Problem 9 - Fundamentals of Mathematics

Question

Estimate each value. After you have made an estimate, find the exact value. Results may vary. \((\) Section 8.2) \(2,660 \div 28\)

Step-by-Step Solution

Answer

Estimate: 90; Exact value: 95.

Step 1: Make an Estimate

Start by rounding the numbers to make the division easier. We can round 2,660 to 2,700 and 28 to 30. Now calculate the estimated quotient: \[2,700 \div 30 = 90\]So, the estimated value is approximately 90.

Step 2: Find the Exact Value

Now, let's calculate the exact value of the division. Divide 2,660 by 28 using long division or a calculator:1. 28 goes into 266 about 9 times (since \(28 \times 9 = 252\)). 2. Subtract 252 from 266, which leaves 14.3. Bring down the 0, making it 140.4. 28 goes into 140 about 5 times (since \(28 \times 5 = 140\)).5. Subtract 140 from 140, which leaves 0.Hence, the exact value is 95.

Step 3: Compare Estimate with Exact Value

Compare your estimated value (90) and the exact value (95). Notice that the estimate of 90 is close to the exact value of 95, which means our estimation was reasonable.

Key Concepts

Rounding Numbers

Rounding numbers is a technique we use to simplify calculations and make them more manageable. When faced with a division problem such as 2,660 divided by 28, the first step is to round these numbers.

• For 2,660, you might round up to 2,700 because it's close and easier to work with.
• For 28, rounding to 30 simplifies the division process further.

The idea is to choose numbers that make mental calculations easier while still being close to the original values. This approximation helps us predict a quotient that is in the right ballpark before performing the exact calculation.
Rounding numbers is particularly useful when you're under time pressure or need a quick estimate without extensive computation.

Exact Division

Once you've made your estimate, it's important to find the exact value using proper division methods. This usually involves long division or using a calculator. For the division of 2,660 by 28, here's a simple approach:

• Start by seeing how many times 28 fits into the first part of 2,660; in this case, 266. It fits approximately 9 times.
• Multiply 28 by 9 to get 252, then subtract this from 266 to get a remainder of 14.
• Bring down the next digit, which results in 140.
• Determine how many times 28 fits into 140, which is exactly 5 times.
• Finally, multiply 28 by 5 to get 140 and subtract this to finish the division, resulting in no remainder.

Thus, the exact result of 2,660 divided by 28 is 95. Exact division ensures accuracy and is essential in verifying the validity of your estimates.

Comparing Estimates

The final step in any division estimation process is to compare your estimated result to the exact value you calculated. This comparison helps you evaluate the correctness of your rounding and estimation methods.

• In this example, the estimated result was 90 and the exact division gave us 95.

The proximity of these two numbers demonstrates that the estimation was close and reasonably accurate. Such comparisons are vital in learning, as they teach us how effective our rounding choices are, and they help in developing better estimation skills.
By understanding and practicing this, students gain confidence in both their mental math abilities and in conducting precise calculations.