[\frac{28}{125} \div \frac{5}{2} = \frac{28}{125} \times \frac{2}{5}.\]

Step 4: Multiply the Fractions

To multiply two fractions, multiply the numerators together and the denominators together:\[\frac{28 \times 2}{125 \times 5} = \frac{56}{625}.\]

Step 5: Simplify the Result (If Necessary)

In this case, \( \frac{56}{625} \) is already in its simplest form because 56 and 625 have no common factors other than 1. Thus, the answer remains \( \frac{56}{625} \).

Key Concepts

Reciprocal of a Fraction

To grasp the concept of dividing fractions, understanding the 'reciprocal' is crucial. A reciprocal is what you get when you swap the numerator and the denominator of a fraction.
For example, if you have the fraction \( \frac{5}{2} \), its reciprocal is \( \frac{2}{5} \). Notice how we just flipped the positions of the two numbers. This switch is fundamental when dealing with division of fractions, as it turns the division problem into a multiplication one.

• Find the reciprocal of \( \frac{a}{b} \) by swapping it to \( \frac{b}{a} \).
• This allows you to replace division with multiplication.

Reciprocals are not just abstract concepts; they play an integral role in transforming and simplifying mathematical expressions involving division of fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves changing a fraction like \( \frac{28}{56} \) into a fraction like \( \frac{1}{2} \). The process involves finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by this number.
When you simplify, the fraction remains equal in value even though the numbers in it change. Simplified fractions are easier to understand and often needed for final answers in math problems.

• Identify the GCD of the numerator and the denominator.
• Divide both by the GCD to get the simplest form.

In the case of \( \frac{56}{625} \), in the provided example, it's already simplified because the numbers don't share a divisor greater than 1. Simplifying makes it easier to compare, add, or subtract fractions in other problems.

Multiplication of Fractions

Multiplying fractions is one of the simplest operations involving fractions, as it only requires multiplying across the numerators and denominators. Given two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), multiplying them involves the following steps:
First, multiply the numerators together to get a new numerator: \( a \times c \).

• For \( \frac{28}{125} \times \frac{2}{5} \), calculate \( 28 \times 2 = 56 \).

Next, multiply the denominators together to get a new denominator: \( b \times d \).

• Again, for our example, \( 125 \times 5 = 625 \).

This gives us a new fraction \( \frac{56}{625} \). If possible, simplify at this stage, ensuring that the fraction is in its simplest form. Multiplication is straightforward and avoids the need for finding common denominators, which is necessary in addition and subtraction.