- Line up digits by place value, not by the final digit in each number. - A trailing zero can be added to a decimal without changing its value. - Add from the smallest place value toward the largest.

1. Align the decimal points. Rewrite \(8.9\) as \(8.90\) so hundredths line up with hundredths. 2. Add from right to left, regrouping when needed. 3. The sum is \(36.36\).

\(36.36\)

- Align the decimal points. - Add the hundredths first. - Then add the tenths; check whether you need to regroup.

1. Add hundredths: \(4+5=9\) hundredths. 2. Add tenths: \(2+3=5\) tenths. 3. The sum is \(0.59\), with no regrouping.

\(0.24+0.35=0.59\). Four hundredths plus five hundredths make nine hundredths; two tenths plus three tenths make five tenths.

- Line up ones, tenths, and hundredths. - Work from the smallest place value to the largest. - Check the result by adding it to the smaller number.

1. Subtract hundredths: \(7-4=3\). 2. Subtract tenths: \(8-3=5\). 3. Subtract ones: \(6-2=4\), giving \(4.53\).

\(6.87-2.34=4.53\): \(7-4=3\) hundredths, \(8-3=5\) tenths, and \(6-2=4\) ones.

- Match place values rather than the rightmost printed digits. - You may add a zero to the end of the shorter decimal. - Check whether your sum should be close to three or thirty.

1. Rewrite \(3.4\) as \(3.40\) so both numbers have hundredths. 2. Add \(3.40+0.28=3.68\). 3. The decimal points remain in one column.

Line up the decimal points: \(3.40+0.28=3.68\).

- Look closely at the hundredths digits. - Which pairs add to a whole number? - You do not need to add the four numbers in the order given.

1. Pair numbers that add to whole numbers: \(0.85 + 3.15\) and \(2.14 + 1.86\). 2. Evaluate the first pair: \(0.85 + 3.15 = 4.00\). 3. Evaluate the second pair: \(2.14 + 1.86 = 4.00\). 4. Add the pair sums: \(4.00\,\text{kg} + 4.00\,\text{kg} = 8.00\,\text{kg}\).

The total mass is \(8.00\,\text{kg}\). One efficient grouping is \((0.85 + 3.15) + (2.14 + 1.86) = 4.00 + 4.00 = 8.00\).

- Look for decimal parts that combine to make whole numbers. - Several subtractions can be rewritten as subtracting the sum of the amounts. - In b), find two numbers with the same tenths digit and two numbers whose decimal parts make a whole.

1. For a), group numbers that add to whole numbers: \((12.35 + 7.65) + (7.8 + 2.2) = 20 + 10 = 30\). 2. For b), rewrite the subtractions as subtracting a total and group conveniently: \((24.7 - 4.7) - (3.9 + 6.1) = 20 - 10 = 10\).

a) \(30\) b) \(10\)

- Look for decimal parts that combine or cancel to make whole numbers. - Rewrite repeated subtraction as subtracting a sum before regrouping. - Keep the signs attached to the terms when you rearrange.

1. Group the terms to create easy differences and sums: \((12.4 - 2.4) - (5.8 + 1.2)\). 2. Evaluate the groups: \(12.4 - 2.4 = 10\) and \(5.8 + 1.2 = 7\). 3. Subtract: \(10 - 7 = 3\).

\(3\)

- Write the two decimals with their decimal points aligned. - Check whether regrouping is needed before subtracting the hundredths. - Subtract by place value from right to left.

1. Align the decimal points: subtract \(5.47\) from \(8.03\). 2. Regroup across the zero so the hundredths place can be subtracted. 3. The standard subtraction gives \(8.03-5.47=2.56\).

\(2.56\)

- Use the inverse relationship between addition and subtraction. - Align the decimal places before subtracting. - Substitute the result into the original equation.

1. Rewrite \(50.2\) as \(50.200\). 2. Subtract to find the missing addend: \(x=50.200-17.806=32.394\). 3. Verification: \(32.394+17.806=50.200=50.2\).

\(x=32.394\)

- Write the first number with a trailing zero so you can line up the hundredths. - Start at the hundredths place and decide whether you need to regroup. - Keep the decimal points aligned in the answer.

1. Line up decimal points: \(50.20-7.86\). 2. In the hundredths column, \(0-6\) needs regrouping: change two tenths to one tenth and ten hundredths. Then \(10-6=4\) hundredths. 3. In the tenths column, \(1-8\) needs regrouping. The ones column is zero, so exchange one ten for ten ones, then one of those ones for ten tenths. Now \(11-8=3\) tenths. 4. In the ones column, \(9-7=2\), and in the tens column, \(4-0=4\). The difference is \(42.34\).

\(42.34\). Regroup one tenth into ten hundredths, then one ten into ten ones and one one into ten tenths.

- Think about the related subtraction fact. - Align tenths and hundredths before subtracting. - Check by adding your result to the known amount.

1. Use subtraction to find the missing amount: \(1.90-1.36=0.54\). 2. Check: \(1.36+0.54=1.90\).

\(0.54\). Check: \(1.36+0.54=1.90\).

- Look at the place value of the extra digit in \(4.70\). - Think about the amount each first addend represents before adding. - Compare the expressions after writing their decimals in aligned columns.

1. The final zero in \(4.70\) means zero extra hundredths. So \(4.7=4.70\); they are the same number. 2. In either expression, the addition is \(4.70+2.05=6.75\). 3. The student is incorrect because adding a zero at the end of a decimal does not change its value or the sum.

No. The final zero in \(4.70\) represents zero hundredths, so \(4.7=4.70\). Both sums equal \(6.75\).

- Decide whether fractions or decimals will be easier to use. - Subtract when water is consumed and add when water is refilled. - For the capacity, identify the greatest amount in the bottle at any time.

1. Convert \(\frac{3}{8}\) to a decimal: \(\frac{3}{8}=0.375\). 2. The final amount is \(1.5-0.375+0.5-0.4=1.225\) liters. 3. After the first drink, the bottle contains \(1.5-0.375=1.125\) liters. After refilling, it contains \(1.125+0.5=1.625\) liters. This is the greatest amount in the bottle. 4. Therefore, the minimum capacity is \(1.625\) liters.

a) \(1.225\) liters b) At least \(1.625\) liters

- Round both values to the same place for a quick check. - Add a trailing zero to the number with fewer decimal places. - Compare the exact answer's size with the estimate.

1. Rounding gives \(103-28=75\) as the estimate. 2. Align decimal places: \(103.005-27.890=75.115\). 3. The exact difference \(75.115\) is close to the estimate \(75\), so the result is reasonable.

Estimate: \(75\); exact difference: \(75.115\)

- Use trailing zeros only to align all four values while calculating. - Identify the greatest and least entries separately from finding the total. - Remove padding zeros that do not communicate stated measurement precision.

1. Align decimal places and add: \(2.475+3.080+1.995+2.610=10.160\), so the total is \(10.16\) minutes. 2. The longest section is \(3.08\) minutes, and the shortest is \(1.995\) minutes. 3. Their difference is \(3.080-1.995=1.085\) minutes.

a) \(10.16\) minutes b) \(1.085\) minutes

- Express the whole number to the thousandths place. - Regroup across the zeros carefully. - Use a nearby whole-number estimate to test the claimed result.

1. Write the whole number as \(40.000\). 2. Subtract with regrouping: \(40.000-6.785=33.215\). 3. The claim is unreasonable because subtracting nearly \(7\) from \(40\) should give a result near \(33\), not near \(35\).

Ingrid's claim is incorrect because \(40\) must be written as \(40.000\) and regrouped by place value. Also, subtracting nearly \(7\) from \(40\) should give a result near \(33\). The correct difference is \(33.215\).

- Follow the events in the order they occur. - Treat water used and water added as different operations. - Add zeros only as needed to align decimal places.

1. Align the decimals as \(100.000\), \(48.907\), and \(3.280\). 2. After the water is used, \(100.000-48.907=51.093\,\text{L}\). 3. After more water is added, \(51.093+3.280=54.373\,\text{L}\).

\(54.373\,\text{L}\)

- Find the full length of the multi-section route first. - Rewrite both totals with equal decimal places before comparing. - Subtract the shorter route from the longer one.

1. Add the Route A sections: \(4.780+6.095+2.400=13.275\,\text{km}\). 2. Compare \(13.275\) and \(13.060\); Route A is longer. 3. The difference is \(13.275-13.060=0.215\,\text{km}\).

a) \(13.275\,\text{km}\) b) Route A c) \(0.215\,\text{km}\)

- Think about what must be subtracted from the starting number to get the stated result. - Align the decimal places to find the missing number. - Substitute your answer into the original equation to check it.

1. Rewrite \(63.5\) as \(63.500\). 2. Find the missing amount: \(z=63.500-27.846=35.654\). 3. \(35.654\) is greater than \(27.846\); substituting gives \(63.5-35.654=27.846\).

\(z=35.654>27.846\). The two parts add to \(63.5\); since \(27.846<31.75\), which is half of \(63.5\), the other part must be larger. Check: \(63.5-35.654=27.846\).

- Start in the hundredths column and note any regrouping. - Use the tenths digit in the sum together with the regrouped amount. - Check the completed addition.

1. In the hundredths column, \(4+8=12\), so write \(2\) and regroup \(1\) tenth. 2. In the tenths column, \(d+6+1\) must end in \(1\) and regroup \(1\) whole. 3. Thus, \(d+7=11\), so \(d=4\). 4. Check: \(7.44+2.68=10.12\).

\(d=4\)

- Find the current total before comparing it with the planned length. - Add zeros only as needed to align place values. - Report the final distances without unnecessary padding zeros.

1. Align the decimals and add: \(6.375+5.800+7.025=19.200\), so the finished length is \(19.2\,\text{km}\). 2. Subtract from the planned total: \(25.000-19.200=5.800\), so \(5.8\,\text{km}\) remains.

a) \(19.2\,\text{km}\) b) \(5.8\,\text{km}\)

- Complete the operation inside the parentheses first. - Align decimal places in both calculations. - Estimate the final subtraction to check the result's size.

1. Evaluate the parentheses: \(12.675+9.040=21.715\). 2. Rewrite \(45.8\) as \(45.800\). 3. Subtract: \(45.800-21.715=24.085\).

\(24.085\)

- Treat the number containing the missing digit as an unknown subtrahend. - Use the minuend and difference to recover that number. - Match the recovered decimal to the given digit pattern.

1. Find the missing subtrahend by subtraction: \(15.200-7.835=7.365\). 2. Compare \(7.365\) with \(7.d65\). 3. The missing tenths digit is \(d=3\). 4. Check: \(15.200-7.365=7.835\).

\(d=3\)

- Check each equation independently. - Add trailing zeros to align place values. - Use an estimate to notice any result with an unreasonable size.

1. Part a) is correct: \(7.050+2.995=10.045\). 2. Part b) is incorrect: \(12.400-5.678=6.722\). 3. Part c) is correct: \(0.906+0.940=1.846\).

a) Correct. b) Incorrect; \(12.4-5.678=6.722\). c) Correct.

- Find each positive difference from the target time. - Compare the two distances from the target. - Subtract those distances to find how much closer one time is.

1. Swimmer A differs from the target by \(25.000-24.683=0.317\,\text{s}\). 2. Swimmer B differs from the target by \(25.412-25.000=0.412\,\text{s}\). 3. Swimmer A is closer because \(0.317<0.412\). 4. The difference in their distances from the target is \(0.412-0.317=0.095\,\text{s}\).

a) \(0.317\,\text{s}\) b) \(0.412\,\text{s}\) c) Swimmer A, by \(0.095\,\text{s}\)

- Look for two addends whose decimal parts combine conveniently. - Addition allows the addends to be grouped in a helpful order. - Use zeros for alignment when needed, but do not pad the final result.

1. Add \(18.675+2.325=21.000\), which is \(21\). 2. Then add \(21+4.09=25.09\). 3. Grouping the first two addends reduces regrouping and gives the same total.

\(25.09\)

- Compare the place value of the digit \(6\) in \(14.6\) and \(14.06\). - A zero may be appended to the right of a decimal without changing its value. - After correcting the equivalent decimal, align decimal points before adding.

1. Adding a zero immediately after the decimal changes \(14.6\) to \(14.06\), so Jordan changed the value. 2. To show hundredths without changing the number, write \(14.6\) as \(14.60\). 3. Align the decimal points and add: \(14.60+3.75=18.35\).

Jordan placed the zero in the wrong place. The correct rewrite is \(14.60\), and the correct sum is \(18.35\).