Problem 90 - Algebra 1: Concepts and Skills

Question

Estimate the answer. Then evaluate the expression. $$ 10.35+5.301 $$

Step-by-Step Solution

Answer

The estimated sum of 10.35 and 5.301 is 15. The actual sum is 15.651.

Step 1: Estimation

To estimate, simplify each of the decimals to a number that can be easily added. In this case, 10.35 can be simplified to 10 and 5.301 can be simplified to 5. The sum would then be estimated as \(10 + 5 = 15\).

Step 2: Align the decimals

Before we perform the addition, it's important to align the decimal points.\10.35\+ 5.301

Step 3: Carry out the addition

Now add the numbers column by column from right to left. If there is any digit in one number but not in the other, then just carry the existing digit down to the sum. The result gives: \10.35\+ 5.301\= 15.651

Key Concepts

Estimating Sums

Understanding how to estimate the sum of decimal numbers is a foundational skill in mathematics. When estimating sums, the goal is to simplify the numbers to make mental calculation easier while retaining a result that is close to the actual sum.

Here's how you do it: First, round each decimal to the nearest whole number or to a decimal place that makes the numbers easier to work with. In our example, the numbers 10.35 and 5.301 can be rounded to 10 and 5, respectively. After rounding, you simply add the estimated numbers together, which in our case provides an estimated sum of 15.

Estimating is particularly useful when you need a quick calculation and an approximate result is sufficient. It is commonly used in everyday life scenarios, such as when shopping and needing to quickly calculate the total cost of items being purchased.

Aligning Decimals

When adding decimals, it's essential to keep the decimal points aligned to ensure that the corresponding place values are correctly matched. To do this, write the numbers one under the other with the decimal points directly in line. If one number has fewer digits, you can add zeros to the end of the number to help maintain the alignment.

Looking at our example, we'd write:

 10.35+  5.301

Alignment ensures that we add tenths to tenths, hundredths to hundredths, and so on, all the way to the whole numbers. If this step is overlooked, it can lead to incorrect calculations and confounding results.

Step-by-Step Addition

After estimating and aligning, the next step in the proper addition of decimals is the actual computation, carried out one column at a time from right to left.

Start with the rightmost digits and add them, carrying over any extra to the next column on the left, just like you would with whole numbers. Here's how our example works out:

 10.35+  5.301------ 15.651

You bring down any digits directly if there's nothing in the corresponding place in the other number. Once you've added all the columns, you should have the correct sum with the decimal in place. Step-by-step addition is critical for achieving accurate results, especially when working with financial data, measurements, or in scientific calculations where precision is paramount.