Question

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Explanation:

Identify the given proportions in Question 1

We need to solve for the unknown variable \(x\) in each of the given ratio equations:

• Part a: \(\frac{2}{15} = \frac{x}{60}\)
• Part b: \(\frac{4}{50} = \frac{x}{200}\)
• Part c: \(\frac{3}{5} = \frac{x}{45}\)

Solve each proportion in Question 1

• For a: Multiply both sides by \(60\):

$$ x = \frac{2}{15} \times 60 = 2 \times 4 = 8 $$

• For b: Multiply both sides by \(200\):

$$ x = \frac{4}{50} \times 200 = 4 \times 4 = 16 $$

• For c: Multiply both sides by \(45\):

$$ x = \frac{3}{5} \times 45 = 3 \times 9 = 27 $$

Identify the given proportions in Question 2

We need to find the unknown distance \(x\) in kilometres:

• Part a: \(\frac{5\text{ km}}{15\text{ min}} = \frac{x}{75\text{ min}}\)
• Part b: \(\frac{30\text{ km}}{10\text{ min}} = \frac{x}{21\text{ min}}\)
• Part c: \(\frac{20\text{ km}}{6\text{ min}} = \frac{x}{72\text{ min}}\)

Solve each proportion in Question 2

• For a: Multiply both sides by \(75\):

$$ x = \frac{5}{15} \times 75 = \frac{1}{3} \times 75 = 25\text{ km} $$

• For b: Multiply both sides by \(21\):

$$ x = \frac{30}{10} \times 21 = 3 \times 21 = 63\text{ km} $$

• For c: Multiply both sides by \(72\):

$$ x = \frac{20}{6} \times 72 = 20 \times 12 = 240\text{ km} $$

Analyze Question 3

Question 3 asks to find the distance between several cities using the provided map. However, because the map image is highly blurry and lacks a readable scale bar or grid coordinates, we cannot measure the exact physical map distances to calculate the real-world distances. We will provide the mathematical method used to solve such map scale problems.

Answer:

Question 1

• a. \(x = 8\)
• b. \(x = 16\)
• c. \(x = 27\)

Question 2

• a. \(x = 25\text{ km}\)
• b. \(x = 63\text{ km}\)
• c. \(x = 240\text{ km}\)

Question 3

Due to the low resolution and blurriness of the map image, the precise scale factor and measured map distances cannot be determined. To solve these:

1. Measure the straight-line distance between the two cities on the map using a ruler (in centimetres or inches).
2. Measure the length of the scale bar at the bottom right of the map.
3. Set up a proportion:

$$ \frac{\text{Measured Distance on Map}}{\text{Measured Scale Bar Length}} = \frac{\text{Actual Distance}}{\text{Scale Bar Real Distance}} $$

1. Solve for the actual distance.