Problem 100 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{-55.02}{-0.7} $$

Step-by-Step Solution

Answer

The result of the division is 78.6.

Step 1: Understanding the Problem

We are asked to perform the division operation with two negative numbers: \(-55.02\) divided by \(-0.7\). According to the rules of division, dividing two numbers with the same sign (both negative) results in a positive quotient.

Step 2: Simplify the Division

To simplify \(\frac{-55.02}{-0.7}\), convert the division of two negative numbers into a division of positives: \(\frac{55.02}{0.7}\).

Step 3: Dividing the Numbers

Perform the division \(\frac{55.02}{0.7}\). We can make this division simpler by removing the decimal from the divisor. Multiply both the numerator and the denominator by 10 to get whole numbers: \(\frac{550.2}{7}\).

Step 4: Perform the Division

Divide 550.2 by 7. Performing this division gives us 78.6.

Key Concepts

Division Rules

When dividing numbers, one of the first things you need to remember is the division rules concerning negative numbers. These rules are quite straightforward:

• Dividing a positive number by a positive number gives a positive quotient.
• Dividing a negative number by a negative number also results in a positive quotient. This is because the two negative signs cancel each other out.
• Dividing a positive number by a negative number (or vice versa) results in a negative quotient.

These division rules are important to ensure that you finalize your answer with the correct sign. In the given exercise, dividing two negative numbers \(-55.02\) and \(-0.7\) results in a positive answer. Thus, we convert the division problem to that of the positives: \(\frac{55.02}{0.7}\). Remembering the rules of signs in division helps prevent errors and ensures calculations are correct.

Operations with Decimals

Working with decimals can sometimes be tricky, but there are simple strategies to handle them effectively during division. When you need to divide decimals, you have a few options:

• Align the decimal points and proceed with the long division, keeping decimals in mind, and placing the decimal point directly up from the dividend in your quotient.
• Another technique, which we used in this exercise, involves removing the decimal by multiplying both the numerator and the denominator by a power of ten. This turns the divisor into a whole number, making the division simpler. Multiply both \(55.02\) and \(0.7\) by 10 to remove the decimals: \(\frac{550.2}{7}\).

After eliminating decimals, the division becomes more straightforward, allowing you to perform standard long division with whole numbers. This technique simplifies calculations and minimizes errors when dealing with decimals.

Simplifying Fractions

Simplifying fractions is another fundamental aspect when solving division problems, which can help to make your calculations a lot easier. A fraction is simplified when both the numerator and the denominator are divided by their greatest common divisor (GCD).To begin with, ensure that any decimal involved in the fraction is dealt with first, as discussed in the previous section. With the given fraction \(\frac{550.2}{7}\), since we already removed decimals, focus on dividing.However, unlike typical fractions that may still be simplified, here we perform the division directly because \(550.2\) and \(7\) do not share any common factors other than \(1\). Thus, proceeding with direct long division of the whole numbers is the key to getting your answer, which is \(78.6\).Simplify whenever possible, but always check if your operation needs just completing the division step instead, especially when common factors are minimal or nonexistent.