Problem 40 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ z \div 4=5 $$

Step-by-Step Solution

Answer

The solution to the equation is \( z = 20 \).

Step 1: Identify the equation

The given equation is \( z \div 4 = 5 \). Here, it's required to find the value of \( z \). This equation can be rewritten in other ways depending on how it is to be solved, but since this problem is to be solved using mental math, no rewriting is necessary.

Step 2: Identify the operation in reverse

In this case, since \( z \) is being divided by 4 to give 5, reversing the operation means \( z \) will be found by multiplying 5 by 4. Therefore, \( z = 5 \times 4 \).

Step 3: Perform the multiplication

Performing the multiplication, \( z = 20 \). This is the solution to the equation \( z \div 4 = 5 \).

Key Concepts

Division

Division is one of the fundamental operations in math, used to split a number into equal parts. In the exercise, the equation given is \( z \div 4 = 5 \). This means that when we divide an unknown number \( z \) by 4, the result is 5. Understanding division is crucial since it helps us break down numbers easily. When you approach it, think of how many times the divisor fits into the dividend.

• Dividend: The number to be divided, which is the unknown \( z \) in our problem.
• Divisor: The number that divides the dividend, which is 4 in this case.
• Quotient: The result of the division, which equals 5 here.

So, division is about determining how many groups of the divisor can fit into the dividend. When solving equations, it helps to look at division in reverse to make it clearer.

Multiplication

In the step-by-step solution, reversing the division helps us find the value of \( z \). This is done through multiplication, which is essentially repeated addition. Here, multiplying 5 by 4 reverses the division, allowing us to find the original number, \( z \). Therefore, \( z = 5 \times 4 \) results in \( z = 20 \). Multiplication, the cousin of addition, works by adding a number to itself a specified number of times. This operation is represented by the times symbol (\( \times \)).

• The number being multiplied (5) is called the multiplicand.
• The number it is multiplied by (4) is known as the multiplier.
• The result (20) is called the product.

So, think of multiplication as piling groups to get a total, which is very logical when needing to solve problems involving division.

Solving Equations

Solving equations often requires finding a way to make one side of the equation equal to the other. The goal is to find out what value the unknown variable represents. In our practice problem, we solved \( z \div 4 = 5 \) by identifying what value of \( z \) satisfies the equation. The equation was solved by applying two key operations: division and multiplication. Here's a simple way to approach solving equations mentally:

• Identify the operation applied to the unknown variable. In our case, \( z \) was divided by 4.
• Reverse the operation to isolate the variable. We reversed the division by multiplying.
• Apply the operation carefully to find the solution, ensuring both sides of the equation remain balanced.

Using mental math for solving equations isn't just about calculation. It's about logical reasoning to reverse operations, which is key to becoming efficient in math problem-solving.