Problem 93 - Algebra 1: Concepts and Skills

Question

Estimate the answer. Then evaluate the expression. $$ 5.175+1.15 $$

Step-by-Step Solution

Answer

The estimated sum of 5.175 and 1.15 is 7. The exact sum is 6.325.

Step 1: Estimate the answer

Before performing the exact calculation, an estimation can be done. This would involve rounding the decimals to the nearest whole number. Thus, 5.175 can be rounded down to 5, and 1.15 can be rounded up to 2. Hence, an estimated sum would be \(5+2=7\)

Step 2: Exact Calculation

For the exact calculation, the decimals are added similarly to how integers are added, with the difference that the decimal places need to be aligned. Hence, writing the numbers one below the other gives: \n\n5.175\n+1.150\n\nThen, adding them gives the exact answer of 6.325.

Key Concepts

Decimal Addition

Decimal addition is similar to adding whole numbers, but with the added step of aligning decimal points. When adding decimals, such as in the expression \(5.175 + 1.15\), it's crucial to write down the numbers so that the decimal points are in a vertical line. This method ensures that each place value (tenths, hundredths, thousandths, etc.) is correctly matched.For example, you would write the numbers like this to add them:
\[\begin{align*} &\phantom{0}5.175\+ &\phantom{0}1.15\overline{0}\end{align*}\]Notice how an extra zero is added to \(1.15\) to match the thousandths place of \(5.175\). This does not change the value of the number but allows for proper alignment. Once aligned, you add each column starting from the rightmost digit, just like with whole numbers, making sure to carry over any value more than 9, similar to traditional addition. The sum of \(5.175 + 1.15\) is then \(6.325\).

Rounding Decimals

Rounding decimals simplifies numbers by reducing the number of digits while keeping the number's value close to the original. To round a decimal to the nearest whole number, focus on the tenths place. If the tenths digit is 5 or greater, round up. If it's less than 5, round down.

• If the decimal is \(5.175\), look at the hundredths place, which is 7.
• Since 7 is greater than 5, round the whole number up: \(5.175\) rounds to 6.
• For \(1.15\), the tenths place is 1, and the hundredths place is 5, which means you would round it up to 2.

In our exercise, this made the estimate \(5 + 2 = 7\).
It's also possible to round to other decimal places, depending on the level of precision required. For instance, rounding to the nearest tenth, \(5.175\) would be rounded to \(5.2\), and \(1.15\) stays \(1.2\), yielding an estimate of \(5.2 + 1.2 = 6.4\). This technique is commonly used to make complex calculations more manageable and to understand the scale of numbers without needing an exact figure.

Arithmetic Estimation

Arithmetic estimation is a valuable skill that saves time and helps to quickly gauge the correctness of a calculated answer. It's about finding a number that is close enough to the exact answer without being exact itself. This is particularly useful in situations where you need a rough idea of the total, or when precise calculations are not necessary.To estimate a sum accurately, remember two key steps:

• Round each decimal number to the nearest whole number or to a place value that makes the arithmetic easy.
• Perform the arithmetic operation on these rounded numbers.

The goal is not to achieve the perfect answer but to acquire a ballpark figure that is within a reasonable range of it. Our exercise demonstrated how rounding \(5.175\) to 5 and \(1.15\) to 2 provided an estimated sum of 7. This quick estimate helps in verifying that our final calculation of the exact sum, \(6.325\), is sensible. Actively practicing arithmetic estimation empowers you with the ability to validate results and ensures that any major calculation errors can be swiftly identified and rectified.