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Knowing Our Numbers - Estimation

Grade 6CBSE

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

🔑Concepts

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Estimation is the process of finding an approximate value of a quantity that is 'close enough' to the actual value. It is often represented visually on a number line where a number is moved to its nearest 'landmark' value like a multiple of 10,100,10, 100, or 10001000.

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Rounding to the nearest Tens: Look at the digit in the ones place. If it is 55 or more (5,6,7,8,95, 6, 7, 8, 9), round up by adding 11 to the tens digit and replacing the ones digit with 00. If it is less than 55 (0,1,2,3,40, 1, 2, 3, 4), round down by keeping the tens digit the same and replacing the ones digit with 00. Imagine a hill where digits 1−41-4 roll back to the previous ten and 5−95-9 roll forward to the next ten.

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Rounding to the nearest Hundreds: Examine the digit in the tens place. If this digit is 55 or greater, round up to the next hundred. If it is less than 55, round down. For example, 841841 is rounded to 800800 because 4<54 < 5, while 859859 is rounded to 900900 because the tens digit is 55.

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Rounding to the nearest Thousands: Focus on the digit in the hundreds place. If it is ≥5\ge 5, round up; if <5< 5, round down. Visualizing this on a long scale, 4,8004,800 is closer to 5,0005,000 than to 4,0004,000, so it rounds up to 5,0005,000.

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The General Rule for Estimation: To estimate the result of an arithmetic operation (addition, subtraction, multiplication), round each number involved to its greatest place value. For example, in 439+3,342439 + 3,342, round 439439 to the nearest hundred and 3,3423,342 to the nearest thousand.

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Estimating Sums and Differences: Instead of calculating exact totals, we round the numbers first and then perform the operation. This provides a quick check for the reasonableness of a calculation. For instance, 5,290+17,9865,290 + 17,986 can be estimated as 5,000+18,000=23,0005,000 + 18,000 = 23,000.

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Estimating Products: To estimate a product, round each factor to its greatest place value. For 63×18263 \times 182, we round 6363 to 6060 (nearest ten) and 182182 to 200200 (nearest hundred). The estimated product is 60×200=12,00060 \times 200 = 12,000.

📐Formulae

Estimated Sum≈Rounded Number 1+Rounded Number 2\text{Estimated Sum} \approx \text{Rounded Number 1} + \text{Rounded Number 2}

Estimated Difference≈Rounded Number 1−Rounded Number 2\text{Estimated Difference} \approx \text{Rounded Number 1} - \text{Rounded Number 2}

Estimated Product≈Rounded Number 1×Rounded Number 2\text{Estimated Product} \approx \text{Rounded Number 1} \times \text{Rounded Number 2}

Rule for rounding n:If digit at (n−1) place ≥5, then round up.\text{Rule for rounding } n: \text{If digit at } (n-1) \text{ place } \ge 5, \text{ then round up.}

💡Examples

Problem 1:

Estimate the sum: 5,290+17,9865,290 + 17,986.

Solution:

  1. Round 17,98617,986 to the nearest thousand. Since the hundreds digit is 99 (which is >5> 5), it rounds up to 18,00018,000.
  2. Round 5,2905,290 to the nearest thousand. Since the hundreds digit is 22 (which is <5< 5), it rounds down to 5,0005,000.
  3. Add the rounded numbers: 18,000+5,000=23,00018,000 + 5,000 = 23,000.

Explanation:

To estimate the sum of two large numbers, we round both numbers to the same significant place value (in this case, thousands) to simplify the mental addition.

Problem 2:

Estimate the product of 1,291×5921,291 \times 592 using the general rule.

Solution:

  1. Round 1,2911,291 to its greatest place value (thousands). Since the hundreds digit is 22, it rounds to 1,0001,000.
  2. Round 592592 to its greatest place value (hundreds). Since the tens digit is 99, it rounds to 600600.
  3. Multiply the rounded numbers: 1,000×600=6,00,0001,000 \times 600 = 6,00,000.

Explanation:

Using the general rule, we round each number to its own highest place value. This makes the multiplication much easier as we only need to multiply the non-zero digits and count the total number of zeros.