Problem 8 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$16 \div \frac{1}{8}$$

Step-by-Step Solution

Answer

The quotient is 128.

Step 1: Understand the Problem

We need to find the quotient of 16 divided by \( \frac{1}{8} \). In simpler terms, we are determining how many groups of \( \frac{1}{8} \) fit into 16.

Step 2: Replace the Divisor with its Reciprocal

The current division problem is \( 16 \div \frac{1}{8} \). To solve by multiplication, replace \( \frac{1}{8} \) with its reciprocal, which is 8.

Step 3: Multiply with the Reciprocal

Now, multiply 16 by 8. So, the expression becomes \( 16 \times 8 \).

Step 4: Calculate the Product

Perform the multiplication: \( 16 \times 8 = 128 \).

Step 5: Conclusion

The quotient, when dividing 16 by \( \frac{1}{8} \), is 128.

Key Concepts

Understanding Reciprocals

In mathematics, a reciprocal of a number is simply the number you multiply your original number by to get 1. In other words, the product of a number and its reciprocal is always 1. For instance:
\[a \times \frac{1}{a} = 1\]
Where \(a\) is never zero, because dividing by zero is undefined.

When dealing with fractions, the reciprocal is created by swapping the numerator and the denominator. For example:

• The reciprocal of \(\frac{1}{8}\) is 8 (because you swap the numerator 1 with the denominator 8).
• Similarly, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

This simple swap is a powerful tool in arithmetic and algebra.
Knowing how to find and use the reciprocal is key to solving division problems using multiplication. It allows you to transform division problems into multiplication problems, which can often be simpler to handle, as multiplication might be more straightforward than division for most people.

Multiplying after Finding the Reciprocal

Once you've figured out the reciprocal of the divisor, you can transform a division problem into a multiplication one. This is vital because multiplying is sometimes simpler than dividing. Here's how it works:

Take the original division problem, such as \(16 \div \frac{1}{8}\). Once you find the reciprocal of \(\frac{1}{8}\), which is 8, you then multiply the original number, 16, by this reciprocal. So the expression changes from:

• Division: \(16 \div \frac{1}{8}\)
• To Multiplication: \(16 \times 8\)

By performing this multiplication instead, you turn the problem into a simple calculation.

The result of this multiplication gives the same result as the division would have, because multiplying by the reciprocal is mathematically equivalent to dividing by the original number. So, \(16 \times 8 = 128\) gives us the same answer as dividing 16 by \(\frac{1}{8}\), highlighting the power of switching between division and multiplication when fractions are involved.

Grasping the Quotient

The quotient in a division problem is essentially the result you get after dividing one number by another. In this context, for the division = operation \(16 \div \frac{1}{8}\), the process led us to multiply \(16\) by the reciprocal of \(\frac{1}{8}\), which is \(8\). The result of this multiplication gives us our quotient.

The calculation looks like this:

• \(16 \times 8 = 128\)

This means that the quotient of \(16 \div \frac{1}{8}\) is 128. This tells us that there are 128 full groups of \(\frac{1}{8}\) in the number 16.

The conceptual understanding of why and how we use reciprocal to find the quotient underscores a fundamental arithmetic skill. The quotient isn't just the end result; it's a measure of how many times one number is contained within another, made more accessible through multiplication with reciprocals.