Problem 19 - Prealgebra

Question

Perform each of the following divisions. $$29.7 \div 22$$

Step-by-Step Solution

Answer

29.7 divided by 22 equals approximately 1.35.

Step 1: Understanding Division

We need to divide 29.7 by 22. This means we'll find how many times 22 fits into 29.7, or what multiplier of 22 gives us 29.7.

Step 2: Convert to a Fraction

Express the division as a fraction: \( \frac{29.7}{22} \). This is another way to represent the problem.

Step 3: Remove Decimal by Multiplying

To get rid of the decimal, multiply both the numerator and the denominator by 10 to make it \( \frac{297}{220} \). This simplification results in an equivalent fraction without decimal points.

Step 4: Simplify the Fraction

Check if \( \frac{297}{220} \) can be simplified. The greatest common divisor (GCD) of 297 and 220 is 1, meaning the fraction is already in its simplified form.

Step 5: Convert to Decimal Form

Perform the division \( 297 \div 220 \) using long division to convert the fraction to a decimal. This results in approximately 1.35.

Key Concepts

Decimal Conversion

When dealing with division of numbers that include decimals, a handy step is turning those decimals into whole numbers by multiplying. This is called decimal conversion. By converting decimals into whole numbers, calculations become easier. For example, when dividing 29.7 by 22, you express it as a fraction: \( \frac{29.7}{22} \). To eliminate the decimal, multiply both the numerator (the top part of the fraction) and the denominator (the bottom part) by 10. This changes your fraction to \( \frac{297}{220} \), a whole number fraction.

• This step simplifies the process since working with whole numbers is often straightforward.
• Ensure that you multiply both parts of your fraction by the same amount (in this case, 10).

By performing decimal conversion at the start, you set the stage for clearer and simpler arithmetic operations later on.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Once you have the fraction \( \frac{297}{220} \), the next step is to check if it can be simplified.

• To simplify, find the greatest common divisor (GCD) of the numerator and denominator.
• For \( \frac{297}{220} \), the GCD is 1, meaning it's already in its simplest form.

No further reduction is possible in this case.
Remember, a fraction in its simplest form allows for more straightforward comparisons and understanding of its value in relation to others. Simplification helps in identifying the basic ratio represented by the fraction.

Long Division

Long division is a reliable method for dividing numbers, especially when converting a fraction to a decimal. With the fraction simplified to \( \frac{297}{220} \), next, you perform long division to find the decimal form. Here's a simple breakdown:

• Set up the division problem: 297 divided by 220.
• The whole number part of the answer is the number of times 220 fits completely into 297, which is 1 time.
• Subtract 220 from 297 to get 77, and bring down a zero to get 770.
• Determine how many times 220 can fit into 770, which is 3 times.
• Perform additional subtraction and repeat the process to reach the desired decimal accuracy.

This yields 1.35 as a decimal. Long division offers a systematic approach to dividing numbers when more complicated methods aren’t available or when solving by hand, making it an essential skill in arithmetic.