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Large Numbers Around Us - Exact and Approximate Values

Grade 7CBSE

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

🔑Concepts

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Large numbers are represented using two main systems: the Indian System of Numeration and the International System of Numeration. In the Indian system, we use terms like lakhs and crores, whereas the International system uses millions and billions.

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The place value of a digit increases by 1010 times as we move from right to left. For example, 1 ten=10×1 unit1 \text{ ten} = 10 \times 1 \text{ unit} and 1 lakh=10×10,0001 \text{ lakh} = 10 \times 10,000.

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Estimation (Approximation) is the process of finding a number that is close enough to the exact value to be useful, but easier to calculate. This is often done by rounding off.

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Rounding off rules: To round to the nearest ten, look at the ones digit. If it is 55 or more, increase the tens digit by 11. If it is less than 55, leave the tens digit as it is. Replace all digits to the right with 00.

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Estimated sums, differences, and products are used to get a quick idea of the answer without performing lengthy calculations with exact large numbers.

📐Formulae

1 Million=10 Lakhs=1,000,0001 \text{ Million} = 10 \text{ Lakhs} = 1,000,000

10 Millions=1 Crore=1,00,00,00010 \text{ Millions} = 1 \text{ Crore} = 1,00,00,000

1 Billion=1,000 Millions=1,00,00,00,0001 \text{ Billion} = 1,000 \text{ Millions} = 1,00,00,00,000

Estimated Value≈Exact Value\text{Estimated Value} \approx \text{Exact Value}

💡Examples

Problem 1:

Write the number 72,50,38472,50,384 in the International system and round it off to the nearest lakh.

Solution:

In the International system, the number is 7,250,3847,250,384 (Seven million, two hundred fifty thousand, three hundred eighty-four). To round 7,25,0,3847,25,0,384 to the nearest lakh (Indian system equivalent), we look at the ten-thousands digit which is 55. Since it is ≥5\ge 5, we round up. The approximate value is 73,00,00073,00,000.

Explanation:

We place commas after every three digits from the right for the International system. For rounding to the nearest lakh, we check the digit at the 10,00010,000 place.

Problem 2:

Estimate the sum of 4,39,2814,39,281 and 2,73,5142,73,514 by rounding off to the nearest thousand.

Solution:

Rounding 4,39,2814,39,281 to the nearest thousand gives 4,39,0004,39,000 (since 2<52 < 5). Rounding 2,73,5142,73,514 to the nearest thousand gives 2,74,0002,74,000 (since 5≥55 \ge 5).

439000+274000713000\begin{array}{r} 439000 \\ + 274000 \\ \hline 713000 \end{array}

The estimated sum is 7,13,0007,13,000.

Explanation:

Rounding to the nearest thousand means looking at the hundreds digit to decide whether to round up or down.

Problem 3:

Find the difference between the largest 77-digit number and the smallest 66-digit number.

Solution:

Largest 77-digit number = 99,99,99999,99,999. Smallest 66-digit number = 1,00,0001,00,000.

9999999−1000009899999\begin{array}{r} 9999999 \\ - 100000 \\ \hline 9899999 \end{array}

The exact difference is 98,99,99998,99,999.

Explanation:

The largest number of nn digits consists of nn nines. The smallest number of nn digits consists of 11 followed by n−1n-1 zeros.

Exact and Approximate Values Class 7 Notes & Examples