Problem 51 - Prealgebra

Question

Simplify each expression as much as possible. Show that multiplying 3 by 5 is the same as dividing 3 by \(\frac{1}{5}\)

Step-by-Step Solution

Answer

Multiplying 3 by 5 equals 15, and dividing 3 by \(\frac{1}{5}\) also equals 15, confirming equivalence.

Step 1: Understanding Multiplication

We begin by expressing the multiplication operation. We want to multiply 3 by 5, which is written as \(3 \times 5\). The result of this operation is straightforward: \(3 \times 5 = 15\).

Step 2: Understanding Division by a Fraction

Next, we express the division operation. We are dividing 3 by \(\frac{1}{5}\). According to the rules of division involving fractions, dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, dividing by \(\frac{1}{5}\) is the same as multiplying by 5. This gives us: \(3 \div \frac{1}{5} = 3 \times 5\).

Step 3: Evaluating Division by the Fraction

Now, we perform the operation from the divided expression. Since dividing by \(\frac{1}{5}\) is equivalent to multiplying by 5, we have \(3 \times 5\). We already calculated that \(3 \times 5 = 15\), confirming the equivalence.

Key Concepts

Multiplication

Multiplication is one of the four fundamental arithmetic operations. It involves adding a number to itself a certain number of times. For example, when you multiply 3 by 5, you add the number 3 together five times:

• 3 + 3 + 3 + 3 + 3 = 15

But instead of repeatedly adding, multiplication provides a quicker way to reach this result. Thus, we write this as:
\(3 \times 5 = 15\).
This process of grouping numbers simplifies what could otherwise be a long addition process. It’s powerful for working with large numbers or repeated addition, making calculations faster and simpler.
When learning about multiplication, remember it's about finding how many instances you have of a particular number, grouped together.

Division of fractions

The division of fractions might seem daunting at first, but it becomes simpler with the right approach. When you divide by a fraction, you're essentially finding out how many times a fraction fits into the whole.

• For instance, dividing by \(\frac{1}{5}\) is asking, "How many fifths are in the number?"

To ease this operation, we use the concept of the reciprocal.
Instead of dividing, you can multiply by the reciprocal of the fraction. This turns the division into a multiplication problem, which is often easier to solve. For example, dividing 3 by \(\frac{1}{5}\) becomes multiplying 3 by 5.
Thus, \[3 \div \frac{1}{5} \equiv 3 \times 5 = 15\] This equivalence shows how division by a fraction works perfectly with the rules of multiplication.

Reciprocal

A reciprocal is a special kind of number used in division and multiplication. To find the reciprocal of a fraction, you simply flip the numerator and the denominator. So, the reciprocal of \(\frac{1}{5}\) is 5, or \(\frac{5}{1}\), because flipping \(\frac{1}{5}\) results in \(\frac{5}{1}\).

• Reciprocals are vital in simplifying division with fractions.

When you divide by a fraction, using its reciprocal turns the division into multiplication, which is usually simpler. For example, rather than dividing directly by \(\frac{1}{5}\), you multiply by its reciprocal, 5. This approach ties back to some fundamental prealgebra rules:

• Knowing that any number times its reciprocal equals 1 (for example, \(\frac{5}{1} \times \frac{1}{5} = 1\)).

Reciprocals are like the mathematical tool that flips a problem around and makes it easier to handle!