Problem 44 - Algebra A Combined Function

Question

Divide. \(-4.9 \div-0.07\)

Step-by-Step Solution

Answer

The result of the division is 70.

Step 1: Rewrite the Expression

First, rewrite the division expression with numbers in fraction form for clarity. We have \[\frac{-4.9}{-0.07}\]

Step 2: Simplify the Signs

Next, notice that both the numerator and the denominator have negative signs. Dividing a negative by a negative results in a positive:\[\frac{-4.9}{-0.07} = \frac{4.9}{0.07}\]

Step 3: Convert Decimal to Whole Numbers

To simplify, we can eliminate the decimals by multiplying both the numerator and the denominator by 100 to make them whole numbers:\[\frac{4.9 imes 100}{0.07 imes 100} = \frac{490}{7}\]

Step 4: Perform Division

Now, divide 490 by 7. Set up the long division: 7 goes into 49 exactly 7 times (as 7 x 7 = 49) and no remainder for 49. Pull down the next digit, which is 0. 7 goes into 0 exactly 0 times. Thus, the result is 70.

Key Concepts

Simplifying Expressions

When dividing numbers, especially with decimals or negative signs, it's helpful to simplify the expression. This makes the problem easier to work with. Here, the original expression is \(-4.9 \div -0.07\). By rewriting it as a fraction, we transform it into \[ \frac{-4.9}{-0.07} \].
Both the numerator and the denominator have negative signs. When dividing negatives, remember:

• Negative divided by negative equals a positive
• Negative divided by positive equals a negative
• Positive divided by negative equals a negative

So, in this case, dividing a negative by another negative simplifies to a positive fraction: \[ \frac{4.9}{0.07} \]. These steps of simplifying the expression start us off on the right foot.

Long Division

Long division might seem daunting, but it's a straightforward process once you understand it. After simplifying our expression to \[ \frac{4.9}{0.07} \], we convert it to \[ \frac{490}{7} \] by eliminating decimals (more on that next). Now, we perform the division 490 ÷ 7 using long division.
Here's how you tackle it:

• See how many times 7 fits into 49, which it does 7 times.
• Multiply 7 by 7 to get 49, then subtract to find no remainder: 49 - 49 = 0.
• Bring down the next number, which is 0, making it 0.
• Determine that 7 goes into 0, zero times.

Thus, we find the quotient to be 70. Long division helps us systematically find the result, breaking it down into digestible steps.

Converting Decimals to Whole Numbers

Converting decimals into whole numbers simplifies operations. It involves multiplying by powers of 10 to shift the decimal point. For the fraction \[ \frac{4.9}{0.07} \], we multiply both numerator and denominator by 100:

• \(4.9 \times 100\) becomes 490 (move the decimal two places to the right)
• \(0.07 \times 100\) becomes 7 (move the decimal two places to the right)

This transformation results in the fraction \[ \frac{490}{7} \], which simplifies the division process. By eliminating decimals, calculations become more straightforward and prevent potential errors during division. This step is crucial when dealing with decimal numbers and ensures easier handling of mathematical operations moving forward.