Rounding and Estimation

$4,732$ sits between $4,700$ and $4,800$, and the midpoint is $4,750$. Since $4,732$ is below that, it leans toward the lower neighbor. The gap down is $32$ and the gap up is $68$, so the nearest hundred is $\boxed{{\displaystyle 4700}}$. Only a value at or above $4,750$ would tick up to $4,800$.

The thousands digit is the $8$, so the answer is either $68,000$ or $69,000$. The deciding digit is the one just to its right, the hundreds digit $4$. Since $4$ is under $5$, the thousands digit stays put and everything after it drops to zero. $$68,431\approx \boxed{{\displaystyle 68,000}}$$ The midpoint check agrees, since $68,431$ is below $68,500$.

The hundreds on either side of $2,950$ are $2,900$ and $3,000$, and the midpoint is $2,950$ itself. A tie rounds to the larger value, so we go up to $\boxed{{\displaystyle 3000}}$. The hundreds digit is a $9$, so it cannot climb. It rolls to $0$ and carries $1$ into the thousands, turning the $2$ into a $3$.

The tenths on either side of $7.83$ are $7.8$ and $7.9$, and the midpoint is $7.85$. The hundredths digit is $3$, so the reading sits below the midpoint and stays with the lower neighbor. Keep the tenths digit and drop the rest, giving $\boxed{{\displaystyle 7.8}}$.

The hundredths on either side of $3.617$ are $3.61$ and $3.62$, and the midpoint is $3.615$. The thousandths digit is $7$, which is $5$ or more, so round up. The harbor chart records $\boxed{{\displaystyle 3.62}}$. The distances agree, since $3.617$ is $0.007$ above $3.61$ but only $0.003$ below $3.62$.

Keeping two decimal places makes the thousandths digit the decider, and here it is $5$ with nothing behind it, an exact tie. So the hundredths place rounds up. That digit is a $9$, so it becomes $0$ and carries $1$ into the tenths, which is also $9$, so that becomes $0$ and carries into the ones, lifting the $1$ to a $2$. $$1.995\approx \boxed{{\displaystyle 2.00}}$$ Keep both zeros, since the notebook still records two decimal places.

The multiples of ten around $-83$ are $-90$ and $-80$. The gap down to $-90$ is $7$, and the gap up to $-80$ is only $3$, so the technician records $\boxed{{\displaystyle -80}}$ degrees. Negatives use the same midpoint test as positives. The midpoint here is $-85$, and $-83$ sits to the right of it.

The tenths around $-2.35$ are $-2.4$ and $-2.3$, and $-2.35$ is $0.05$ from each, so it is an exact tie. A tie rounds to the larger value, and the larger of two negatives is the one farther right. The display shows $\boxed{{\displaystyle -2.3}}$. It feels backwards because $2.3$ is less than $2.4$, but with the minus signs attached, $-2.3$ is the bigger number.

For $487$, the leftover past the hundred is $87$, above $50$, so it rounds up to $500$. For $213$ the leftover is $13$, so it stays at $200$. For $356$ the leftover is $56$, so it rounds up to $400$. Adding gives $500+200+400=\boxed{{\displaystyle 1100}}$. The exact total is $1,056$, so the estimate tells you instantly whether the van is near its limit.

$29$ rounds to $30$ at the nearest ten, and $6.10$ rounds to $6$ at the nearest dollar, since $10$ cents is nowhere near the $6.50$ midpoint. Multiplying the friendly numbers gives $30\times 6=\boxed{{\displaystyle 180}}$. The printed $1,769$ is about ten times that, the signature of a decimal point that slipped one spot. The exact total is $176.90$.

A reported $4.3$ covers everything from $4.25$ up to just under $4.35$, because $4.35$ is a tie that rounds up to $4.4$ instead. The largest two-decimal value in that stretch is $\boxed{{\displaystyle 4.34}}$, whose hundredths digit $4$ sends it down to $4.3$. The band is half-open, so the lower end is included and the upper end is not.

Rounding to $7$ at the nearest dollar means $6.5\le p<7.5$. Rounding to $6.8$ at the nearest tenth means $6.75\le p<6.85$, which sits entirely inside the first range. The prices to the cent in that band run $6.75$ through $6.84$, and the only one with a hundredths digit of $0$ is $\boxed{{\displaystyle 6.80}}$. Turning each rounding clue into a range and overlapping them is the standard way to pin a value down.

The hundreds around $7,284$ are $7,200$ and $7,300$, and the midpoint is $7,250$. Since $7,284$ is above the midpoint, it climbs, and the press box reports $\boxed{{\displaystyle 7300}}$ fans.

$3,500$ is $500$ from $3,000$ and $500$ from $4,000$, so neither thousand is closer. That is an exact tie, and our rule sends a tie to the larger value, so the banner reads $\boxed{{\displaystyle 4000}}$.

The tenths around $6.47$ are $6.4$ and $6.5$, and the midpoint is $6.45$. The hundredths digit is $7$, which is $5$ or more, so the tenths digit goes up by one and $6.47$ rounds to $\boxed{{\displaystyle 6.5}}$.

The hundredths around $0.382$ are $0.38$ and $0.39$, and the midpoint is $0.385$. The thousandths digit is $2$, below $5$, so the hundredths digit stays and everything after it is dropped, giving $\boxed{{\displaystyle 0.38}}$. The gaps confirm it, $0.002$ down versus $0.008$ up.

$9.5$ is $0.5$ from $9$ and $0.5$ from $10$, so it is an exact tie. A tie rounds to the larger value, so the schedule lists the trip as $\boxed{{\displaystyle 10}}$ minutes.

$0.65$ is $0.05$ from $0.6$ and $0.05$ from $0.7$, so it is a perfect tie. A tie rounds to the larger value, so the fitness band shows $\boxed{{\displaystyle 0.7}}$.

The hundredths digit of $0.396$ is the $9$, and the thousandths digit $6$ is at least $5$, so that $9$ rounds up. A $9$ cannot become a single-digit $10$, so it turns into $0$ and carries $1$ into the tenths, taking $3$ to $4$. That gives $\boxed{{\displaystyle 0.40}}$. Keep the trailing zero so the answer still shows the hundredths place the spec sheet asked for.

The hundredths digit of $8.97$ is $7$, which is at least $5$, so the tenths place rounds up. The tenths digit is already $9$, so it rolls over to $0$ and carries $1$ into the ones, taking the $8$ to $9$. The trip computer shows $\boxed{{\displaystyle 9.0}}$. Write the trailing zero to show the value was rounded to the nearest tenth.

The multiples of ten around $-46$ are $-50$ and $-40$. The gap to $-50$ is $4$ and the gap to $-40$ is $6$, so $-46$ rounds to $\boxed{{\displaystyle -50}}$. The midpoint is $-45$, and $-46$ sits to the left of it, so there is no tie to break.

$-3.5$ is exactly halfway between $-4$ and $-3$, so closeness cannot decide and the half-way rule takes over. Larger means farther right on the number line, and $-3$ is right of $-4$, so the elevation rounds to $\boxed{{\displaystyle -3}}$ meters. Between two negatives, the larger one is always the one closer to zero.

$-1.25$ is $0.05$ from $-1.3$ and $0.05$ from $-1.2$, so it is an exact tie. A tie goes to the larger value, and $-1.2$ lies to the right of $-1.3$ on the number line, so the app displays $\boxed{{\displaystyle -1.2}}$.

$41$ has a ones digit of $1$, so it rounds down to $40$. $19.80$ has $80$ cents, past the $50$ mark, so it rounds up to $20$ dollars. Multiplying gives $40\times 20=\boxed{{\displaystyle 800}}$. The printed $81.18$ is nowhere near that ballpark, so the receipt is almost certainly missing a digit.

Round each weight to the nearest hundred one at a time. For $596$, the tens digit is $9$, which is well past halfway, so it rounds up to $600$. For $207$, the tens digit is $0$, so it rounds down to $200$. For $398$, the tens digit is $9$, so it rounds up to $400$. Now add the friendly numbers, $600+200+400=1200$. The estimate clears the $1000$ gram mark with room to spare. $\boxed{{\displaystyle 1200}}$

The tenths around the stored reading are $5.2$ and $5.3$, and the midpoint is $5.25$. By the half-way rule that midpoint rounds up to $5.3$, while $5.24$ would drop to $5.2$. The smallest qualifying value is $\boxed{{\displaystyle 5.25}}$.