Problem 42 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(85 \div 5 \cdot 5-85\)

Step-by-Step Solution

Answer

The value is 0.

Step 1: Solve Division

First, perform the division operation. Divide 85 by 5. We calculate: \[ 85 \div 5 = 17 \]

Step 2: Solve Multiplication

Next, multiply the result from Step 1 by 5. We calculate: \[ 17 \cdot 5 = 85 \]

Step 3: Solve Subtraction

Now, subtract 85 from the result of Step 2. We calculate: \[ 85 - 85 = 0 \]

Step 4: Verify with Calculator

Use a calculator to verify the steps 1. Input 85 divide by 5, then multiply by 5, and subtract 85. 2. Check if the result matches the calculated value. Upon verification, the result is 0.

Key Concepts

Division

Division is a mathematical concept that splits a quantity into equal parts. It's essentially the opposite of multiplication. In this exercise, we start by dividing 85 by 5. This means we want to know how many times 5 can "fit" into 85.

• Think of division as "grouping," where you're creating groups of a certain size.
• Division can be written using the symbol \( \div \) or a fraction bar, e.g., \( \frac{85}{5} \).
• The number being divided is called the "dividend" (85), and the number you divide by is the "divisor" (5).

After performing the division, we find that 85 divided by 5 equals 17. This means if you have 85 apples and want to share them equally among 5 people, each person gets 17 apples.
Understanding division is crucial as it often serves as the starting point in multi-step arithmetic problems like ours.

Multiplication

Multiplication is a mathematical operation where we combine equal groups. It's repeated addition. In our problem, after finding that 85 divided by 5 is 17, we multiply this result by 5. This means we are adding 17 together five times.

• Use the symbol \( \cdot \) or simply place numbers next to each other, like \( 2x \), to represent multiplication.
• The numbers being multiplied are called "factors." In this case, 17 and 5 are factors that produce 85.
• Think about multiplication like arranging objects in rows and columns. For example, 17 times 5 can be visualized as 5 rows of 17 objects.

When we multiply 17 by 5, the result is 85. This step often confirms that you correctly performed the previous operation, especially if you're working backward from a known total.

Subtraction

Subtraction is a basic operation where we calculate the difference between numbers. It involves taking away one number from another. In our sequence of calculations, after multiplying and getting 85, we subtract 85 from 85.

• Subtraction is represented by the minus sign, \( - \).
• The first number is the "minuend" (the amount you start with), and the second is the "subtrahend" (what you are taking away).
• In many problems, subtraction can tell you what's left over after removing a portion of a total.

Here, subtracting 85 from 85 leaves 0. This step shows a balance is achieved, confirming our earlier calculations were correct. Subtraction is key because it often concludes the computation, giving us a final result or answer.