Question

1. match the expressions to their resulting units.

\\(\frac{8\text{ yd}}{1}\\), \\(\frac{3\text{ ft}}{1\text{ yd}}\\), \\(\frac{0.305\text{ m}}{1\text{ ft}}\\) \\(\square\\) \\(\square\\) yards
\\(\frac{5\text{ km}}{1}\\), \\(\frac{1\text{ mi}}{1600\text{ km}}\\), \\(\frac{5280\text{ ft}}{1\text{ mi}}\\), \\(\frac{1\text{ yd}}{3\text{ ft}}\\) \\(\square\\) \\(\square\\) centimeters
\\(\frac{7000\text{ m}}{1}\\), \\(\frac{3\text{ ft}}{1\text{ yd}}\\), \\(\frac{0.305\text{ m}}{1\text{ ft}}\\), \\(\frac{1\text{ km}}{1000\text{ m}}\\) \\(\square\\) \\(\square\\) kilometers
\\(\frac{4\text{ yd}}{1}\\), \\(\frac{3\text{ ft}}{1\text{ yd}}\\), \\(\frac{0.305\text{ m}}{1\text{ ft}}\\), \\(\frac{100\text{ cm}}{1\text{ m}}\\) \\(\square\\) \\(\square\\) meters
\\(\frac{7\text{ m}}{1}\\), \\(\frac{1\text{ ft}}{0.305\text{ m}}\\), \\(\frac{12\text{ in}}{1\text{ ft}}\\) \\(\square\\) \\(\square\\) inches

Explanation:

To solve this, we analyze each expression by canceling units:

First Expression: $\boldsymbol{\frac{8\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}}$

• Cancel yd: $8 \cdot 3\ \text{ft} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}$
• Cancel ft: $8 \cdot 3 \cdot 0.305\ \text{m}$
• Result: meters (matches "meters").

Second Expression: $\boldsymbol{\frac{5\ \text{km}}{1} \cdot \frac{1\ \text{mi}}{1000\ \text{km}} \cdot \frac{5280\ \text{ft}}{1\ \text{mi}} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}}$

• Cancel km: $5 \cdot \frac{1\ \text{mi}}{1000} \cdot 5280\ \text{ft} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$
• Cancel mi: $5 \cdot \frac{5280\ \text{ft}}{1000} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$
• Cancel ft: $5 \cdot \frac{5280}{1000 \cdot 3}\ \text{yd}$
• Result: yards (matches "yards").

Third Expression: $\boldsymbol{\frac{7000\ \text{m}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000\ \text{m}}}$

• Cancel m (first and last): $7000 \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000}$
• Cancel ft: $7000 \cdot \frac{3 \cdot 0.305}{1\ \text{yd}} \cdot \frac{1\ \text{km}}{1000}$ (Note: This step simplifies, but key is canceling to km)
• Result: kilometers (matches "kilometers").

Fourth Expression: $\boldsymbol{\frac{4\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}}$

• Cancel yd: $4 \cdot 3\ \text{ft} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$
• Cancel ft: $4 \cdot 3 \cdot 0.305\ \text{m} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$
• Cancel m: $4 \cdot 3 \cdot 0.305 \cdot 100\ \text{cm}$
• Result: centimeters (matches "centimeters").

Fifth Expression: $\boldsymbol{\frac{7\ \text{m}}{1} \cdot \frac{1\ \text{ft}}{0.305\ \text{m}} \cdot \frac{12\ \text{in}}{1\ \text{ft}}}$

• Cancel m: $7 \cdot \frac{1\ \text{ft}}{0.305} \cdot \frac{12\ \text{in}}{1\ \text{ft}}$
• Cancel ft: $7 \cdot \frac{12}{0.305}\ \text{in}$
• Result: inches (matches "inches").

Final Matches:

1. $\frac{8\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}$ → meters
2. $\frac{5\ \text{km}}{1} \cdot \frac{1\ \text{mi}}{1000\ \text{km}} \cdot \frac{5280\ \text{ft}}{1\ \text{mi}} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$ → yards
3. $\frac{7000\ \text{m}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000\ \text{m}}$ → kilometers
4. $\frac{4\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$ → centimeters
5. $\frac{7\ \text{m}}{1} \cdot \frac{1\ \text{ft}}{0.305\ \text{m}} \cdot \frac{12\ \text{in}}{1\ \text{ft}}$ → inches

(Note: If the question expects matching each expression to the unit, the above analysis shows the correct pairings.)

Answer:

To solve this, we analyze each expression by canceling units:

First Expression: $\boldsymbol{\frac{8\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}}$

• Cancel yd: $8 \cdot 3\ \text{ft} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}$
• Cancel ft: $8 \cdot 3 \cdot 0.305\ \text{m}$
• Result: meters (matches "meters").

Second Expression: $\boldsymbol{\frac{5\ \text{km}}{1} \cdot \frac{1\ \text{mi}}{1000\ \text{km}} \cdot \frac{5280\ \text{ft}}{1\ \text{mi}} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}}$

• Cancel km: $5 \cdot \frac{1\ \text{mi}}{1000} \cdot 5280\ \text{ft} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$
• Cancel mi: $5 \cdot \frac{5280\ \text{ft}}{1000} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$
• Cancel ft: $5 \cdot \frac{5280}{1000 \cdot 3}\ \text{yd}$
• Result: yards (matches "yards").

Third Expression: $\boldsymbol{\frac{7000\ \text{m}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000\ \text{m}}}$

• Cancel m (first and last): $7000 \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000}$
• Cancel ft: $7000 \cdot \frac{3 \cdot 0.305}{1\ \text{yd}} \cdot \frac{1\ \text{km}}{1000}$ (Note: This step simplifies, but key is canceling to km)
• Result: kilometers (matches "kilometers").

Fourth Expression: $\boldsymbol{\frac{4\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}}$

• Cancel yd: $4 \cdot 3\ \text{ft} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$
• Cancel ft: $4 \cdot 3 \cdot 0.305\ \text{m} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$
• Cancel m: $4 \cdot 3 \cdot 0.305 \cdot 100\ \text{cm}$
• Result: centimeters (matches "centimeters").

Fifth Expression: $\boldsymbol{\frac{7\ \text{m}}{1} \cdot \frac{1\ \text{ft}}{0.305\ \text{m}} \cdot \frac{12\ \text{in}}{1\ \text{ft}}}$

• Cancel m: $7 \cdot \frac{1\ \text{ft}}{0.305} \cdot \frac{12\ \text{in}}{1\ \text{ft}}$
• Cancel ft: $7 \cdot \frac{12}{0.305}\ \text{in}$
• Result: inches (matches "inches").

Final Matches:

1. $\frac{8\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}}$ → meters
2. $\frac{5\ \text{km}}{1} \cdot \frac{1\ \text{mi}}{1000\ \text{km}} \cdot \frac{5280\ \text{ft}}{1\ \text{mi}} \cdot \frac{1\ \text{yd}}{3\ \text{ft}}$ → yards
3. $\frac{7000\ \text{m}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{1\ \text{km}}{1000\ \text{m}}$ → kilometers
4. $\frac{4\ \text{yd}}{1} \cdot \frac{3\ \text{ft}}{1\ \text{yd}} \cdot \frac{0.305\ \text{m}}{1\ \text{ft}} \cdot \frac{100\ \text{cm}}{1\ \text{m}}$ → centimeters
5. $\frac{7\ \text{m}}{1} \cdot \frac{1\ \text{ft}}{0.305\ \text{m}} \cdot \frac{12\ \text{in}}{1\ \text{ft}}$ → inches

(Note: If the question expects matching each expression to the unit, the above analysis shows the correct pairings.)