Rounding and Estimation: Additional Practice (Set 1)

The hundredths digit is the second $9$ in $0.79\underline{9}51$, and the deciding digit, the thousandths $9$, is $5$ or more, so round up. Raising $99$ hundredths carries.

$0.79951 \approx 0.80$

The ten hundredths become one tenth, so $0.79$ rolls up to $0.80$. Keep two decimal places.

A number rounds to $5.0$ to the nearest tenth when it lies from the midpoint $4.95$ up to (but not including) $5.05$.

$4.95 \le x < 5.05$

The values $4.96$, $5.04$, and $4.99$ fall in this band, but $5.06$ is past $5.05$, so it rounds to $5.1$ instead.

Write $\tfrac{5}{6}$ as the repeating decimal $0.8333\ldots$. The hundredths digit is $3$, and the deciding digit, the thousandths $3$, is less than $5$.

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

Keep the hundredths digit, giving $0.83$. The repeating $3$s never reach the midpoint $0.835$.

A whole number rounds to $300$ to the nearest hundred when it lies from the midpoint $250$ up to (but not including) $350$.

$250 \le n < 350$

That band runs from $250$ through $349$, which is $349 - 250 + 1 = 100$ whole numbers.

The tenths digit is $4$, and the deciding digit, the hundredths $9$, is $5$ or more, so round up.

$9 \ge 5 \;\Rightarrow\; \text{round up}$

Raise the tenths from $4$ to $5$ and keep one decimal place, giving $0.5$. The endless $9$s push the value past the midpoint $0.45$.

Rounding to the nearest tenth, only the single digit just to the right of the tenths place decides it, which is the hundredths digit $4$.

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

So $0.8462$ rounds to $0.8$. The student's error was reading the whole tail $462$; the digits past the hundredths place cannot move the rounding up.

Rounding to the nearest hundred thousand, the deciding digit is the ten-thousands digit, which is $4$.

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

The digits past the ten-thousands place do not matter, so $2{,}349{,}500$ rounds down to $2{,}300{,}000$.

Round each factor: $612 \approx 600$ and $0.49 \approx 0.5$.

$612 \times 0.49 \approx 600 \times 0.5 = 300$

Here $612$ rounds down and $0.49$ rounds up, so the rounding directions disagree and you cannot tell the side from them alone. Computing exactly, $612 \times 0.49 = 299.88$, which is less than $300$, so the estimate is $300$ and the exact value is less.

A whole number rounds to $70$ to the nearest ten when it lies from the midpoint $65$ up to (but not including) $75$.

$65 \le n < 75$

The smallest whole number in that band is $65$, which rounds up to $70$ by the round-half-up convention.

Round each price to the nearest dollar: $19.95 \to 20$, $4.89 \to 5$, and $12.10 \to 12$.

$20 + 5 + 12 = 37$

The total is about $37$ dollars. The exact total is $36.94$ dollars, so the estimate is within a few cents.

Estimate the total before trusting the note: round $1.85$ to $2$ cups, with $7$ batches.

$7 \times 1.85 \approx 7 \times 2 = 14$

The true total is $12.95$ cups, close to the estimate of $14$. The note's $1.295$ is ten times too small, so the decimal point landed one place too far left.

Round each number to one nonzero digit: $7{,}812 \approx 8{,}000$ and $38 \approx 40$, since $8{,}000$ divides nicely by $40$.

$7{,}812 \div 38 \approx 8{,}000 \div 40 = 200$

The answer is about $200$. The exact quotient is about $206$, so the estimate is in the right ballpark.