- Identify the digit in the hundredths place. - Look one place to the right. - Decide whether the hundredths digit stays the same or increases by \(1\).

1. The hundredths digit is \(4\). 2. The thousandths digit is \(7\), so round the hundredths digit up. 3. \(6.347\) rounds to \(6.35\).

\(6.35\)

- Read the value of one small interval on the number line. - Identify the two neighboring tenths. - Decide which labeled tenth is closer to the marker.

1. \(A=2.64\). 2. It lies closer to \(2.6\) than to \(2.7\). 3. Therefore, \(2.64\) rounds to \(2.6\) to the nearest tenth.

\(2.6\)

- Read the marker from the tenths ticks. - Compare its position with the halfway point between \(6\) and \(7\). - Choose the nearer whole number.

1. The marker is at \(6.4\). 2. \(6.4\) is less than halfway from \(6\) to \(7\). 3. Therefore, it rounds to \(6\).

\(6\)

- Identify the digit in the place you are rounding to. - Look at the digit immediately to its right. - Digits \(0\) through \(4\) keep the rounding digit the same; digits \(5\) through \(9\) increase it by \(1\).

1. For a), the tenths digit is \(3\), so \(12.354\) rounds to \(12\). 2. For b), the hundredths digit is \(5\), so the tenths digit rounds up: \(12.4\). 3. For c), the thousandths digit is \(4\), so the hundredths digit stays the same: \(12.35\).

a) \(12\) b) \(12.4\) c) \(12.35\)

- Look at the digit immediately to the right of the rounding place. - When a \(9\) rounds up, regroup into the place to its left. - Preserve trailing zeros that show the requested precision.

1. For a), the thousandths digit is \(6\), so the hundredths digit rounds up. Regrouping gives \(0.40\). 2. For b), the hundredths digit is \(5\), so the tenths digit rounds up. Regrouping gives \(2.0\). 3. For c), the thousandths digit is \(4\), so the hundredths digit stays the same. The result is \(7.00\). 4. For d), the hundredths digit is \(9\), so the tenths digit rounds up. Regrouping gives \(1.0\).

a) \(0.40\) b) \(2.0\) c) \(7.00\) d) \(1.0\)

- Look at the digit immediately to the right of the requested place. - Use the same place-value rule for each decimal. - Regroup when a \(9\) must round up.

1. For a), the hundredths digit is \(6\), so the tenths digit rounds up. The result is \(13.0\). 2. For b), the thousandths digit is \(5\), so the hundredths digit rounds up. The result is \(0.05\). 3. For c), the thousandths digit is \(9\), so the hundredths digit rounds up. Regrouping across the \(9\)s gives \(6.00\).

a) \(13.0\) b) \(0.05\) c) \(6.00\)

- For each part, underline the place you are rounding to. - Look exactly one place to the right of that digit. - Watch for regrouping when a rounded digit becomes \(10\).

1. a) The hundredths digit is \(2\), so \(4.126\) rounds to \(4.1\). 2. b) The thousandths digit is \(5\), so the hundredths place rounds up through regrouping: \(0.995\) rounds to \(1.00\). 3. c) The thousandths digit is \(4\), so \(7.444\) rounds to \(7.44\).

a) \(4.1\) b) \(1.00\) c) \(7.44\)

- Find the value of one small interval. - Read the marker's value to the hundredths place. - Apply the stated rounding rule to the next digit.

1. The consecutive tenths are \(4.2\) and \(4.3\). 2. \(B\) is at \(4.25\). 3. Because the hundredths digit is \(5\), \(4.25\) rounds up to \(4.3\).

\(B=4.25\), and it rounds to \(4.3\).

- Identify each marker's value from the hundredths ticks. - Think about the values that round to \(3.7\). - Pay special attention to values ending in \(5\) hundredths.

1. The marker values are \(A=3.64\), \(B=3.65\), \(C=3.74\), and \(D=3.75\). 2. \(3.64\) rounds to \(3.6\), while \(3.65\) rounds to \(3.7\). 3. \(3.74\) rounds to \(3.7\), while \(3.75\) rounds to \(3.8\). 4. Therefore, points \(B\) and \(C\) round to \(3.7\).

Points \(B\) and \(C\).

- Round each addend before estimating. - Keep the exact decimals for the exact sum. - Compare the estimate and exact result by subtraction.

1. \(3.47\) rounds to \(3\), and \(5.62\) rounds to \(6\). 2. The estimated sum is \(3+6=9\). 3. The exact sum is \(3.47+5.62=9.09\). 4. The exact sum is \(0.09\) greater than the estimate.

Rounded values: \(3\) and \(6\) Estimate: \(9\) Exact sum: \(9.09\) The exact sum is \(0.09\) greater than the estimate.

- Which digit in the original number controls rounding to the nearest tenth? - Do not round to an intermediate place first. - Compare the direct rounding result with the two-stage result.

1. To round directly to the nearest tenth, look at the hundredths digit in the original number. 2. In \(0.445\), the hundredths digit is \(4\). 3. Since \(4<5\), the tenths digit stays \(4\), so \(0.445\) rounds to \(0.4\). 4. Jordan rounded in stages. The first rounding changed the hundredths digit and caused the second rounding to produce a different result. Rounding should be done directly from the original number to the requested place.

Jordan’s method is incorrect because rounding in stages can change the final result. Rounded directly to the nearest tenth, \(0.445\) is \(0.4\).

- To round to the nearest hundredth, inspect the thousandths digit. - Test each number independently. - Keep track of which hundredth each number is closest to.

1. A rounds to \(3.42\) because the thousandths digit is \(5\). 2. B rounds to \(3.42\) because the thousandths digit is \(4\). 3. C rounds to \(3.43\) because the thousandths digit is \(5\). 4. D rounds to \(3.41\) because the thousandths digit is \(4\). 5. E is already \(3.420\), so it rounds to \(3.42\).

A, B, and E

- In a), decide whether the amount you buy may be less than the amount needed. - In b), decide whether the total cost may be more than the money available. - Use the situation, not only the usual rounding rule, to decide whether to round up or down.

1. In a), \(2\) cans provide only \(2\) gallons, which is not enough. You need \(3\) cans. 2. In b), \(3\) tickets cost \(3\times\$6.80=\$20.40\), which is over the budget. You can buy \(2\) tickets.

a) \(3\) cans b) \(2\) tickets

- The hundredths digit controls rounding to the nearest tenth. - Find the lower endpoint of the interval that rounds to \(4.7\). - Apply the rounding rule directly to \(4.75\).

1. In \(4.649\), the hundredths digit is \(4\), so it rounds to \(4.6\). 2. In \(4.65\), the hundredths digit is \(5\), so it rounds to \(4.7\). 3. The least two-decimal-place number that rounds to \(4.7\) is \(4.65\). 4. The number \(4.75\) has a hundredths digit of \(5\), so its tenths digit rounds up from \(7\) to \(8\). Therefore, it rounds to \(4.8\), not \(4.7\).

\(4.649\) rounds to \(4.6\), and \(4.65\) rounds to \(4.7\). a) \(4.65\) b) \(4.75\) rounds to \(4.8\) because its hundredths digit is \(5\).

- Find the halfway points between the two neighboring whole-acre values. - Decide which tenths at the two ends still round to the reported whole number. - Check each endpoint by rounding it to the nearest acre.

1. Values from \(6.5\) acres up to, but not including, \(7.5\) acres round to \(7\) acres. 2. The least possible tenth-acre value is \(6.5\) acres. 3. The greatest possible tenth-acre value before \(7.5\) is \(7.4\) acres.

a) \(6.5\) acres b) \(7.4\) acres

- Identify the digit immediately to the right of the tenths place. - Recall which digits cause rounding up. - Do not round in stages from right to left.

1. When rounding to the nearest tenth, the hundredths digit controls the rounding decision. 2. To round \(4.3\square9\) up to \(4.4\), the digit in the box must be \(5\), \(6\), \(7\), \(8\), or \(9\). 3. The final digit \(9\) is in the thousandths place. It does not affect direct rounding to the nearest tenth because only the hundredths digit is inspected.

\(5, 6, 7, 8, 9\). The thousandths digit does not affect direct rounding to the nearest tenth.

- Which digit determines whether the rounding digit stays the same or increases? - What happens when increasing a \(9\) requires regrouping? - Think about what the number of decimal places communicates.

1. For \(0.495\), the thousandths digit is \(5\), so it rounds to \(0.50\) to the nearest hundredth. To the nearest tenth, it rounds to \(0.5\). 2. For \(12.997\), the thousandths digit is \(7\), so regrouping gives \(13.00\) to the nearest hundredth and \(13.0\) to the nearest tenth. 3. The zeros in \(13.00\) show that the value was rounded to the hundredths place.

\(0.495 \rightarrow 0.50\) to the nearest hundredth and \(0.5\) to the nearest tenth. \(12.997 \rightarrow 13.00\) to the nearest hundredth and \(13.0\) to the nearest tenth. The trailing zeros in \(13.00\) show hundredths precision.

- Find the smallest hundredths digit that makes \(5.3\) round up to \(5.4\). - Find the largest hundredths digit that keeps \(5.4\) from rounding up to \(5.5\). - Remember that the number must have exactly two decimal places.

1. A number rounds to \(5.4\) to the nearest tenth when it is at least \(5.35\) but less than \(5.45\). 2. With exactly two decimal places, the least possible number is \(5.35\). 3. The greatest possible number is \(5.44\).

Least: \(5.35\) Greatest: \(5.44\)

- Identify the lower endpoint of each set of numbers that rounds to the given result. - A hundredth-sized rounding interval is narrower than a tenth-sized interval. - Subtract the smaller lower endpoint from the larger one.

1. Numbers that round to \(5.0\) to the nearest tenth begin at \(4.95\). Thus, the least possible number in Case A is \(4.95\). 2. Numbers that round to \(5.00\) to the nearest hundredth begin at \(4.995\). Thus, the least possible number in Case B is \(4.995\). 3. Their difference is \(4.995-4.95=0.045\).

1. Case A: \(4.95\); Case B: \(4.995\) 2. \(0.045\)

- Find the lower endpoint of the rounding interval for each number. - A tenth has a half-unit of \(0.05\), and a hundredth has a half-unit of \(0.005\). - Add the two least possible values.

1. The least number that rounds to \(7.4\) to the nearest tenth is \(7.35\). 2. The least number that rounds to \(2.15\) to the nearest hundredth is \(2.145\). 3. The least possible sum is \(7.35+2.145=9.495\).

The lower bounds are \(x=7.35\) and \(y=2.145\), so the least possible sum is \(9.495\).

- Identify the lower and upper boundaries for numbers that round to \(2.46\). - The lower boundary is included, but the upper boundary is not. - Count by thousandths from the least value to the greatest value.

1. The rounding interval begins at \(2.455\) and ends just before \(2.465\). 2. With exactly three decimal places, the least possible number is \(2.455\), and the greatest is \(2.464\). 3. The possible values are \(2.455, 2.456, 2.457, 2.458, 2.459, 2.460, 2.461, 2.462, 2.463, 2.464\). 4. There are \(10\) values.

a) Least: \(2.455\); greatest: \(2.464\) b) \(10\) decimals

- First determine the full interval of numbers that round to \(5.0\). - Consider separately numbers beginning with \(4.9\) and \(5.0\). - Use the digit-sum condition together with the rounding condition. - Each digit must be from \(0\) through \(9\).

1. A number that rounds to \(5.0\) to the nearest tenth must satisfy \(4.950 \le x < 5.050\). 2. A number beginning with \(4.9\) would need a hundredths digit of at least \(5\) to round up. But then the three decimal digits could not have sum \(10\), because the tenths digit is already \(9\). 3. Therefore, the number begins with \(5.0\). Let the hundredths and thousandths digits be \(a\) and \(b\). Then \(a + b = 10\), and \(a < 5\) so the number rounds down to \(5.0\). 4. The possible digit pairs are \((1, 9)\), \((2, 8)\), \((3, 7)\), and \((4, 6)\). 5. The possible numbers are \(5.019\), \(5.028\), \(5.037\), and \(5.046\).

\(5.019\), \(5.028\), \(5.037\), or \(5.046\)

- Find the two halfway points around the rounded value. - Think about which side of each halfway point still rounds to the reported value. - Because the recorded weight uses hundredths, test hundredth-ton values at the ends of the allowed range.

1. A hundredth-ton value rounds to \(2.4\) when it is at least \(2.35\) and less than \(2.45\). 2. The least possible hundredth-ton value is \(2.35\). 3. The greatest possible hundredth-ton value below \(2.45\) is \(2.44\).

Least: \(2.35\) tons Greatest: \(2.44\) tons

- Start with the values that round to \(4.53\) to the nearest hundredth. - Check that the endpoints also round to \(4.5\) to the nearest tenth. - Work with thousandths rather than interval notation.

1. Values that round to \(4.53\) to the nearest hundredth begin at \(4.525\) and stop before \(4.535\). 2. Every thousandth from \(4.525\) through \(4.534\) also rounds to \(4.5\) to the nearest tenth. 3. Therefore, the least possible recorded distance is \(4.525\) miles and the greatest is \(4.534\) miles.

Least: \(4.525\) miles Greatest: \(4.534\) miles