Problem 64 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{-86}{2.5}\)

Step-by-Step Solution

Answer

The result of \(-86/2.5\) is \(-34\).

Step 1: Understand the problem

The problem requires us to perform the division of -86 by 2.5, which means we need to calculate how many times 2.5 fits into -86.

Step 2: Convert to easier numbers (optional)

To make the division easier, we can multiply both the numerator (-86) and the denominator (2.5) by 10 to eliminate the decimal. This gives us \(-860/25\).

Step 3: Perform the division

Now, divide -860 by 25. First, calculate how many whole times 25 fits into 860. 25 goes into 860 exactly 34 times because 25 multiplied by 34 is 850. This means we have \(-34\) as the result since we are dividing with a negative numerator.

Step 4: Verify your result

To verify, multiply 25 by 34 to check the accuracy: 25 \times 34 = 850. This confirms that our division calculation is correct. Hence, our end result \(-34\) is accurate.

Key Concepts

Integer division

Integer division is a fundamental math operation used when you divide two whole numbers. The process involves determining how many times the divisor can "fit" into the dividend without exceeding it. In simpler terms, integer division reveals how many full groups of the divisor can be formed from the dividend. This operation often leaves a remainder. For instance, dividing 10 by 3 results in 3 full groups, with a remainder of 1. This remainder indicates the portion of the dividend not completely divided by the divisor. In cases such as our original exercise where we multiply to eliminate decimals, we transform the problem into a clearer integer division as shown with \(-860/25\), making the division straightforward. Key points to remember during integer division:

• Identify the whole number divisor and dividend.
• Determine how many times the divisor fits into the dividend.
• Check for any remainders through multiplication verification.

Integer division is crucial, especially when working with certain computer science applications where only whole numbers are used.

Division with decimals

Division with decimals might seem daunting at first, but it can be simplified with a few strategies. When dealing with decimal numbers, the key is to eliminate the decimal point by multiplying both the dividend and divisor by the same power of ten. This technique allows you to work with whole numbers instead. In the example from our exercise, \(-86/2.5\), we opted to multiply both numbers by 10 to become \(-860/25\). This shift to integers simplifies the calculation process.
The essential steps include:

• Identify the decimal places in the divisor and dividend.
• Multiply both by a power of 10 to convert into whole numbers.
• Perform the division as you would with integers.

Once the division is complete, converting back is unnecessary if the result is straightforward as seen with an integer output. This simple technique is applicable for any decimal division problem, improving accuracy and comprehension.

Negative numbers

When it comes to operations involving negative numbers, it's imperative to understand how signs affect mathematical processes. In math, a negative sign indicates a number less than zero. The rules for division remain consistent whether you're working with positive or negative numbers, but the sign of the result depends on the signs of the numbers involved.
Here's a quick recap of these rules:

• A positive divided by a positive equals a positive.
• A negative divided by a positive equals a negative.
• A positive divided by a negative equals a negative.
• A negative divided by a negative results in a positive.

For example, in our exercise \(-860/25\), we have a negative numerator. Following the rules, the result is negative. Understanding these sign rules is crucial not only in division but in adding, subtracting, and multiplying as well.
Mastering how signs interact assists in accurately solving a wide range of mathematical problems.