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Number - Operations with Decimals

Grade 6IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Place Value and the Decimal Anchor: Visualize a place value chart where the decimal point acts as a central fixed anchor. Digits to the left represent whole numbers (Ones, Tens, Hundreds), while digits to the right represent fractional parts: Tenths (110\frac{1}{10}), Hundredths (1100\frac{1}{100}), and Thousandths (11000\frac{1}{1000}). Moving one position to the right always represents a value that is 10 times smaller.

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Vertical Alignment in Addition and Subtraction: To add or subtract decimals, visualize drawing a straight vertical line through all decimal points to ensure like place values are stacked correctly. If numbers have a different number of digits after the decimal, use 'placeholder zeros' (e.g., writing 5.25.2 as 5.205.20) to fill the empty columns and maintain visual balance.

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The Multiplication Jump Rule: When multiplying decimals, treat the numbers as whole numbers first. After finding the product, count the total number of decimal places in all original factors. In the final result, visualize starting the decimal point at the far right and 'jumping' it to the left by that total count.

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Dividing Decimals by Whole Numbers: When dividing a decimal (dividend) by a whole number (divisor), place the decimal point in the quotient (answer) directly above the decimal point in the dividend. Use a long division bracket and ensure that each digit of the answer is written precisely above the corresponding digit of the dividend to maintain place value accuracy.

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Dividing by Decimals using Equivalent Scales: To divide by a decimal, you must first convert the divisor into a whole number by shifting its decimal point to the right. To keep the expression equivalent, you must shift the dividend's decimal point by the exact same number of places. This is conceptually the same as multiplying both the numerator and denominator of a fraction by a power of 1010 (e.g., 10,100,10, 100, or 10001000).

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Powers of 10 and Decimal Shifting: Multiplying a decimal by 10n10^n (where nn is the number of zeros) slides the decimal point nn places to the right, making the number larger. Dividing by 10n10^n slides the decimal point nn places to the left, making the number smaller. Visualize the decimal point 'sliding' along the number line based on the number of zeros.

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Rounding and Estimation: When rounding to a specific place value, look at the digit to the immediate right (the 'neighbor'). If the neighbor is 5,6,7,8,5, 6, 7, 8, or 99, round the target digit up. If it is 0,1,2,3,0, 1, 2, 3, or 44, keep the target digit the same. Visualize a number line to determine which tenth or hundredth the value is physically closer to.

📐Formulae

Total Decimal Places in Product=Places in Factor 1+Places in Factor 2\text{Total Decimal Places in Product} = \text{Places in Factor 1} + \text{Places in Factor 2}

DividendDivisor=Dividend×10nDivisor×10n\frac{\text{Dividend}}{\text{Divisor}} = \frac{\text{Dividend} \times 10^n}{\text{Divisor} \times 10^n}

a.bc×101=abc.0 (Shift 1 place right)a.bc \times 10^1 = abc.0 \text{ (Shift 1 place right)}

a.bc÷101=0.abc (Shift 1 place left)a.bc \div 10^1 = 0.abc \text{ (Shift 1 place left)}

💡Examples

Problem 1:

Calculate 24.7+3.8224.7 + 3.82.

Solution:

  1. Align the decimal points vertically.
  2. Add a placeholder zero to 24.724.7 to make it 24.7024.70.
  3. Perform the addition: 24.7024.70 +03.82+ 03.82 −−−−−−−------- 28.5228.52
  4. The sum is 28.5228.52.

Explanation:

By adding the placeholder zero, we ensure that the hundredths place (00 and 22) and the tenths place (77 and 88) are correctly added together. The decimal point in the answer stays aligned with the others.

Problem 2:

Solve 1.5×0.041.5 \times 0.04.

Solution:

  1. Multiply the numbers as if they were whole numbers: 15×4=6015 \times 4 = 60.
  2. Count the total decimal places in the factors:
    • 1.51.5 has 11 decimal place.
    • 0.040.04 has 22 decimal places.
    • Total = 1+2=31 + 2 = 3 places.
  3. Start at the right of 6060 and move the decimal 33 places to the left:
    • Move 1: 6.06.0
    • Move 2: .60.60
    • Move 3: .060.060 (adding a zero as a placeholder).
  4. The final answer is 0.060.06.

Explanation:

Since there were three total decimal places in the factors, our product must also have three places. We add a leading zero to 6060 to allow for the third decimal jump.