Problem 15 - Intermediate Algebra

Question

By what number should a. 5 be divided to obtain \(1 ?\) b. 5 be multiplied to obtain \(1 ?\)

Step-by-Step Solution

Answer

a. Divide by 5; b. Multiply by \( \frac{1}{5} \).

Step 1: Understanding Division

To find the number by which 5 should be divided to get 1, we set up the equation as \( \frac{5}{x} = 1 \). Here, \( x \) is the number we are looking for.

Step 2: Solving for Division

To solve \( \frac{5}{x} = 1 \), we can multiply both sides by \( x \) to get \( 5 = x \). Therefore, 5 must be divided by 5 to obtain 1.

Step 3: Understanding Multiplication

To find the number by which 5 should be multiplied to get 1, we set up the equation as \( 5 \times x = 1 \). Here, \( x \) is the number we are looking for.

Step 4: Solving for Multiplication

To solve \( 5 \times x = 1 \), divide both sides by 5 to isolate \( x \). This gives us \( x = \frac{1}{5} \). Therefore, 5 must be multiplied by \( \frac{1}{5} \) to obtain 1.

Key Concepts

Division

Division is the process of distributing a number into equal parts. It's one of the four basic operations in arithmetic, alongside addition, subtraction, and multiplication. When we talk about division in algebra, we refer to dividing one quantity by another to see how many times the divisor fits into the dividend. Let's break down how division works using our example:

• The equation we started with is \( \frac{5}{x} = 1 \).
• Here, 5 is the dividend, and \( x \) is the divisor.
• We want our result, or quotient, to be 1.

By multiplying both sides of the equation \( \frac{5}{x} = 1 \) by \( x \), we can isolate the variable, showing us that \( x = 5 \). This means to make the quotient 1 when dividing 5, we must divide by 5 itself. This process exemplifies how division helps us understand the relationship between numbers in a clear and systematic way.

Multiplication

Multiplication is like adding a number to itself a certain number of times. This process is faster than repetitive addition and is fundamental in algebra. In our exercise, we aim to find out what number multiplied by 5 yields 1. Let's consider the equation:

• We start with \( 5 \times x = 1 \).
• Here, 5 is the known factor, and \( x \) is what we're solving for.
• The result, or product, is 1.

To solve this, you divide both sides of the equation by 5, which gives us \( x = \frac{1}{5} \). This tells us that multiplying 5 by \( \frac{1}{5} \) gives the product 1. This concept highlights the inverse relationship between multiplication and division, where multiplying by a fraction is equivalent to dividing by its reciprocal.

Equation Solving

Equation solving is the process of finding the unknown variable that makes an equation true. In basic algebra, this involves using operations like addition, subtraction, multiplication, and division to isolate the variable. Let's explore this using both parts of the exercise:

• For division, \( \frac{5}{x} = 1 \) required us to multiply both sides by \( x \) to solve for \( x \).
• This step isolates the variable, leading to \( x = 5 \).
• For multiplication, \( 5 \times x = 1 \) required dividing both sides by 5 to solve for \( x \).

The resulting solution was \( x = \frac{1}{5} \), showcasing how different operations are applied to balance and solve equations. Mastering these operations is key to understanding equations and building a strong foundation in algebra.