Question

arty says that if you change a scale so that a unit represents a longer distance than in an original scale, then the lengths in the new scale drawing will be longer. do you agree? give an example of a scale and some measurements to support your answer.

Explanation:

Step1: Understand Scale Concept

A scale relates the drawing length to actual length. If a unit on the new scale represents a longer actual distance than the original scale, the drawing length for the same actual length will be shorter (since we need fewer units to represent the same actual length). Wait, but the question says "if you change a scale so that a unit represents a longer distance than in an original scale, then the lengths in the new scale drawing will be longer." Wait, maybe I misread. Wait, no—let's take an example. Let original scale be 1 cm : 1 m (so 1 cm on drawing = 1 m actual). New scale: 1 cm : 2 m (1 cm on drawing = 2 m actual). Now, take an actual length of 2 m. Original drawing length: since 1 cm = 1 m, 2 m would be 2 cm. New drawing length: 1 cm (since 1 cm = 2 m). So new drawing length is shorter. Wait, but the question says "will be longer"—so maybe the question is reversed? Wait, no, maybe I made a mistake. Wait, maybe the scale is "1 unit represents longer distance"—so if original scale is 1 cm : 0.5 m (1 cm = 0.5 m actual), new scale is 1 cm : 1 m (1 cm = 1 m actual). Now, actual length of 1 m: original drawing length is 2 cm (since 0.5 m per cm, so 1 m / 0.5 m per cm = 2 cm). New drawing length is 1 cm. Still shorter. Wait, so maybe the question is incorrect? But the task is to answer. Wait, maybe I misinterpret "a unit represents a longer distance"—maybe the scale is like 1 inch represents 1 foot (original) and new scale is 1 inch represents 0.5 feet (so unit represents shorter distance), but no. Wait, the question says "change a scale so that a unit represents a longer distance than in an original scale"—so unit (drawing unit) represents more actual distance. So drawing unit is "bigger" in terms of actual distance. So for the same actual length, the number of drawing units needed is less. So drawing length is shorter. But the question says "lengths in the new scale drawing will be longer"—so maybe the question has a typo, or I'm misunderstanding. Wait, maybe the scale is "1 unit on drawing represents longer distance"—so if original scale is 1:100 (1 cm = 1 m), new scale is 1:50 (1 cm = 0.5 m). Then, a unit (1 cm) represents shorter distance (0.5 m vs 1 m). Wait, no—1:100 means 1 cm = 100 cm = 1 m. 1:50 means 1 cm = 50 cm = 0.5 m. So 1:100 has unit representing longer distance (1 m) than 1:50 (0.5 m). So if we change from 1:50 (unit represents 0.5 m) to 1:100 (unit represents 1 m), then for actual length of 1 m: original drawing length (1:50) is 2 cm (1 m / 0.5 m per cm = 2 cm), new drawing length (1:100) is 1 cm. So shorter. So the statement "if you change a scale so that a unit represents a longer distance than in an original scale, then the lengths in the new scale drawing will be longer" is incorrect. But the question asks "Do you agree? Give an example..." Wait, maybe I messed up. Let's take another example. Original scale: 1 inch = 1 foot (1:12). New scale: 1 inch = 2 feet (1:24). Actual length: 2 feet. Original drawing length: 2 inches (since 1 inch = 1 foot, 2 feet needs 2 inches). New drawing length: 1 inch (1 inch = 2 feet, so 2 feet needs 1 inch). So new drawing length is shorter. So the statement is wrong. But maybe the question meant "shorter"? Or maybe I have the scale definition reversed. Wait, scale can be represented as drawing : actual. So 1:100 means 1 unit drawing = 100 units actual. So if we have a larger actual per drawing unit (e.g., 1:200 instead of 1:100), then drawing length for same actual is smaller. So the statement is incorrect. But the question asks to agree or not. Let's proceed.

Step2: Provide Example

Let original scale be \( 1 \text{ cm} : 1 \text{ m} \) (so \( 1 \text{ cm}_{\text{drawing}} = 1 \text{ m}_{\text{actual}} \)).
New scale: \( 1 \text{ cm} : 2 \text{ m} \) ( \( 1 \text{ cm}_{\text{drawing}} = 2 \text{ m}_{\text{actual}} \) ).

Take an actual length of \( 2 \text{ m} \).
- Original drawing length: Since \( 1 \text{ cm} = 1 \text{ m} \), \( 2 \text{ m} \) requires \( \frac{2 \text{ m}}{1 \text{ m/cm}} = 2 \text{ cm} \).
- New drawing length: Since \( 1 \text{ cm} = 2 \text{ m} \), \( 2 \text{ m} \) requires \( \frac{2 \text{ m}}{2 \text{ m/cm}} = 1 \text{ cm} \).

Here, the new scale has a unit (1 cm) representing a longer distance (2 m vs 1 m), but the new drawing length (1 cm) is shorter than the original (2 cm). Thus, the statement is incorrect—lengths in the new scale drawing are shorter, not longer.

Answer:

I do not agree with Arty's statement. For example, let the original scale be \( 1 \text{ cm} : 1 \text{ m} \) (1 cm on the drawing represents 1 m in reality) and the new scale be \( 1 \text{ cm} : 2 \text{ m} \) (1 cm on the drawing represents 2 m in reality). For an actual length of 2 m:

• Original drawing length: \( \frac{2 \text{ m}}{1 \text{ m/cm}} = 2 \text{ cm} \).
• New drawing length: \( \frac{2 \text{ m}}{2 \text{ m/cm}} = 1 \text{ cm} \).

The new scale’s unit represents a longer distance (2 m vs 1 m), but the new drawing length (1 cm) is shorter than the original (2 cm). Thus, lengths in the new scale drawing are shorter, not longer.