Estimate the quotient: [tex]$6.31 \times 10^5+2.085 \times 10^3$[/tex]
Answer (2)
The estimated quotient of 6.31 × 1 0 5 divided by 2.085 × 1 0 3 is approximately 300 . This is found by estimating the coefficients to simplify the division and applying the properties of exponents. Therefore, the final result is 300 .
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Divide the coefficients: 2.085 6.31 ≈ 3 .
Divide the powers of 10: 1 0 3 1 0 5 = 1 0 2 .
Combine the results: 3 × 1 0 2 = 300 .
The estimated quotient is 300 .
Explanation
Understanding the Problem We are asked to estimate the quotient of 6.31 × 1 0 5 and 2.085 × 1 0 3 . This means we want to find an approximate value for the division of these two numbers.
Setting up the Quotient First, let's write the expression for the quotient: 2.085 × 1 0 3 6.31 × 1 0 5 We can rewrite this as: 2.085 6.31 × 1 0 3 1 0 5
Dividing Powers of 10 Now, let's divide the powers of 10. Using the rule 1 0 b 1 0 a = 1 0 a − b , we have: 1 0 3 1 0 5 = 1 0 5 − 3 = 1 0 2
Dividing the Coefficients Next, we need to divide the coefficients: 2.085 6.31 . To estimate this, we can round the numbers to make the division easier. We can approximate 6.31 as 6.3 and 2.085 as 2.1 . So, we have: 2.085 6.31 ≈ 2.1 6.3 Now, we can divide: 2.1 6.3 = 3
Combining the Results Finally, we combine the results: 2.085 6.31 × 1 0 3 1 0 5 ≈ 3 × 1 0 2 Since 1 0 2 = 100 , we have: 3 × 1 0 2 = 3 × 100 = 300
Final Answer Therefore, the estimated quotient is 300.
Examples
Estimating quotients is useful in everyday situations, such as when you're trying to quickly calculate how many items you can buy with a certain amount of money. For example, if you have $631 and each item costs $2.085, you can estimate how many items you can buy by dividing 631 by 2.085. By estimating, you can quickly determine that you can buy approximately 300 items. This kind of estimation is also helpful in science and engineering when dealing with very large or very small numbers.
