Lesson 5: Estimation and Reasonableness Checks

Property

To estimate the sum of decimals, round each decimal to a nearby whole number or another convenient place value (like the nearest tenth). Then, add the rounded numbers to find an approximate sum. This can be represented as: $a + b \approx \text{round}(a) + \text{round}(b)$.

Examples

• To estimate $4.9 + 7.2$, round $4.9$ to $5$ and $7.2$ to $7$. The estimated sum is $5 + 7 = 12$. The actual sum is $12.1$.
• To estimate $15.85 + 3.12$, round $15.85$ to $16$ and $3.12$ to $3$. The estimated sum is $16 + 3 = 19$. The actual sum is $18.97$.

Explanation

Estimating before you calculate helps you make sense of the numbers and predict a reasonable answer. By rounding decimals to the nearest whole numbers, you can perform a simpler addition problem in your head. This estimate serves as a valuable check to see if your final, precise answer is correct. If your calculated sum is very different from your estimate, you may have made a calculation error, like misaligning the decimal points.

Property

To estimate the difference between two decimals, first round each number to the same, convenient place value (such as the nearest whole number or tenth). Then, subtract the rounded numbers to find the estimated difference. If $a \approx a_{rounded}$ and $b \approx b_{rounded}$, then $a - b \approx a_{rounded} - b_{rounded}$.

Examples

To estimate the value of $12.82 - 4.19$ by rounding to the nearest whole number:

• The estimated difference is $13 - 4 = 9$.

To estimate the value of $7.58 - 3.91$ by rounding to the nearest tenth:

• The estimated difference is $7.6 - 3.9 = 3.7$.

Explanation

Estimating differences helps you quickly check if an answer is reasonable without performing a complex calculation. By rounding the numbers in a subtraction problem first, you can work with simpler, whole numbers or tenths. This mental math strategy is useful for everyday situations, like calculating change or comparing prices. The closer your rounding is to the original numbers (e.g., rounding to tenths vs. whole numbers), the more precise your estimate will be.

Property

To determine if an estimate is an overestimate (greater than the actual answer) or an underestimate (less than the actual answer), analyze how each number was rounded.

• Addition: If both numbers are rounded up, the sum is an overestimate. If both are rounded down, the sum is an underestimate.
• Subtraction ($A - B$): If $A$ is rounded up and $B$ is rounded down, the difference is an overestimate. If $A$ is rounded down and $B$ is rounded up, the difference is an underestimate.

Examples

Property

Most conversions between customary and metric systems are approximate ($\approx$), meaning they are rounded. An exact conversion ($=$) is a defined relationship. The primary exact conversion between systems is for inches and centimeters.

Examples