Problem 73 - Prealgebra

Question

Work Problems 73–80 mentally, without pencil and paper or a calculator. The answer to the problem 251 + 249 is closest to which of the following numbers? a. 500 b. 0 c. 500

Step-by-Step Solution

Answer

The answer is 500.

Step 1: Understand the Problem

We need to find a number that 251 + 249 is closest to from the given choices.

Step 2: Estimate Each Number

Notice that 251 is very close to 250, and 249 is also close to 250. Let's use these rounded numbers to estimate the sum.

Step 3: Perform Mental Math

Add the rounded numbers: 250 + 250 = 500.

Step 4: Compare the Estimate to the Choices

Our estimated sum of 500 matches one of the provided choices. Since 251 + 249 is exactly 500, the closest answer is indeed 500.

Key Concepts

Estimation

Estimation is a useful strategy for quickly finding an approximate answer to mathematical problems without needing exact calculations. Think of estimation as a way to anticipate what a full calculation should give you once you solve it. This helps in making faster decisions and checking the reasonableness of your final answers.

When using estimation in mental math, focus on two main techniques:

• Identify closeness: Look at each number in a problem and identify numbers they are close to. For instance, 251 is close to 250. Consider what happens when you round each number.
• Add rounded numbers: Once rounded, it's easier to perform quick additions or subtractions, as smaller, simpler numbers are involved. Like rounding 249 to 250 as well. The numbers are easier and faster to work with.

Remember, estimation won’t always give you the exact answer, but it will often give you a close enough approximation to be useful in basic problem solving.

Arithmetic Problem Solving

Arithmetic problem solving involves using logical thinking and mathematical operations to resolve problems. Doing it mentally encourages a deeper understanding and sharpens your cognitive skills.

To enhance your arithmetic problem-solving skills:

• Break down the problem: Instead of dealing with complex numbers, simplify them into components that are easier to manage. This makes addition, subtraction, multiplication, or division less intimidating.
• Use strategies: Techniques like estimation simplify the arithmetic process. If thinking of 251 + 249 as 250 + 250 seems easier, go that route. Simplified methods lead to quicker solutions.

When practiced regularly, mental arithmetic boosts number sense, enabling quick calculations in everyday scenarios, such as shopping or budgeting.

Rounding Numbers

Rounding is the process of adjusting numbers to make them simpler, keeping them close to their true value. It’s a fundamental technique in mental math that aids in estimation and quick calculations.

To round numbers mentally:

• Identify place value: Look at the number, and decide which place value to round to (like tens, hundreds, etc.). For 251 and 249, rounding to the nearest ten simplifies both to 250.
• Check midpoint: If the digit after the rounding place is 5 or higher, round the target digit up. If it's less than 5, it stays the same. This is a straightforward rule for rounding numbers.

By mastering rounding, you simplify mental arithmetic. It’s a skill particularly valuable in scenarios requiring quick reasoning or when precision isn’t essential.