Addition and subtraction of length exercises with step-by-step solutions.

Problem Analysis:

The image contains two sections:
1. Addition of Lengths: Four problems involving the addition of lengths in different units (m, dm, cm, km, hm, dam).
2. Subtraction of Lengths: Four problems involving the subtraction of lengths in different units.

We will solve each problem step by step, ensuring clarity and correctness.

---

Section 1: Addition of Lengths

#### (a)
Units: \( \text{m}, \text{dm}, \text{cm} \)

\[
\begin{array}{r r r}
& \text{m} & \text{dm} & \text{cm} \\
& 2 & 6 & 2 \\
+ & 4 & 1 & 5 \\
\hline
& 7 & 4 & 5 \\
\end{array}
\]

- Step-by-step explanation:
1. Add the centimeters: \( 2 + 5 = 7 \) cm.
2. Add the decimeters: \( 6 + 1 = 7 \) dm.
3. Add the meters: \( 2 + 4 = 6 \) m.
4. Combine the results: \( 6 \, \text{m} \, 7 \, \text{dm} \, 7 \, \text{cm} \).

- Final Answer: \( 6 \, \text{m} \, 7 \, \text{dm} \, 7 \, \text{cm} \).

#### (b)
Units: \( \text{m}, \text{dm}, \text{cm} \)

\[
\begin{array}{r r r}
& \text{m} & \text{dm} & \text{cm} \\
& 2 & 4 & 5 \\
+ & 5 & 6 & 7 \\
\hline
& 14 & 9 & 0 \\
\end{array}
\]

- Step-by-step explanation:
1. Add the centimeters: \( 5 + 7 = 12 \) cm. Carry over 1 to the decimeters.
2. Add the decimeters: \( 4 + 6 + 1 (\text{carry}) = 11 \) dm. Carry over 1 to the meters.
3. Add the meters: \( 2 + 5 + 1 (\text{carry}) = 8 \) m.
4. Combine the results: \( 8 \, \text{m} \, 11 \, \text{dm} \, 12 \, \text{cm} \). Convert \( 11 \, \text{dm} \) to \( 1 \, \text{m} \, 1 \, \text{dm} \) and add it to the meters: \( 8 + 1 = 9 \, \text{m} \), leaving \( 1 \, \text{dm} \). Convert \( 12 \, \text{cm} \) to \( 1 \, \text{dm} \, 2 \, \text{cm} \) and add it to the decimeters: \( 1 + 1 = 2 \, \text{dm} \), leaving \( 2 \, \text{cm} \).
5. Final result: \( 9 \, \text{m} \, 2 \, \text{dm} \, 2 \, \text{cm} \).

- Final Answer: \( 9 \, \text{m} \, 2 \, \text{dm} \, 2 \, \text{cm} \).

#### (c)
Units: \( \text{km}, \text{hm}, \text{dam}, \text{m} \)

\[
\begin{array}{r r r r}
& \text{km} & \text{hm} & \text{dam} & \text{m} \\
& 4 & 0 & 0 & 8 \\
+ & 7 & 2 & 5 & 0 \\
\hline
& 11 & 2 & 5 & 8 \\
\end{array}
\]

- Step-by-step explanation:
1. Add the meters: \( 8 + 0 = 8 \) m.
2. Add the decameters: \( 0 + 5 = 5 \) dam.
3. Add the hectometers: \( 0 + 2 = 2 \) hm.
4. Add the kilometers: \( 4 + 7 = 11 \) km.
5. Combine the results: \( 11 \, \text{km} \, 2 \, \text{hm} \, 5 \, \text{dam} \, 8 \, \text{m} \).

- Final Answer: \( 11 \, \text{km} \, 2 \, \text{hm} \, 5 \, \text{dam} \, 8 \, \text{m} \).

#### (d)
Units: \( \text{km}, \text{hm}, \text{dam}, \text{m} \)

\[
\begin{array}{r r r r}
& \text{km} & \text{hm} & \text{dam} & \text{m} \\
& 5 & 6 & 0 & 5 \\
+ & 4 & 2 & 3 & 3 \\
\hline
& 13 & 4 & 3 & 8 \\
\end{array}
\]

- Step-by-step explanation:
1. Add the meters: \( 5 + 3 = 8 \) m.
2. Add the decameters: \( 0 + 3 = 3 \) dam.
3. Add the hectometers: \( 6 + 2 = 8 \) hm.
4. Add the kilometers: \( 5 + 4 = 9 \) km.
5. Combine the results: \( 9 \, \text{km} \, 8 \, \text{hm} \, 3 \, \text{dam} \, 8 \, \text{m} \).

- Final Answer: \( 9 \, \text{km} \, 8 \, \text{hm} \, 3 \, \text{dam} \, 8 \, \text{m} \).

---

Section 2: Subtraction of Lengths

#### (a)
Units: \( \text{m}, \text{dm}, \text{cm} \)

\[
\begin{array}{r r r}
& \text{m} & \text{dm} & \text{cm} \\
& 5 & 7 & 2 \\
- & 2 & 4 & 4 \\
\hline
& 3 & 2 & 8 \\
\end{array}
\]

- Step-by-step explanation:
1. Subtract the centimeters: \( 2 - 4 \). Borrow 1 from the decimeters, making it \( 12 - 4 = 8 \) cm.
2. Subtract the decimeters: \( 6 - 4 = 2 \) dm (after borrowing).
3. Subtract the meters: \( 5 - 2 = 3 \) m.
4. Combine the results: \( 3 \, \text{m} \, 2 \, \text{dm} \, 8 \, \text{cm} \).

- Final Answer: \( 3 \, \text{m} \, 2 \, \text{dm} \, 8 \, \text{cm} \).

#### (b)
Units: \( \text{m}, \text{dm}, \text{cm} \)

\[
\begin{array}{r r r}
& \text{m} & \text{dm} & \text{cm} \\
& 7 & 8 & 4 \\
- & 4 & 9 & 7 \\
\hline
& 2 & 8 & 7 \\
\end{array}
\]

- Step-by-step explanation:
1. Subtract the centimeters: \( 4 - 7 \). Borrow 1 from the decimeters, making it \( 14 - 7 = 7 \) cm.
2. Subtract the decimeters: \( 7 - 9 \). Borrow 1 from the meters, making it \( 17 - 9 = 8 \) dm.
3. Subtract the meters: \( 6 - 4 = 2 \) m (after borrowing).
4. Combine the results: \( 2 \, \text{m} \, 8 \, \text{dm} \, 7 \, \text{cm} \).

- Final Answer: \( 2 \, \text{m} \, 8 \, \text{dm} \, 7 \, \text{cm} \).

#### (c)
Units: \( \text{m}, \text{dm}, \text{cm} \)

\[
\begin{array}{r r r}
& \text{m} & \text{dm} & \text{cm} \\
& 5 & 7 & 0 \\
- & 3 & 2 & 5 \\
\hline
& 2 & 4 & 5 \\
\end{array}
\]

- Step-by-step explanation:
1. Subtract the centimeters: \( 0 - 5 \). Borrow 1 from the decimeters, making it \( 10 - 5 = 5 \) cm.
2. Subtract the decimeters: \( 6 - 2 = 4 \) dm (after borrowing).
3. Subtract the meters: \( 5 - 3 = 2 \) m.
4. Combine the results: \( 2 \, \text{m} \, 4 \, \text{dm} \, 5 \, \text{cm} \).

- Final Answer: \( 2 \, \text{m} \, 4 \, \text{dm} \, 5 \, \text{cm} \).

#### (d)
Units: \( \text{km}, \text{m} \)

\[
\begin{array}{r r}
& \text{km} & \text{m} \\
& 45 & 300 \\
- & 14 & 200 \\
\hline
& 31 & 100 \\
\end{array}
\]

- Step-by-step explanation:
1. Subtract the meters: \( 300 - 200 = 100 \) m.
2. Subtract the kilometers: \( 45 - 14 = 31 \) km.
3. Combine the results: \( 31 \, \text{km} \, 100 \, \text{m} \).

- Final Answer: \( 31 \, \text{km} \, 100 \, \text{m} \).

---

Final Answers:

1. Addition of Lengths:
- (a) \( 6 \, \text{m} \, 7 \, \text{dm} \, 7 \, \text{cm} \)
- (b) \( 9 \, \text{m} \, 2 \, \text{dm} \, 2 \, \text{cm} \)
- (c) \( 11 \, \text{km} \, 2 \, \text{hm} \, 5 \, \text{dam} \, 8 \, \text{m} \)
- (d) \( 9 \, \text{km} \, 8 \, \text{hm} \, 3 \, \text{dam} \, 8 \, \text{m} \)

2. Subtraction of Lengths:
- (a) \( 3 \, \text{m} \, 2 \, \text{dm} \, 8 \, \text{cm} \)
- (b) \( 2 \, \text{m} \, 8 \, \text{dm} \, 7 \, \text{cm} \)
- (c) \( 2 \, \text{m} \, 4 \, \text{dm} \, 5 \, \text{cm} \)
- (d) \( 31 \, \text{km} \, 100 \, \text{m} \)

\boxed{
\begin{aligned}
&\text{Addition:} \\
&(a) \, 6 \, \text{m} \, 7 \, \text{dm} \, 7 \, \text{cm}, \, (b) \, 9 \, \text{m} \, 2 \, \text{dm} \, 2 \, \text{cm}, \, (c) \, 11 \, \text{km} \, 2 \, \text{hm} \, 5 \, \text{dam} \, 8 \, \text{m}, \, (d) \, 9 \, \text{km} \, 8 \, \text{hm} \, 3 \, \text{dam} \, 8 \, \text{m} \\
&\text{Subtraction:} \\
&(a) \, 3 \, \text{m} \, 2 \, \text{dm} \, 8 \, \text{cm}, \, (b) \, 2 \, \text{m} \, 8 \, \text{dm} \, 7 \, \text{cm}, \, (c) \, 2 \, \text{m} \, 4 \, \text{dm} \, 5 \, \text{cm}, \, (d) \, 31 \, \text{km} \, 100 \, \text{m}
\end{aligned}
}