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Chapter 1: Problem 57

Estimate each quotient by first rounding to the nearest hundred. Show your work. $$ 1165 \div 236 $$

Short Answer

Expert verified
Estimated quotient: 6

Step by step solution

01

Round the Divisor to the Nearest Hundred

First, round the divisor 236 to the nearest hundred. Since 36 is closer to zero than 100, round 236 down to 200.
02

Round the Dividend to the Nearest Hundred

Next, round the dividend 1165 to the nearest hundred. Since 65 is closer to 100 than zero, round 1165 up to 1200.
03

Perform the Division

Now, divide the rounded numbers. Divide 1200 by 200: \[ 1200 \div 200 = 6. \]
04

Verify the Estimation

Make sure the estimated result makes sense. The original problem 1165 divided by 236 is approximately equal to 5, so the estimation is reasonable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rounding Numbers
Rounding numbers is a technique used to reduce the digits in a number while keeping its value close to the original. This makes large numbers easier to work with. To round a number to the nearest hundred, you look at the tens digit. If the tens digit is 5 or greater, you round up. If it's less than 5, you round down.

In our problem, we need to round 236 and 1165 to the nearest hundred.
  • For 236, the tens digit is 3 (less than 5), so we round down to 200.
  • For 1165, the tens digit is 6 (5 or greater), so we round up to 1200.
This simplified version of the numbers helps make division easier and quicker.
Division
Division is a mathematical operation where you split a number (the dividend) into equal parts. The number you divide by is called the divisor, and the result is the quotient.

In the estimation process, after rounding the numbers, you perform a simpler division. For instance, our rounded numbers were 1200 and 200. The division is performed as: \(1200 \div 200 = 6\). This gives us the estimated quotient.

To ensure our estimation is sensible, we compare it to the actual problem. Here, \(1165 \div 236\) approximately equals 5.
Math Estimation
Math estimation involves making an educated guess about a calculation's result, and rounding is a key part of estimation. It helps when exact values are unnecessary, and simplifies complex calculations.

In our example:
  • First, we rounded 236 to 200.
  • Then, we rounded 1165 to 1200.
  • We divided the rounded numbers to get the estimate, \(1200 \div 200 = 6\).
Estimation can help verify the reasonableness of answers, making it a valuable skill in everyday math activities. When actual calculations seem tricky, estimating can provide a quick approximation to guide understanding.

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