Problem 45 - Prealgebra

Question

The following problems review material from a previous section. Reviewing these problems will help you with the next section. Write as a decimal. $$\frac{88}{0.5}$$

Step-by-Step Solution

Answer

The decimal is 176.

Step 1: Understand the Problem Formulation

We are given the fraction \( \frac{88}{0.5} \). Our goal is to convert this fraction to a decimal.

Step 2: Identify the Process to Simplify the Fraction

To simplify \( \frac{88}{0.5} \), we need to divide 88 by 0.5. Dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 3: Find the Reciprocal of the Denominator

The reciprocal of \( 0.5 \) is \( \frac{1}{0.5} = 2 \).

Step 4: Perform the Multiplication

Multiply the numerator by the reciprocal of the denominator: \( 88 \times 2 = 176 \).

Step 5: Write the Result as a Decimal

The result of the multiplication, 176, is already in decimal form because it is a whole number.

Key Concepts

Reciprocals

The concept of a reciprocal is crucial in mathematics, especially when working with fractions. A reciprocal is essentially a flipped version of a fraction. It allows us to change operations from division to multiplication, which is often easier to compute. Consider a simple fraction \( \frac{a}{b} \). Its reciprocal is \( \frac{b}{a} \).
For a whole number such as 5, its reciprocal is \( \frac{1}{5} \). Similarly, for a decimal like 0.5, which can be expressed as the fraction \( \frac{1}{2} \), the reciprocal is \( \frac{2}{1} \) or simply 2.
Using reciprocals makes arithmetic more straightforward, leading to simpler and quicker calculations. Trust in the power of reciprocals to convert complex division operations into simple multiplication tasks!

Multiplication

Multiplication is one of the most basic yet powerful operations in mathematics. It allows us to increase numbers by repeated addition. When multiplying fractions, we multiply across the numerators and denominators.
For example, with \( \frac{3}{4} \) and \( \frac{2}{5} \), you multiply: \( 3 \times 2 = 6 \) and \( 4 \times 5 = 20 \), resulting in \( \frac{6}{20} \).
But when dealing with a whole number or a decimal, such as in the problem \( 88 \times 2 \), it's straightforward. Multiply directly: \( 88 \times 2 = 176 \). This shows how multiplication, particularly when connected with reciprocals, converts a potentially tricky division into a simple calculation.

Division by Fractions

Dividing by fractions can be confusing, but there's a neat trick to simplify it. The golden rule is "divide by a fraction" by "multiplying by its reciprocal". For instance, dividing by \( \frac{1}{2} \) is the same as multiplying by 2.
Here's how it works: suppose you have \( \frac{88}{0.5} \). Normally, dividing 88 by half might seem challenging, but by using the reciprocal (in this case, converting 0.5 to its reciprocal 2), the problem quickly simplifies.
Multiply 88 by 2, transforming the problem into \( 88 \times 2 = 176 \), which is both straightforward and clearly demonstrates the power of this method. This conversion reduces the need for tricky arithmetic and enhances computational speed.