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A Peek Beyond the Point - Locating and Comparing Decimals

Grade 7CBSE

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

🔑Concepts

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A decimal number consists of a whole number part and a fractional part, separated by a decimal point. The digits to the left of the decimal represent whole numbers (ones,tens,hundredsones, tens, hundreds), and the digits to the right represent parts of a whole (tenths,hundredths,thousandthstenths, hundredths, thousandths).

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Locating on Number Line: To represent a decimal like 0.70.7 on a number line, divide the space between 00 and 11 into 1010 equal parts. Each part represents 110\frac{1}{10} or 0.10.1. The 7th7^{th} division from 00 is 0.70.7.

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Like and Unlike Decimals: Decimals having the same number of decimal places are called like decimals (e.g., 2.352.35 and 8.128.12). Decimals with different numbers of decimal places are unlike decimals (e.g., 0.50.5 and 0.520.52).

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Comparing Decimals: To compare two decimals, first compare the whole number parts. If they are equal, compare the tenths digits. If those are equal, compare the hundredths, and so on. It is helpful to convert unlike decimals to like decimals by adding trailing zeros (e.g., compare 0.60.6 as 0.600.60 against 0.580.58).

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Expanded Form: A decimal can be written as the sum of the values of its digits. For example, 15.24=10+5+210+410015.24 = 10 + 5 + \frac{2}{10} + \frac{4}{100}.

📐Formulae

Place Value of nth digit after decimal=110n\text{Place Value of } n^{th} \text{ digit after decimal} = \frac{1}{10^n}

Expanded Form: ab.cde=(a×10)+(b×1)+c10+d100+e1000\text{Expanded Form: } ab.cde = (a \times 10) + (b \times 1) + \frac{c}{10} + \frac{d}{100} + \frac{e}{1000}

Conversion: Numerator10k=0.0...0⏟k−1Numerator\text{Conversion: } \frac{\text{Numerator}}{10^k} = 0.\underbrace{0...0}_{k-1} \text{Numerator}

💡Examples

Problem 1:

Which is greater: 2.052.05 or 2.52.5?

Solution:

2.5>2.052.5 > 2.05

Explanation:

First, convert them into like decimals. 2.052.05 has two decimal places, so we write 2.52.5 as 2.502.50. Comparing the whole numbers: 2=22 = 2. Comparing the tenths place: 5>05 > 0. Therefore, 2.50>2.052.50 > 2.05.

Problem 2:

Find the value of 12.35−7.812.35 - 7.8.

Solution:

12.35−7.804.55\begin{array}{r} 12.35 \\ - 7.80 \\ \hline 4.55 \end{array}

Explanation:

To subtract, we first convert 7.87.8 to the like decimal 7.807.80. We align the decimal points vertically and subtract normally, carrying over where necessary.

Problem 3:

Express 4+310+710004 + \frac{3}{10} + \frac{7}{1000} as a decimal.

Solution:

4.3074.307

Explanation:

The whole number is 44. The tenths place (110\frac{1}{10}) has 33. There is no hundredths place (1100\frac{1}{100}) mentioned, so we put 00. The thousandths place (11000\frac{1}{1000}) has 77. Combining these, we get 4.3074.307.