Rounding and Estimation: Practice

The hundredths digit is the second $9$ in $4.99\underline{6}2$, and the next digit, the thousandths $6$, is $5$ or more, so round up. Raising $99$ hundredths by one hundredth carries twice.

$4.9962 \approx 5.00$

The ten hundredths roll into the tenths, which were also $9$, so those roll into the ones: $4.99$ becomes $5.00$. Keep two decimal places.

A number rounds to $7.3$ to the nearest tenth when it lies from the midpoint $7.25$ up to (but not including) the midpoint $7.35$.

$7.25 \le x < 7.35$

The values $7.34$, $7.27$, and $7.349$ all fall in this band, but $7.351$ is past $7.35$, so it rounds to $7.4$ instead.

The hundredths digit is $6$, and the deciding digit is the next one, the thousandths digit, which is also $6$.

$6 \ge 5 \;\Rightarrow\; \text{round up}$

Raise the hundredths from $6$ to $7$ and keep two decimal places, giving $0.77$. The endless $6$s after the thousandths place cannot change the decision.

Round each factor to the nearest whole number: $39.6$ rounds to $40$ and $5.2$ rounds to $5$.

$39.6 \times 5.2 \approx 40 \times 5 = 200$

Now check the exact product against the estimate.

$39.6 \times 5.2 = 205.92$

The exact value $205.92$ is more than the estimate $200$, so the answer is $200$ and the exact product is more.

Rounding to the nearest thousand, the deciding digit is the hundreds digit, which is $4$.

$4 < 5 \;\Rightarrow\; \text{round down}$

The tens digit $5$ does not matter, because only the single digit just to the right of the thousands place decides it. So $12{,}450$ rounds to $12{,}000$.

The hundredths digit is $3$, and the deciding digit is the next one, the thousandths digit, which is $3$.

$3 < 5 \;\Rightarrow\; \text{round down}$

Keep the hundredths digit and stop, giving $0.03$. The repeating $3$s never reach the midpoint, so the value stays below $0.035$.

Round to the nearest hundred on top and the nearest whole number below: $612 \approx 600$, $389 \approx 400$, and $4.9 \approx 5$. Add the top, then divide.

$\frac{612 + 389}{4.9} \approx \frac{600 + 400}{5} = \frac{1000}{5} = 200$

The estimate is about $200$. The exact value is near $204$, confirming the estimate.

The tenths digit is $5$, and the deciding digit is the next one, the hundredths digit, which is $4$.

$4 < 5 \;\Rightarrow\; \text{round down}$

Keep the tenths digit and stop, giving $0.5$. The repeating $4$s stay below the midpoint $0.55$, so the value rounds down.

Round each factor to a single nonzero digit: $0.0488$ rounds to $0.05$ and $612$ rounds to $600$.

$0.0488 \times 612 \approx 0.05 \times 600 = 30$

The answer is about $30$. The exact product is about $29.9$, so the estimate is close and quickly rules out an answer near $3$ or $300$.

First round $5.55$ to the nearest tenth. The hundredths digit is $5$, so round the tenths up: $5.55 \approx 5.6$.

$5.55 \approx 5.6 \approx 6$

Now round $5.6$ to the nearest whole number. The tenths digit $6$ is $5$ or more, so round up to $6$. The final answer is $6$.

Round each number to make the division easy: $846 \approx 800$ and $19 \approx 20$, since $800$ divides nicely by $20$.

$846 \div 19 \approx 800 \div 20 = 40$

The answer is about $40$. The exact quotient is about $44.5$, so the estimate is in the right ballpark.

Write $\tfrac{2}{3}$ as the repeating decimal $0.6666\ldots$. The thousandths digit is $6$, and the deciding digit is the next $6$.

$6 \ge 5 \;\Rightarrow\; \text{round up}$

Raise the thousandths from $6$ to $7$ and keep three decimal places, giving $0.667$. The endless $6$s past that place do not change the decision.