Problem 78 - Algebra 1

Question

Write a question that can be represented by the equation. Then use mental math to solve the equation. $$6 x=18$$

Step-by-Step Solution

Answer

A suitable question could be: 'A grocery store sells bananas in batches of 6. If a customer purchases 18 bananas, how many batches of bananas did the customer buy?' The solution to the equation is \(x = 3\), meaning the customer bought 3 batches of bananas.

Step 1: Formulate a Real World Problem

Firstly, a real world problem that could be represented by the equation \(6x = 18\) needs to be formulated. An example of such a problem could be: 'A grocery store sells bananas in batches of 6. If a customer purchases 18 bananas, how many batches of bananas did the customer buy?' This problem can be represented by the equation \(6x = 18\), where \(x\) is the number of banana batches.

Step 2: Solve the Equation using Mental Math

Once we have the equation, we can use mental math to solve for \(x\) . In this case, the operation between \(6\) and \(x\) is multiplication, so to isolate \(x\) and solve for it, the opposite operation, which is division, will be used. So, \(x\) can be found by dividing \(18\) by \(6\), which gives \(x = 3\).

Key Concepts

Understanding Real World Problems

Real world problems provide a practical way to apply and understand algebraic equations. Imagine a scenario where you're at a grocery store. You discover that bananas are sold in packs of 6. If you were to buy a total of 18 bananas, it becomes a real-world situation that can be expressed using the algebraic equation \(6x = 18\). This equation helps us identify how many packs you need to buy. By translating real situations into mathematical equations, we learn how to tackle practical problems with ease.

• Translating scenarios into equations.
• Understanding components such as packs, quantities, or different entities like time and cost.
• Fostering problem-solving skills applicable outside just mathematics.

Using Mental Math for Quick Solutions

Mental math is a powerful tool that lets you solve basic arithmetic problems quickly and with minimal use of pen and paper. In our grocery store example, once the equation \(6x = 18\) is established, mental math allows us to swiftly determine the solution. To solve for \(x\), recognize that the operation here is multiplication and then apply division as its inverse.

Steps for Mental Math:

• Identify the current operation (in this case, multiplication).
• Apply the inverse operation (division) to isolate \(x\).
• Perform the operation in your head: divide 18 by 6 to find \(x = 3\).

This showcases how mental math fosters confidence, helps develop number sense, and saves time during quick calculations.

The Role of Multiplication and Division in Solving Equations

Multiplication and division are fundamental operations that enable us to solve equations like \(6x = 18\). Understanding these operations is vital for tackling algebraic expressions.
Starting with \(6x = 18\), the goal is to find \(x\). Here, \(6x\) signifies 6 multiplied by \(x\). To undo this, we use the inverse operation—division. By dividing both sides of the equation by 6, we isolate \(x\).

Importance of Multiplication and Division:

• They are complementary operations; one can reverse the effect of the other.
• They simplify problems, making it possible to figure out unknowns in equations.
• Proficiency in these operations supports advanced mathematical learning and applications.

Thus, multiplication and division are pivotal not only in algebra but also in solving real-world problems efficiently.