Problem 79 - Algebra 1

Question

Write a question that can be represented by the equation. Then use mental math to solve the equation. $$\frac{y}{8}=4$$

Step-by-Step Solution

Answer

The number that divided by 8 equals 4 is 32.

Step 1: Formulate the Question

A question that can be represented by the equation \(\frac{y}{8}=4\) could be: 'What number divided by 8 equals 4?'

Step 2: Solve for Unknown 'y'

We need to find the value for y. Since the equation says that \(y\) divided by \(8\) equals \(4\), the reverse operation would be to multiply \(8\) by \(4\) to find the value of \(y\). So, \(y = 8 \times 4\)

Step 3: Compute 'y' Value

Calculate the multiplication operation to find the value of \(y\). Doing the calculation, \(y= 8 \times 4 = 32\). Thus, the answer to the question 'What number divided by 8 equals 4?' is 32

Key Concepts

Division Equations

Division equations are a type of linear equation where a number (the unknown) is divided by another number to result in a given quotient. For example, if you have the equation \( \frac{y}{8} = 4 \), it means dividing \( y \) by 8 equals 4. To solve this type of equation, you generally perform the opposite operation—in this case, multiplication.
This technique is useful across various real-world problems, such as calculating averages or splitting quantities into equal parts. Understanding division in equations helps establish how parts relate to a whole, providing valuable insights into proportionality and balance in mathematical relationships.

Mental Math

Mental math involves performing calculations in your head without the need for paper, a calculator, or any digital assistance. It is highly beneficial for quick problem-solving and improves overall cognitive abilities.
When faced with an equation like \( \frac{y}{8} = 4 \), you can use mental math to easily find \( y \). Ask yourself what number multiplied by 8 equals 32. If you know multiplication tables well, you know that 8 multiplied by 4 equals 32. This skill helps in efficiently handling everyday math-related tasks and boosts confidence in dealing with numbers.
Consistent practice with mental math sharpens your thinking speed and accuracy, making arithmetic seem effortless over time.

Solving for Unknowns

When solving for unknowns in equations, the goal is to isolate the unknown variable on one side of the equation. In the equation \( \frac{y}{8} = 4 \), \( y \) is the unknown we need to solve for. To isolate \( y \), perform the inverse operation of what is being done to it. Here, since \( y \) is being divided by 8, the inverse is to multiply both sides of the equation by 8. This process is called "undoing" the operation so we can find the correct value of the unknown.
Here's the step-by-step breakdown:

• Identify the operation being performed on the unknown, here it's division by 8.
• Perform the reverse operation: multiply both sides by 8.
• Simplify to find \( y \), leading to \( y = 32 \).

Mastering how to solve for unknowns helps tremendously in understanding not only math but also logical problem-solving processes in different subjects and real-life situations.