Free math worksheets

- Read the wall length from the plan. - Use the stated scale once for that length.

1. The wall measures \(3.5\,\text{cm}\) on the plan. 2. Each plan centimeter represents \(2\,\text{m}\), so the actual length is \(3.5\cdot2=7\,\text{m}\).

\(7\,\text{m}\)

- Decide whether the drawing should be larger or smaller than the actual pencil. - At a scale of \(1{:}2\), each \(1\) unit in the drawing represents \(2\) units in reality. - Find one-half of the actual length.

1. A scale of \(1{:}2\) means the drawing length is one-half of the actual length. 2. Divide the actual length by \(2\): \(14\,\text{cm} \div 2 = 7\,\text{cm}\).

\(7\,\text{cm}\)

- In a scale of \(1:50\), each unit in the drawing represents how many units in the actual object? - How many centimeters are in \(1\) meter?

1. Multiply the drawing length by the scale factor: \(9\,\text{cm} \cdot 50 = 450\,\text{cm}\). 2. Convert centimeters to meters: \(450\,\text{cm} \div 100 = 4.5\,\text{m}\).

The actual car is \(450\,\text{cm}\), or \(4.5\,\text{m}\), long.

- First express the actual wingspan in centimeters. - The drawing is \(100\) times as small as the actual airplane. Which operation gives the drawing length?

1. Convert the actual wingspan to centimeters: \(18\,\text{m} \cdot 100 = 1800\,\text{cm}\). 2. Divide by the scale factor: \(1800\,\text{cm} \div 100 = 18\,\text{cm}\).

The wingspan in the drawing is \(18\,\text{cm}\).

- Convert the two heights to the same unit before comparing them. - How many times as great is the actual height as the drawing height? - Write a scale in the form \(1:\text{scale factor}\).

1. Express both heights in centimeters: \(250\,\text{m} = 25{,}000\,\text{cm}\). 2. Find the scale factor: \(25{,}000 \div 25 = 1000\). 3. Write the scale as \(1:1000\).

The scale is \(1:1000\).

- What does the \(20\) in the scale tell you about the actual dimensions? - How many centimeters are in \(1\) meter? - Should the actual building be larger or smaller than the drawing?

1. For a), multiply the drawing width by the scale factor: \(18\,\text{cm} \cdot 20 = 360\,\text{cm}\). This is \(3.6\,\text{m}\). 2. For b), multiply the drawing height by the scale factor: \(11\,\text{cm} \cdot 20 = 220\,\text{cm}\). This is \(2\,\text{m}\ 20\,\text{cm}\).

a) The wall of windows is \(360\,\text{cm}\), or \(3.6\,\text{m}\), wide. b) The door is \(220\,\text{cm}\), or \(2\,\text{m}\ 20\,\text{cm}\), high.

- Express the plan length and actual length in the same unit before applying the scale. - To go from an actual length to a plan length, should you multiply or divide by the scale factor? - Check whether each result is reasonable for a scale drawing.

1. For a), find the actual length in centimeters: \(9\,\text{cm} \cdot 500 = 4500\,\text{cm}\). Convert to meters: \(4500\,\text{cm} \div 100 = 45\,\text{m}\). 2. For b), convert the actual length to centimeters: \(45\,\text{m} = 4500\,\text{cm}\). Divide by the scale factor: \(4500\,\text{cm} \div 500 = 9\,\text{cm}\).

a) The climbing trail is actually \(45\,\text{m}\) long. b) The zip line should be \(9\,\text{cm}\) long on the plan.

- Decide whether the value should become greater or less when you move between the drawing and the actual object. - Convert meters to centimeters before applying the scale. - Which operation takes you from the actual object to the drawing?

1. For a), multiply by the scale factor: \(22\,\text{cm} \cdot 40 = 880\,\text{cm}\). Convert to meters: \(880\,\text{cm} = 8.8\,\text{m}\). 2. For b), first convert the actual width: \(2.40\,\text{m} = 240\,\text{cm}\). Then divide by the scale factor: \(240\,\text{cm} \div 40 = 6\,\text{cm}\).

a) The actual truck is \(8.8\,\text{m}\) long. b) The truck is \(6\,\text{cm}\) wide in the drawing.

- Work with one part at a time. - How does a length change when you move from the drawing to the actual object? - Convert each result to meters to check whether it is reasonable.

1. For the kitchen table, \(16\,\text{cm} \cdot 10 = 160\,\text{cm}\), which is \(1.6\,\text{m}\). 2. For the closet, \(9\,\text{cm} \cdot 20 = 180\,\text{cm}\), which is \(1.8\,\text{m}\). 3. For the garden path, \(7\,\text{cm} \cdot 50 = 350\,\text{cm}\), which is \(3.5\,\text{m}\).

a) \(160\,\text{cm}\), or \(1.6\,\text{m}\) b) \(180\,\text{cm}\), or \(1.8\,\text{m}\) c) \(350\,\text{cm}\), or \(3.5\,\text{m}\)

- Convert the actual tower height to centimeters first. - How many times as great is the actual height as the drawing height? - Use the same scale factor to locate the clock in the drawing.

1. For a), convert the actual height to centimeters: \(40\,\text{m} = 4000\,\text{cm}\). Then \(4000 \div 8 = 500\), so the scale is \(1:500\). 2. For b), convert the actual clock height: \(30\,\text{m} = 3000\,\text{cm}\). Divide by the scale factor: \(3000\,\text{cm} \div 500 = 6\,\text{cm}\).

a) The drawing uses a scale of \(1:500\). b) The clock should be \(6\,\text{cm}\) above the ground in the drawing.

- What actual length corresponds to \(1\,\text{cm}\) on the plan? - How many centimeters are in \(1\) meter? - Find each actual length in centimeters before converting it to meters. - What operation converts centimeters to meters?

1. Multiply each plan length by \(50\) to find the actual length in centimeters. 2. \(4\,\text{cm} \cdot 50 = 200\,\text{cm} = 2\,\text{m}\). 3. \(8\,\text{cm} \cdot 50 = 400\,\text{cm} = 4\,\text{m}\). 4. \(12\,\text{cm} \cdot 50 = 600\,\text{cm} = 6\,\text{m}\). 5. \(20\,\text{cm} \cdot 50 = 1000\,\text{cm} = 10\,\text{m}\). 6. \(30\,\text{cm} \cdot 50 = 1500\,\text{cm} = 15\,\text{m}\).

The missing values, in order, are \(2\,\text{m}\), \(4\,\text{m}\), \(6\,\text{m}\), \(10\,\text{m}\), and \(15\,\text{m}\).

- A drawing length is less than its corresponding actual length, so divide by the scale factor. - Convert each measurement to centimeters before applying the scale. - Which operation takes you from an actual length to a drawing length?

1. Convert the actual lengths to centimeters: \(1\,\text{m} = 100\,\text{cm}\), \(2\,\text{m} = 200\,\text{cm}\), \(3\,\text{m} = 300\,\text{cm}\), and \(5\,\text{m} = 500\,\text{cm}\). 2. Divide each actual length by \(20\). 3. \(100\,\text{cm} \div 20 = 5\,\text{cm}\). 4. \(200\,\text{cm} \div 20 = 10\,\text{cm}\). 5. \(300\,\text{cm} \div 20 = 15\,\text{cm}\). 6. \(500\,\text{cm} \div 20 = 25\,\text{cm}\). 7. \(60\,\text{cm} \div 20 = 3\,\text{cm}\).

The missing drawing lengths, in order, are \(5\,\text{cm}\), \(10\,\text{cm}\), \(15\,\text{cm}\), \(25\,\text{cm}\), and \(3\,\text{cm}\).

- First determine how many actual meters correspond to \(1\,\text{cm}\) on the plan. - Multiply when moving from a plan length to an actual length. - Divide when moving from an actual length to a plan length. - Keep track of whether each value is in centimeters or meters.

1. A scale of \(1:200\) means \(1\,\text{cm}\) on the plan represents \(200\,\text{cm}\), or \(2\,\text{m}\), in the actual object. 2. For \(3\,\text{cm}\), the actual length is \(3 \cdot 2\,\text{m} = 6\,\text{m}\). 3. For \(12\,\text{m}\), the plan length is \(12\,\text{m} \div 2\,\text{m} = 6\,\text{cm}\). 4. For \(10\,\text{cm}\), the actual length is \(10 \cdot 2\,\text{m} = 20\,\text{m}\). 5. For \(50\,\text{m}\), the plan length is \(50\,\text{m} \div 2\,\text{m} = 25\,\text{cm}\).

The missing values from left to right are \(6\,\text{m}\), \(6\,\text{cm}\), \(20\,\text{m}\), and \(25\,\text{cm}\).

- How do you convert meters to centimeters? - A map scale compares lengths measured in the same unit. - After converting, identify the actual length represented by \(1\,\text{cm}\) on the map.

1. Convert the actual distance to centimeters: \(250\,\text{m} \cdot 100 = 25{,}000\,\text{cm}\). 2. Since \(1\,\text{cm}\) on the map represents \(25{,}000\,\text{cm}\) in the city, the scale is \(1:25{,}000\).

a) \(25{,}000\,\text{cm}\) b) \(1:25{,}000\)

- Determine how many groups of \(10\,\text{km}\) are in each actual distance. - What happens to a number when you divide it by \(10\)? - A map length can include a decimal.

1. Divide each actual distance by \(10\,\text{km}\) per centimeter. 2. For a), \(20\,\text{km} \div 10\,\text{km} = 2\), so the line is \(2\,\text{cm}\) long. 3. For b), \(50\,\text{km} \div 10\,\text{km} = 5\), so the line is \(5\,\text{cm}\) long. 4. For c), \(85\,\text{km} \div 10\,\text{km} = 8.5\), so the line is \(8.5\,\text{cm}\) long. 5. For d), \(120\,\text{km} \div 10\,\text{km} = 12\), so the line is \(12\,\text{cm}\) long.

a) \(2\,\text{cm}\) b) \(5\,\text{cm}\) c) \(8.5\,\text{cm}\) d) \(12\,\text{cm}\)

- Convert meters to centimeters. - A model at \(1:24\) is one twenty-fourth of the actual size. - Divide the actual length by \(24\).

1. Convert the actual length: \(4.80\,\text{m}=480\,\text{cm}\). 2. Divide by the scale factor: \(480\div 24=20\,\text{cm}\).

The model is \(20\,\text{cm}\) long.

- Convert each actual length to centimeters. - Divide by the scale denominator. - Check that each model is smaller than the actual vehicle.

1. The truck is \(15\,\text{m}=1500\,\text{cm}\) long. Its model length is \(1500\div 50=30\,\text{cm}\). 2. The bicycle is \(1.80\,\text{m}=180\,\text{cm}\) long. Its model length is \(180\div 15=12\,\text{cm}\).

a) \(30\,\text{cm}\) b) \(12\,\text{cm}\)

- How many groups of \(5\,\text{m}\) fit in each actual dimension? - Find the plan length and plan width separately. - The plan dimensions should be smaller than the actual dimensions.

1. Find the plan length: \(30\,\text{m} \div 5\,\text{m} = 6\), so the length is \(6\,\text{cm}\). 2. Find the plan width: \(20\,\text{m} \div 5\,\text{m} = 4\), so the width is \(4\,\text{cm}\).

The rectangle on the plan should be \(6\,\text{cm}\) long and \(4\,\text{cm}\) wide.

- Read both drawing dimensions from the diagram before converting. - Use the real-world length represented by each centimeter of the drawing.

1. Read the drawing dimensions: \(3.2\,\text{cm}\) by \(2.4\,\text{cm}\). 2. Convert the longer side: \(3.2 \cdot 2.5 = 8\), so the actual length is \(8\,\text{m}\). 3. Convert the shorter side: \(2.4 \cdot 2.5 = 6\), so the actual width is \(6\,\text{m}\).

The classroom is \(8\,\text{m}\) long and \(6\,\text{m}\) wide.

- Read the two plan dimensions from the diagram. - Scale lengths before computing the real floor area. - Keep the area unit squared.

1. The plan dimensions are \(4\,\text{cm}\) by \(2.5\,\text{cm}\). 2. The actual dimensions are \(4\cdot150=600\,\text{cm}=6\,\text{m}\) and \(2.5\cdot150=375\,\text{cm}=3.75\,\text{m}\). 3. The actual floor area is \(6\cdot3.75=22.5\,\text{m}^2\).

\(22.5\,\text{m}^2\)

- Distinguish a length scale factor from an area scale factor. - Imagine scaling both the length and the width of a square by the same factor.

1. The linear scale factor from Drawing A to Drawing B is \(k=1.5\). 2. Areas scale by \(k^2\), so the area factor is \((1.5)^2=2.25\).

The area in Drawing B is \(2.25\) times the area in Drawing A.

- Decide whether the actual bed is larger or smaller than its drawing. - Use the scale factor to convert the drawing length to the actual length. - Recall that \(100\,\text{cm} = 1\,\text{m}\).

1. A scale of \(1{:}10\) means each \(1\) unit on the drawing represents \(10\) units in the actual room. 2. Multiply the drawing length by \(10\): \(20\,\text{cm} \cdot 10 = 200\,\text{cm}\). 3. Convert centimeters to meters: \(200\,\text{cm} = 2\,\text{m}\).

\(200\,\text{cm}\), or \(2\,\text{m}\)

- Use each map scale to find the corresponding actual length. - Pay attention to place value when multiplying by \(2000\) and \(5000\). - Convert both actual lengths to the same unit before comparing them.

1. Find the actual length of the Beech Trail: \(8\,\text{cm} \cdot 2000 = 16{,}000\,\text{cm} = 160\,\text{m}\). 2. Find the actual length of the Pine Trail: \(3\,\text{cm} \cdot 5000 = 15{,}000\,\text{cm} = 150\,\text{m}\). 3. Compare the lengths: \(160\,\text{m} > 150\,\text{m}\).

The Beech Trail is longer. It is \(160\,\text{m}\) long, while the Pine Trail is \(150\,\text{m}\) long.

- Convert the two quantities in a scale to the same unit. - A scale compares corresponding lengths measured in the same unit. - A map length becomes greater when you find the actual length. - An actual length becomes less when you find the map length.

1. For a), convert \(50\,\text{m}\) to centimeters: \(50\,\text{m} = 5000\,\text{cm}\). Therefore, the scale is \(1:5000\). 2. For b), divide the actual length by the length represented by \(1\,\text{cm}\): \(150\,\text{m} \div 50\,\text{m} = 3\). The map length is \(3\,\text{cm}\). 3. For c), multiply the map length by \(50\,\text{m}\) per centimeter: \(12 \cdot 50\,\text{m} = 600\,\text{m}\).

a) The scale is \(1:5000\). b) The elephant habitat is \(3\,\text{cm}\) long on the map. c) The actual path is \(600\,\text{m}\) long.

- What does the second number in the scale tell you? - How many centimeters are in \(1\) meter? - How many meters are in \(1\) kilometer? - Convert the units one step at a time.

1. Multiply the map length by the scale factor: \(5\,\text{cm} \cdot 20{,}000 = 100{,}000\,\text{cm}\). 2. Convert centimeters to meters: \(100{,}000\,\text{cm} \div 100 = 1000\,\text{m}\). 3. Convert meters to kilometers: \(1000\,\text{m} \div 1000 = 1\,\text{km}\).

a) \(100{,}000\,\text{cm}\) b) \(1000\,\text{m}\) c) \(1\,\text{km}\)

- Convert \(2\,\text{km}\) to meters before using either scale. - Determine how many groups of \(200\,\text{m}\) and \(500\,\text{m}\) fit in the trail length. - If each centimeter represents a shorter actual distance, will the map line be longer or shorter?

1. Convert the actual trail length to meters: \(2\,\text{km} = 2000\,\text{m}\). 2. For Map A, \(2000\,\text{m} \div 200\,\text{m} = 10\), so the line is \(10\,\text{cm}\) long. 3. For Map B, \(2000\,\text{m} \div 500\,\text{m} = 4\), so the line is \(4\,\text{cm}\) long. 4. Since \(10\,\text{cm} > 4\,\text{cm}\), the line is longer on Map A. Each centimeter on Map A represents a shorter actual distance, so more centimeters are needed for the same trail.

a) The trail is \(10\,\text{cm}\) long on Map A. b) The trail is \(4\,\text{cm}\) long on Map B. c) The line is longer on Map A because each centimeter represents a shorter actual distance.

- Decide whether each representation enlarges or reduces the real object. - Apply the scale factor to the ladybug's length. - Convert the wall length to centimeters before using the plan scale.

1. A scale of \(10:1\) enlarges an object, so it belongs to the ladybug drawing. A scale of \(1:200\) reduces an object, so it belongs to the site plan. 2. The ladybug's drawing length is \(8\,\text{mm}\cdot 10=80\,\text{mm}=8\,\text{cm}\). 3. Convert the wall length: \(12\,\text{m}=1200\,\text{cm}\). On the plan, its length is \(1200\div 200=6\,\text{cm}\).

a) Ladybug: \(10:1\); school-site plan: \(1:200\) b) \(8\,\text{cm}\) c) \(6\,\text{cm}\)

- Express the drawing and actual lengths in the same unit. - Simplify the ratio of drawing length to actual length. - Multiply the plan measurement by the scale factor.

1. Convert the actual room length: \(4\,\text{m}=400\,\text{cm}\). 2. Compare drawing length with actual length: \(8:400=1:50\). The scale is \(1:50\). 3. The cabinet's actual width is \(1.5\,\text{cm}\cdot 50=75\,\text{cm}\).

a) \(1:50\) b) \(75\,\text{cm}\)

- Express both lengths in the same unit. - Form the ratio of map length to actual length. - Simplify the ratio so its first term is \(1\).

1. Convert the actual length to centimeters: \(18\,\text{km}=18{,}000\,\text{m}=1{,}800{,}000\,\text{cm}\). 2. Form the map-to-actual ratio: \(6:1{,}800{,}000\). 3. Divide both terms by \(6\): \(1:300{,}000\).

The map scale is \(1:300{,}000\).

- Decide whether the missing value should be larger or smaller than the given value. - Convert all compared lengths to the same unit. - Use \(100\,\text{cm}=1\,\text{m}\) and \(1000\,\text{m}=1\,\text{km}\). - A scale tells how much the actual distance has been reduced.

1. For a), \(8\,\text{cm}\cdot 500=4000\,\text{cm}=40\,\text{m}\). 2. For b), \(4\,\text{km}=400{,}000\,\text{cm}\). The map length is \(400{,}000\div 20{,}000=20\,\text{cm}\). 3. For c), \(10\,\text{km}=1{,}000{,}000\,\text{cm}\). The ratio \(5:1{,}000{,}000\) simplifies to \(1:200{,}000\).

a) \(40\,\text{m}\) b) \(20\,\text{cm}\) c) \(1:200{,}000\)

- Actual lengths must be larger than plan lengths at this scale. - Convert units before applying the scale factor. - At \(1:250\), one unit on the plan represents \(250\) of the same units in reality.

1. For the first column, \(8\,\text{cm}\cdot 250=2000\,\text{cm}=20\,\text{m}\). 2. For the second column, \(50\,\text{m}=5000\,\text{cm}\). The plan length is \(5000\div 250=20\,\text{cm}\). 3. For the third column, \(14\,\text{mm}\cdot 250=3500\,\text{mm}=3.5\,\text{m}\).

First column: \(20\,\text{m}\) Second column: \(20\,\text{cm}\) Third column: \(3.5\,\text{m}\)

- Use the number after the colon as the scale factor. - Convert centimeters, meters, and kilometers carefully. - Multiply when moving from map to reality. - Divide when moving from reality to map.

1. For a), \(11\,\text{cm}\cdot 50{,}000=550{,}000\,\text{cm}=5500\,\text{m}=5.5\,\text{km}\). 2. For b), \(4\,\text{km}=400{,}000\,\text{cm}\). The map distance is \(400{,}000\div 50{,}000=8\,\text{cm}\).

a) \(5.5\,\text{km}\) b) \(8\,\text{cm}\)

- Track the requested unit in each part. - Multiply model measurements by \(400\) to find actual measurements. - Divide actual measurements by \(400\) to find model measurements.

1. The actual building height is \(65\,\text{cm}\cdot 400=26{,}000\,\text{cm}=260\,\text{m}\). 2. Convert the antenna height: \(12\,\text{m}=1200\,\text{cm}\). Its model height is \(1200\div 400=3\,\text{cm}\). 3. The actual window width is \(4\,\text{mm}\cdot 400=1600\,\text{mm}=1.6\,\text{m}\).

a) \(260\,\text{m}\) b) \(3\,\text{cm}\) c) \(1.6\,\text{m}\)

- Convert meters to centimeters first. - Divide each actual dimension by the scale factor. - The plan dimensions must be smaller than the actual dimensions.

1. Convert the actual dimensions: \(60\,\text{m}=6000\,\text{cm}\) and \(45\,\text{m}=4500\,\text{cm}\). 2. The plan length is \(6000\div 150=40\,\text{cm}\). 3. The plan width is \(4500\div 150=30\,\text{cm}\).

The courtyard should be drawn \(40\,\text{cm}\) long and \(30\,\text{cm}\) wide.

- Calculate the drawing dimensions for each proposed scale. - Compare both dimensions with the poster dimensions. - Choose the smallest scale denominator that still fits.

1. Convert the pool dimensions: \(50\,\text{m}=5000\,\text{cm}\) and \(25\,\text{m}=2500\,\text{cm}\). 2. At \(1:50\), the drawing would be \(100\,\text{cm}\times 50\,\text{cm}\). The \(100\)-centimeter side does not fit on the \(90\)-centimeter poster side. 3. At \(1:100\), the drawing would be \(50\,\text{cm}\times 25\,\text{cm}\), which fits. 4. At \(1:200\), the drawing would be \(25\,\text{cm}\times 12.5\,\text{cm}\), which also fits but is smaller. 5. Therefore, \(1:100\) gives the largest drawing that fits.

The best scale is \(1:100\).

- Convert the actual distance to centimeters. - Simplify the scale ratio to start with \(1\). - Multiply the map length by the scale denominator.

1. Convert the actual distance: \(5\,\text{km}=500{,}000\,\text{cm}\). 2. The ratio \(10:500{,}000\) simplifies to \(1:50{,}000\). 3. The actual length of a \(12\)-centimeter route is \(12\cdot 50{,}000=600{,}000\,\text{cm}=6\,\text{km}\).

a) \(1:50{,}000\) b) \(6\,\text{km}\)

- Convert kilometers to centimeters. - A map distance must be smaller than the actual distance. - Divide the actual length by the scale factor.

1. Convert the actual distance: \(45\,\text{km}=4{,}500{,}000\,\text{cm}\). 2. Divide by the scale denominator: \(4{,}500{,}000\div 300{,}000=15\,\text{cm}\).

The towns are \(15\,\text{cm}\) apart on the map.

- Multiply the first map distance by its scale denominator. - Convert centimeters to kilometers. - A larger scale denominator produces a shorter map distance.

1. The actual distance is \(12\,\text{cm}\cdot 25{,}000=300{,}000\,\text{cm}=3\,\text{km}\). 2. On the second map, the distance is \(300{,}000\,\text{cm}\div 50{,}000=6\,\text{cm}\). Equivalently, doubling the scale denominator halves the map distance.

1) \(3\,\text{km}\) 2) \(6\,\text{cm}\)

- Multiply a model length by its scale factor to find the actual length. - Keep measurements in centimeters until the scale calculation is complete. - A larger denominator produces a smaller model.

1. The actual length is \(15\,\text{cm}\cdot 200=3000\,\text{cm}=30\,\text{m}\). 2. At a scale of \(1:500\), the model length is \(3000\,\text{cm}\div 500=6\,\text{cm}\).

1) \(30\,\text{m}\) 2) \(6\,\text{cm}\)

- Express both distances in the same unit. - Divide the actual distance by the map distance. - A longer map segment for the same actual distance gives a smaller denominator.

1. Convert the actual distance: \(1.6\,\text{km}=160{,}000\,\text{cm}\). Then \(160{,}000\div 8=20{,}000\), so the first scale is \(1:20{,}000\). 2. For the second map, \(160{,}000\div 20=8000\), so the scale is \(1:8000\).

1) \(1:20{,}000\) 2) \(1:8000\)

- Use the scale denominator as a multiplication or division factor. - Convert between centimeters and kilometers carefully. - For part b), express one map length in each of the two requested units.

1. For a), \(6.5\,\text{cm} \cdot 12{,}000{,}000 = 78{,}000{,}000\,\text{cm} = 780\,\text{km}\). 2. For b), \(1020\,\text{km} = 102{,}000{,}000\,\text{cm}\). Divide by the scale denominator: \(102{,}000{,}000 \div 12{,}000{,}000 = 8.5\,\text{cm}\). 3. Convert centimeters to millimeters: \(8.5\,\text{cm} = 85\,\text{mm}\).

a) \(780\,\text{km}\) b) \(8.5\,\text{cm}\); \(85\,\text{mm}\)

- Convert kilometers to meters before dividing. - Divide by the scale denominator. - Convert the decimal part of a meter to centimeters.

1. Using the stated approximate distance, \(384{,}000\,\text{km}=384{,}000{,}000\,\text{m}\). 2. Dividing the rounded value by the scale denominator gives \(384{,}000{,}000\div 10{,}000{,}000=38.4\,\text{m}\), so the model distance is approximately \(38.4\,\text{m}\). 3. Convert the decimal part: \(0.4\,\text{m}=40\,\text{cm}\).

The model distance is approximately \(38\,\text{m}\ 40\,\text{cm}\).

- Use the first map to find the actual distance. - Divide the actual distance by the second map distance. - A longer map distance for the same route gives a smaller denominator.

1. Find the actual distance from the first map: \(9\,\text{cm}\cdot 200{,}000=1{,}800{,}000\,\text{cm}\). 2. Divide the actual distance by the second map distance: \(1{,}800{,}000\div 12=150{,}000\). 3. Therefore, the second map scale is \(1:150{,}000\).

\(1:150{,}000\)

- Remember that \(1\,\text{km} = 1000\,\text{m}\). - For part b), convert kilometers to meters before applying the scale. - Decide whether the actual length should be greater or less than the map length.

1. For a), multiply by the distance represented by each centimeter: \(12 \cdot 200\,\text{m} = 2400\,\text{m}\). 2. Convert to kilometers: \(2400\,\text{m} = 2.4\,\text{km}\). 3. For b), convert the actual length: \(3\,\text{km} = 3000\,\text{m}\). 4. Divide by \(200\,\text{m}\) per centimeter: \(3000\,\text{m} \div 200\,\text{m} = 15\), so the map length is \(15\,\text{cm}\).

a) The trail is \(2400\,\text{m}\), or \(2.4\,\text{km}\), long. b) The trail would be \(15\,\text{cm}\) long on the map.

- Convert the actual width to centimeters. - Use the same scale for every length on the sketch. - Use the actual dimensions, not the sketch dimensions, to find actual area.

1. Convert the actual width: \(40\,\text{m}=4000\,\text{cm}\). Since \(4000\div 4=1000\), the scale is \(1:1000\). 2. Convert the actual length: \(60\,\text{m}=6000\,\text{cm}\). The sketch length is \(6000\div 1000=6\,\text{cm}\). 3. The actual area is \(60\,\text{m}\cdot 40\,\text{m}=2400\,\text{m}^2\).

a) \(1:1000\) b) \(6\,\text{cm}\) c) \(2400\,\text{m}^2\)

- Use the total horizontal and vertical spans to determine the two unlabeled plan lengths. - Find the full perimeter on the drawing before applying the scale. - Convert the complete drawing perimeter using the stated scale.

1. From the plan, the unlabeled inner vertical length is \(4-2=2\,\text{cm}\), and the unlabeled lower horizontal length is \(5-2=3\,\text{cm}\). 2. The perimeter on the plan is \(5+2+2+2+3+4=18\,\text{cm}\). 3. Apply the scale: \(18 \cdot 1.5 = 27\). The actual boundary length is \(27\,\text{m}\).

\(27\,\text{m}\) of edging are needed.

- Use the total width and total height to determine the two unlabeled sides of the original L-shape. - Compare the two scale denominators to determine how every drawing length changes. - Keep the side order fixed as you transfer all six lengths to the new drawing.

1. On the original plan, the inner vertical side is \(5-3=2\,\text{cm}\), and the upper horizontal side is \(7-2=5\,\text{cm}\). 2. Starting at the lower-left corner and moving clockwise, the original side lengths are \(5,5,2,2,3,7\) cm. 3. Changing from scale \(1:300\) to \(1:150\) doubles every drawing length because \(300\div150=2\). 4. The new side lengths in clockwise order are \(10,10,4,4,6,14\) cm.

\(10\,\text{cm}, 10\,\text{cm}, 4\,\text{cm}, 4\,\text{cm}, 6\,\text{cm}, 14\,\text{cm}\), starting at the lower-left corner and moving clockwise.

- Ask how many dimensions are being scaled when an area changes. - Apply the scale factor once to each dimension. - Convert square centimeters to square meters only after finding the actual area.

1. The scale factor \(500\) applies to lengths, not directly to areas. 2. Area scales by \(500^2=250{,}000\). 3. The actual area is \(12\cdot250{,}000=3{,}000{,}000\,\text{cm}^2\). 4. Since \(1\,\text{m}^2=10{,}000\,\text{cm}^2\), the actual area is \(300\,\text{m}^2\).

Jordan used the length factor instead of its square. The actual area is \(300\,\text{m}^2\).

- Use the first scale to find the actual side lengths. - Calculate actual area from the actual dimensions. - Use the actual dimensions with the second scale. - Find the second plan's area from its new side lengths.

1. The actual dimensions are \(15\,\text{cm}\cdot 200=3000\,\text{cm}=30\,\text{m}\) and \(10\,\text{cm}\cdot 200=2000\,\text{cm}=20\,\text{m}\). 2. The actual area is \(30\,\text{m}\cdot 20\,\text{m}=600\,\text{m}^2\). 3. On the \(1:500\) plan, the side lengths are \(3000\div 500=6\,\text{cm}\) and \(2000\div 500=4\,\text{cm}\). 4. The area of the rectangle on the second plan is \(6\,\text{cm}\cdot 4\,\text{cm}=24\,\text{cm}^2\).

a) \(30\,\text{m}\) by \(20\,\text{m}\) b) \(600\,\text{m}^2\) c) \(6\,\text{cm}\) by \(4\,\text{cm}\), with area \(24\,\text{cm}^2\)