Problem 43 - Algebra A Combined Function

Question

Divide. \(-3.2 \div-0.02\)

Step-by-Step Solution

Answer

The quotient is 160.

Step 1: Recognize the Division Operation

The problem \(-3.2 \div -0.02\) requires us to divide -3.2 by -0.02. Division by a negative number will result in a positive number.

Step 2: Convert Decimals to Fractions

Convert the division problem into fractions: \(-3.2 \div -0.02\) becomes \(-\frac{32}{10} \div -\frac{2}{100}\). We do this to make division easier by dealing with whole numbers.

Step 3: Apply Multiplication Instead of Division

According to the rule for dividing fractions, we multiply by the reciprocal. So \(-\frac{32}{10} \div -\frac{2}{100}\) becomes \(-\frac{32}{10} \times -\frac{100}{2}\).

Step 4: Simplify the Fraction

Multiply the numerators and the denominators: \(-32 \times -100 = 3200\) and \(10 \times 2 = 20\). This results in \(\frac{3200}{20}\).

Step 5: Perform the Division

Divide 3200 by 20. \(3200 \div 20 = 160\), resulting in a positive value since we multiplied two negative numbers earlier.

Key Concepts

Fractions Conversion

When working with decimals in division problems, converting them to fractions can simplify calculations. This method helps us use integer numbers, which are typically easier to work with than decimals.

To convert a decimal to a fraction:

• Look at the decimal. Identify its place value (e.g., tenths, hundredths, thousandths).
• For example, \(-3.2\) can be seen as \(-\frac{32}{10}\), because the 3 represents 30 tenths (or 3 * 10) and the 2 represents 2 tenths, leading to a total of 32 tenths.
• Similarly, \(-0.02\) can be converted into \(-\frac{2}{100}\), since the 2 is in the hundredths place.
• Once both decimals are converted to fractions, proceed with your arithmetic operations.

Converting decimals into fractions allows for a straightforward application of division rules and helps avoid errors that could arise from complex decimal arithmetic.

Negative Numbers Division

Dividing negative numbers follows straightforward rules, but can be somewhat confusing if you're not careful.

• When dividing two negative numbers, the result is always positive. This is because the division of negatives cancels out the negative signs.
• Consider the example \(-3.2 \div -0.02\). While both numbers are negative, dividing them results in a positive value.
• Here’s why: Positive and negative signs have a predictable pattern in multiplication and division: \((-\times - = +)\).

Understanding this rule makes it easier to tackle problems that involve negative numbers, ensuring your final answers are accurate and logically consistent.

Multiplication with Reciprocals

The reciprocal of a number is the number that, when multiplied with the original number, results in 1. It is essentially flipping a fraction upside down.

• To find the reciprocal of a fraction like \(-\frac{2}{100}\), swap the numerator and the denominator, resulting in \(-\frac{100}{2}\).
• In division problems, multiplying by the reciprocal is a key step. It turns a division problem into a multiplication one, which is often easier to solve.
• With our example, changing \(-\frac{32}{10} \div -\frac{2}{100}\) to \(-\frac{32}{10} \times -\frac{100}{2}\) makes it simpler to compute the result.

Using reciprocals changes the division operation into a multiplication, allowing us to apply straightforward multiplication skills and simplifying the problem-solving process.