Estimate the quotient: [tex]$6.31 \times 10^5+2.085 \times 10^3$[/tex]
Answer (2)
Rewrite the expression as a fraction: 2.085 × 1 0 3 6.31 × 1 0 5 = 2.085 631 .
Approximate the denominator: 2.085 631 ≈ 2 631 .
Calculate the approximate quotient: 2 631 = 315.5 .
Calculate the accurate quotient and round to the nearest integer: 2.085 631 ≈ 303 . The final answer is 303 .
Explanation
Understanding the Problem We are asked to estimate the quotient of 6.31 × 1 0 5 and 2.085 × 1 0 3 . Let a = 6.31 × 1 0 5 and b = 2.085 × 1 0 3 . We want to estimate b a .
Rewriting the Expression We can rewrite a as 631 × 1 0 3 . So, we have 2.085 × 1 0 3 6.31 × 1 0 5 = 2.085 × 1 0 3 631 × 1 0 3 .
Canceling Common Factors We can cancel out the common factor of 1 0 3 to get 2.085 631 .
Approximation Now, let's approximate 2.085 as 2 . Then the expression becomes 2 631 .
Calculating the Quotient Calculating the division, we get 2 631 = 315.5 .
More Accurate Calculation To get a more accurate estimate, we can directly calculate 2.085 631 ≈ 302.64 . Rounding this to the nearest whole number, we get 303 .
Final Answer Therefore, the estimated quotient is approximately 303 .
Examples
Estimating quotients is useful in everyday situations, such as when you're trying to quickly figure out how many items you can buy with a certain amount of money. For example, if you have 6.31 × 1 0 5 dollars and each item costs 2.085 × 1 0 3 dollars, estimating the quotient helps you determine you can buy approximately 303 items. This kind of estimation is also used in science and engineering to make quick calculations and check the reasonableness of results.
To estimate the quotient of 6.31 × 1 0 5 divided by 2.085 × 1 0 3 , we rewrite it as 2085 631000 . Approximating the denominator to 2000 gives an estimated result of approximately 315500 .
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