Problem 58 - Prealgebra

Question

Evaluate each expression if \(a=6, b=4,\) and \(c=5\). $$10 a \div c$$

Step-by-Step Solution

Answer

The expression evaluates to 12.

Step 1: Substitute Values

Replace the variables in the expression with the given values. In the expression \(10a \div c\), substitute \(a = 6\) and \(c = 5\). This gives us \(10 \times 6 \div 5\).

Step 2: Perform Multiplication

Calculate the multiplication part of the expression. Multiply 10 by 6 to get 60, so the expression becomes \(60 \div 5\).

Step 3: Perform Division

Divide 60 by 5 to simplify the expression. \(60 \div 5 = 12\). Thus, the evaluated expression is 12.

Key Concepts

Understanding Substitution

Substitution is an essential math skill that involves replacing variables with their given numerical values. When evaluating expressions, this step allows us to work with concrete numbers rather than abstract symbols.
To substitute properly, pay attention to each variable in the equation. Given the expression:

• \(10a \div c\)

with the values \(a = 6\) and \(c = 5\), you'll replace each variable with its assigned number.
After substitution, the expression transforms into:

• \(10 \times 6 \div 5\)

This formula will now involve real numbers which simplifies further calculations. Substitution is a crucial first step in solving mathematical problems because getting the correct numerical setup leads directly to accurate results.

Multiplying the Numbers

Multiplication is the second step in solving expressions after substitution. This mathematical operation is about adding a number to itself a certain number of times. In our expression:

• \(10 \times 6 \div 5\)

First, you focus on the multiplication part: \(10 \times 6\). Here's how it works:

• Take the number 10 and add it to itself 6 times.
• The result of this is the product. Hence, \(10 \times 6 = 60\).

Multiplication gives you a new simplified expression:

• \(60 \div 5\)

This operation condenses multiple additions into one, speeding up the process of solving mathematical expressions.

Mastering Division

Division is the final step after substitution and multiplication. It involves splitting a number into equal parts. In our running example:

• \(60 \div 5\)

Division requires you to distribute the first number evenly across the second. Here's how it breaks down:

• Ask, "How many times does 5 fit into 60?"
• Since 5 goes into 60 exactly 12 times, the result is \(60 \div 5 = 12\).

Through division, you reach the final answer to the expression. This operation is a method to simplify or resolve parts into a single, known amount. Mastering division helps in understanding how numbers relate when split into equal sections, making it a critical skill in everyday and complex mathematical applications.