1:47:22
Arithmetic Course--Class 20: Operations with Decimals (Part II)
The College Prep School
Overview
This video lesson focuses on the division of decimal numbers, building upon previous lessons on addition, subtraction, and multiplication. It explains that the long division algorithm for whole numbers remains the same for decimals, with the primary difference being the placement of the decimal point in the final answer. The video demonstrates various examples, highlighting common pitfalls and special cases, such as when to add zeros, how to handle remainders, and the placement of the decimal when dividing by whole numbers or when dealing with repeating decimals. The lesson concludes with word problems that require decimal division.
How was this?
Save this permanently with flashcards, quizzes, and AI chat
Chapters
- Decimal division uses the same algorithm as whole number division.
- The main difference is correctly placing the decimal point in the quotient.
- Homework from the previous class is essential for understanding this lesson.
Understanding the core algorithm for decimal division is crucial as it forms the foundation for more complex calculations involving decimals.
0.7218 / 0.6
- Set up the division problem with the dividend inside and the divisor outside.
- Temporarily ignore decimals and leading zeros to perform the division as with whole numbers.
- Each cycle of division involves dividing, multiplying, subtracting, and bringing down the next digit.
- If the divisor does not go into the current number after bringing down a digit, place a zero in the quotient above that digit.
Mastering the step-by-step process of long division ensures accuracy and prevents common errors, especially when dealing with multi-digit divisors.
Performing the division of 7218 by 6 to find the quotient for 0.7218 / 0.6.
- To place the decimal, determine how many places the decimal in the divisor must be moved to make it a whole number.
- Move the decimal in the dividend the same number of places.
- Bring the decimal from the dividend straight up into the quotient's position.
- Incorrect decimal placement leads to an incorrect final answer.
Correctly aligning and moving the decimal point is the most critical step in decimal division, directly impacting the magnitude and accuracy of the result.
Moving the decimal one place in 0.6 to make it 6, and then moving the decimal one place in 0.7218 to make it 7.218, and bringing the decimal straight up in the quotient.
- Unlike whole number division, decimal division continues even after all digits are brought down if there is a remainder.
- Add a zero to the dividend and continue the division process to eliminate remainders or achieve a desired level of precision.
- Remainders in decimal division are represented by continuing the division process, not by writing 'R' followed by the remainder.
Understanding that decimal division continues until a zero remainder is achieved or a repeating pattern emerges is key to accurately representing decimal values.
Continuing the division of 4.172 / 0.07 after reaching a remainder by adding zeros and dividing further.
- When dividing by a whole number, the decimal point in the divisor is already in the correct position (to the right of the ones place).
- The decimal point in the dividend is moved to align with the divisor's decimal position (which requires no movement).
- The decimal point in the quotient is placed directly above the decimal point in the dividend.
- If the dividend is a whole number, its decimal is invisible but located to the right of the ones place.
Dividing by a whole number simplifies the process of placing the decimal in the quotient, as no initial movement of the decimal in the divisor is required.
21.04 divided by 8, where the decimal in the quotient is placed directly above the decimal in the dividend.
- Some divisions result in remainders that continue indefinitely, creating repeating decimals.
- A bar is placed over the repeating digit(s) in the quotient to indicate it repeats.
- The division process continues until a repeating pattern is observed or a sufficient number of decimal places are calculated.
Recognizing and representing repeating decimals accurately is important for understanding the nature of rational numbers and for precise calculations.
The division resulting in a repeating '6' (e.g., 2.315 / 5 resulting in 0.463, or a problem like 12.6 / 6 resulting in 2.1, but if it were 12.7 / 6 it would be 2.11666...).
- Dividing by two-digit or larger numbers requires more careful estimation and multiplication.
- The process of moving the decimal in the divisor and dividend remains the same.
- Special cases include adding multiple zeros to the dividend when the divisor is larger than initial parts of the dividend, or when the divisor has many decimal places.
- When the dividend is a whole number and the divisor has decimal places, the invisible decimal in the dividend must be moved accordingly.
These advanced scenarios test the learner's ability to apply the core algorithm consistently, even when faced with more complex calculations and numerous decimal places.
3.809 divided by 0.013, requiring estimation for a two-digit divisor and careful decimal placement.
- The fraction bar signifies division; to convert a fraction to a decimal, divide the numerator by the denominator.
- The same long division algorithm is used, including handling remainders and repeating decimals.
- The decimal point in a whole number numerator is already in place, simplifying decimal placement in the quotient.
- This method is useful for understanding the decimal representation of rational numbers.
Understanding that fractions are simply another way to represent division allows for conversion between fractional and decimal forms, enhancing mathematical flexibility.
Converting 7/8 to a decimal by dividing 7 by 8, resulting in 0.875.
- Identify the unknown quantity (e.g., time, distance, rate) to determine the correct division operation.
- Use estimation or the factor-label method to decide which number to divide by which.
- The units of the numbers involved (e.g., miles per hour, miles) help determine the correct order of division.
- Ensure the final answer has the correct units and is presented in decimal form.
Applying decimal division to real-world scenarios demonstrates its practical utility and reinforces problem-solving skills.
A turtle crawling 0.03 miles per hour for 0.744 miles requires dividing 0.744 by 0.03 to find the time.
Key takeaways
- The core algorithm for dividing decimals is identical to dividing whole numbers; the key is correctly placing the decimal point.
- To place the decimal in the quotient, align the decimal in the dividend with the decimal in the divisor after making the divisor a whole number.
- Decimal division continues until a remainder of zero is achieved or a repeating pattern is identified, often requiring the addition of zeros to the dividend.
- When dividing by a whole number, the decimal point in the quotient is placed directly above the decimal point in the dividend.
- Fractions can be converted to decimals by performing division: numerator divided by the denominator.
- Word problems involving decimals often require careful analysis of units and context to determine the correct order of division.
- Repeating decimals are indicated by a bar over the repeating sequence of digits.
Key terms
Decimal DivisionLong Division AlgorithmQuotientDividendDivisorDecimal Point PlacementRemainderRepeating DecimalPlaceholder ZeroWhole Number DivisorFraction to Decimal Conversion
Test your understanding
- What is the fundamental rule for placing the decimal point when dividing decimals?
- How does the process of decimal division differ from whole number division when a remainder is encountered?
- Explain the steps involved in converting a fraction like 5/8 into its decimal equivalent.
- Why is it important to correctly identify the divisor and dividend in word problems involving decimal division?
- What does it mean for a decimal to be 'repeating,' and how is it represented?