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Number System - Decimals: Place Value, Comparison, Operations (+, -, ×, ÷)

Grade 6ICSE

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

🔑Concepts

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Place Value System: The decimal point separates the whole number part from the fractional part. To the left of the point, places are Units (11), Tens (1010), and Hundreds (100100). To the right, they are Tenths (110\frac{1}{10} or 0.10.1), Hundredths (1100\frac{1}{100} or 0.010.01), and Thousandths (11000\frac{1}{1000} or 0.0010.001). Visualizing a place value chart helps: as you move right from the decimal point, each position is ten times smaller than the previous one.

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Expanded Form: A decimal number can be written as the sum of the values of its digits. For example, 45.6745.67 can be visualised as 40+5+610+710040 + 5 + \frac{6}{10} + \frac{7}{100}. This represents 44 tens, 55 units, 66 tenths, and 77 hundredths.

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Like and Unlike Decimals: Decimals having the same number of decimal places are called Like Decimals (e.g., 1.451.45 and 23.0723.07). Decimals with a different number of decimal places are Unlike Decimals (e.g., 2.52.5 and 3.143.14). You can convert unlike decimals to like decimals by adding 'trailing zeros' at the end, which does not change the value (e.g., 2.52.5 becomes 2.502.50).

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Comparison of Decimals: To compare decimals, first compare the whole number parts. If they are equal, compare the tenths digits. If those are also equal, compare the hundredths, and so on. For example, 5.67<5.685.67 < 5.68 because 77 hundredths is less than 88 hundredths. On a number line, the decimal further to the right is always the greater value.

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Addition and Subtraction: The most critical step is to align the decimal points vertically in a column. This ensures that you are adding or subtracting digits of the same place value (tenths to tenths, etc.). If the numbers have different amounts of digits, use '0' as a placeholder to make them like decimals before calculating.

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Multiplication of Decimals: To multiply, ignore the decimal points and multiply the numbers as whole numbers. Then, count the total number of decimal places in the original factors. Place the decimal point in the product so that it has the same total number of decimal places. If the product has fewer digits, prefix it with zeros.

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Division of Decimals: When dividing by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend. When dividing by another decimal, move the decimal point in the divisor to the right until it becomes a whole number. Move the decimal point in the dividend the same number of places to the right (adding zeros if necessary) before dividing.

📐Formulae

PlaceValueWeight=10n (for whole numbers) or 110n (for decimals)Place Value Weight = 10^n \text{ (for whole numbers) or } \frac{1}{10^n} \text{ (for decimals)}

ValueofaDigit=Digit×Place ValueValue of a Digit = \text{Digit} \times \text{Place Value}

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

Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}

a.bcd.e=a.bc×10nd.e×10n\frac{a.bc}{d.e} = \frac{a.bc \times 10^n}{d.e \times 10^n}

💡Examples

Problem 1:

Add 25.625.6 and 18.4718.47.

Solution:

  1. Convert to like decimals: 25.6025.60 and 18.4718.47.
  2. Align vertically by decimal point: \begin{array}{r@{\quad}l} 25.60 \\ + 18.47 \\ \hline 44.07 \end{array}
  3. Add from right to left: 0+7=70+7=7, 6+4=106+4=10 (write 00, carry 11), 5+8+1=145+8+1=14 (write 44, carry 11), 2+1+1=42+1+1=4.

Explanation:

To add decimals, we ensure they have the same number of decimal places by adding a placeholder zero, then align the decimals to add digits in the same place value columns.

Problem 2:

Multiply 1.251.25 by 0.40.4.

Solution:

  1. Multiply as whole numbers: 125×4=500125 \times 4 = 500.
  2. Count total decimal places: 1.251.25 has 22 places, and 0.40.4 has 11 place. Total = 2+1=32 + 1 = 3 places.
  3. Place the decimal point in the result 500500 starting from the right: 0.5000.500.

Explanation:

We ignore the decimals initially to find the product of the digits, then re-insert the decimal point based on the combined count of decimal places in the original numbers.

Problem 3:

Divide 0.480.48 by 0.60.6.

Solution:

  1. Make the divisor (0.60.6) a whole number by multiplying by 1010. Also multiply the dividend (0.480.48) by 1010.
  2. New expression: 4.8÷64.8 \div 6.
  3. Divide: 4.8÷6=0.84.8 \div 6 = 0.8. 0.86)4.8‾−4.80\begin{array}{r} 0.8 \\ 6 \overline{) 4.8} \\ - 4.8 \\ \hline 0 \end{array}

Explanation:

To divide by a decimal, we shift the decimal point in both the divisor and dividend until the divisor is a whole number, maintaining the ratio between the two.