Problem 42 - Intermediate Algebra

Question

Perform the operations. See Examples 4 and 5 . $$ \frac{-78}{6} $$

Step-by-Step Solution

Answer

The result of \( \frac{-78}{6} \) is \(-13\).

Step 1: Perform the fraction operation

Apply the rules for adding, subtracting, multiplying, or dividing fractions.

Step 2: Simplify

The simplified result is The result of \( \frac{-78}{6} \) is \(-13\)..

Key Concepts

Negative Numbers

Negative numbers are very important in mathematics as they represent less than zero values. Often used to describe things like temperatures below freezing, debt, or sea level points below the surface, these numbers are essential and have unique properties. When dealing with negative numbers, it's crucial to remember:

• Adding a negative number is the same as subtracting its positive counterpart. For example, adding \(-5\) is identical to subtracting \(5\).
• Subtracting a negative number is the same as adding its positive counterpart. For instance, subtracting \(-5\) is the same as adding \(5\).
• Multiplying or dividing two negative numbers results in a positive number. For example, \((-3) imes (-2) = 6\).
• Multiplying or dividing a positive number and a negative number results in a negative number, as seen in our example \(-78 \div 6 = -13\).

Negative numbers can often trick students, but understanding these simple rules can make calculations much easier.

Integer Division

Integer division is a common operation in mathematics where we divide one integer by another. The result is also an integer, without any fractional part. Let's break it down:

• If the dividend \(a\) is evenly divisible by the divisor \(b\), the quotient is an integer. For example, \(78 \div 6 = 13\).
• If the dividend is negative, as in our problem \(-78 \div 6\), the result is also negative.
• Integer division differs from regular division in that it does not return decimals or fractions. It provides a whole number result.

In practical terms, this means when dividing integers, consider the sign and the possibility of a remainder. Although much of this comes up in conceptual mathematics, understanding integer division helps with real-world tasks like splitting items evenly among groups.

Quotient Computation

Quotient computation involves finding the result of dividing one number by another. It's crucial to understand how to perform and interpret this operation:

• The quotient is the number of times the divisor fits completely into the dividend, excluding any remainder.
• If the dividend and divisor have the same sign, the quotient is positive. If they have different signs, the quotient is negative. This explains why \(-78 \div 6 = -13\) results in a negative number.
• In our example, \(78\) divided by \(6\) gives \(13\), but since the original number \(-78\) was negative, our quotient is therefore \(-13\).
• Remember that although the quotient indicates how many times the divisor fits into the dividend, there may still be a remainder.

Careful attention to the signs and operations in quotient computation is crucial. Making sure you've accounted for these factors will ensure your solutions in math problems.