Respan Dataset Explorer

The diagram represents a reduction of a triangle by using a scale factor of 0.8.

A larger triangle has a base of 10 inches and height of 6 inches.

What is the height of the reduced triangle?
4.0 inches
4.8 inches
5.2 inches
7.5 inches

To find the height of the reduced triangle using the scale factor of 0.8, you need to multiply the height of the larger triangle by the scale factor.

Given:
- Height of the larger triangle = 6 inches
- Scale factor = 0.8

Using the formula:

\[ \text{Height of the reduced triangle} = \text{Height of the larger triangle} \times \text{Scale factor} \]

\[ \text{Height of the reduced triangle} = 6 \text{ inches} \times 0.8 = 4.8 \text{ inches} \]

So, the height of the reduced triangle is **4.8 inches**. 

The correct answer is: **4.8 inches**.

The triangle represents a scale drawing that was created by using a factor of One-half.

A triangle with sides lengths of 5 inches.

Which is true of the measures of the sides of the original triangle?
Each side of the original triangle is One-half the length of each side of the scale drawing.
Each side of the original triangle is 2 times the length of each side of the scale drawing.
Each side of the original triangle is the same length as each side of the scale drawing.
Each side of the original triangle is 2 inches shorter than the scale drawing.

The correct statement regarding the measures of the sides of the original triangle is:

**Each side of the original triangle is 2 times the length of each side of the scale drawing.**

Since the scale factor is one-half, it means that each side of the original triangle is double the length of each side of the scale drawing. Therefore, if the scaled triangle has sides of 5 inches, the original triangle would have sides of \(5 \, \text{inches} \times 2 = 10 \, \text{inches}\).

A scale drawing of a rectangle is made by using a scale factor of StartFraction 5 over 8 EndFraction. The original and the scale drawing are shown below.

A rectangle has a length of 18 inches and width of 12 inches.
[Not drawn to scale]

Which method can be used to find the dimensions of the original rectangle?
18 minus StartFraction 5 over 8 EndFraction = 17 and StartFraction 3 over 8 EndFraction inches and 12 minus StartFraction 5 over 8 EndFraction = 11 and StartFraction 3 over 8 EndFraction inches
StartFraction 5 over 8 EndFraction divided by 18 = StartFraction 5 over 144 EndFraction inches and StartFraction 5 over 8 EndFraction divided by 12 = StartFraction 5 over 96 EndFraction inches
18 times StartFraction 5 over 8 EndFraction = 11 and one-fourth inches and 12 times StartFraction 5 over 8 EndFraction = 7 and one-half inches
18 divided by StartFraction 5 over 8 EndFraction = 28 and four-fifths inches and 12 divided by StartFraction 5 over 8 EndFraction = 19 and one-fifth inches
Mark this and return   

To find the dimensions of the original rectangle based on a scale drawing using a scale factor of \( \frac{5}{8} \), you would need to use the inverse of the scale factor to calculate the dimensions of the original rectangle from the dimensions of the scale drawing.

To find the original dimension, you need to divide the dimensions of the scale drawing by the scale factor, which is equivalent to multiplying by the reciprocal of the scale factor.

The correct method to find the dimensions is:

**18 divided by \( \frac{5}{8} \) = 28 and \( \frac{4}{5} \) inches, and 12 divided by \( \frac{5}{8} \) = 19 and \( \frac{1}{5} \) inches.**

So the correct answer is:

**18 divided by \( \frac{5}{8} \) = 28 and \( \frac{4}{5} \) inches and 12 divided by \( \frac{5}{8} \) = 19 and \( \frac{1}{5} \) inches.**

The diagram represents an enlargement of a rectangle by a scale factor of 4.

A small rectangle has a length of 4 inches and width of 2 inches.

What are the dimensions of the enlarged rectangle?
1 inch by One-half inch
8 inches by 6 inches
16 inches by One-half inch
16 inches by 8 inches
Mark this and return   

To find the dimensions of the enlarged rectangle using a scale factor of 4, you need to multiply the dimensions of the small rectangle by the scale factor.

Given:
- Length of the small rectangle = 4 inches
- Width of the small rectangle = 2 inches
- Scale factor = 4

Now, calculate the dimensions of the enlarged rectangle:

1. Enlarged Length:
   \[ 
   \text{Enlarged Length} = \text{Small Length} \times \text{Scale Factor} = 4 \, \text{inches} \times 4 = 16 \, \text{inches} 
   \]

2. Enlarged Width:
   \[ 
   \text{Enlarged Width} = \text{Small Width} \times \text{Scale Factor} = 2 \, \text{inches} \times 4 = 8 \, \text{inches} 
   \]

Therefore, the dimensions of the enlarged rectangle are:

**16 inches by 8 inches.** 

So the correct answer is: **16 inches by 8 inches.**

Each side of the regular hexagon below has the same measure.

A hexagon with side lengths of 2.5 feet.

If a model of the hexagon is made by using a scale factor of 6, which applies to the model? Select two options.
The model represents a reduction.
Each side of the model is 15 feet long.
The model is proportional to the original hexagon.
One side of the model can be 8.5 feet.
The scale factor is divided by 2.5 to get the dimensions of the model.
Mark this and return   

To determine which statements apply to the model of the hexagon using a scale factor of 6, let's analyze the options based on the information given.

1. Each side of the original hexagon is 2.5 feet long.
2. If the scale factor is 6, then the dimensions of the model will be calculated by multiplying the original dimensions by the scale factor.

Calculating the side length of the model:
- Model side length = Original side length × Scale factor
- Model side length = 2.5 feet × 6 = 15 feet

Now, let's evaluate the options:

1. **The model represents a reduction.**
   - **Incorrect**: The model is an enlargement, not a reduction.

2. **Each side of the model is 15 feet long.**
   - **Correct**: Each side of the model is indeed 15 feet long.

3. **The model is proportional to the original hexagon.**
   - **Correct**: The model maintains the same shape and proportions as the original hexagon.

4. **One side of the model can be 8.5 feet.**
   - **Incorrect**: Each side of the model is 15 feet, not 8.5 feet.

5. **The scale factor is divided by 2.5 to get the dimensions of the model.**
   - **Incorrect**: The scale factor is used to multiply the original dimensions, not divided.

Therefore, the two correct options are:
- **Each side of the model is 15 feet long.**
- **The model is proportional to the original hexagon.**

Beth is designing a card. She brings to a print shop a drawing that is 6 inches by 3 inches, and asks them to print the card after applying a scale factor of mc019-1.jpg. When she returns to the print shop, they hand her a box of cards, each of which is 9 inches by 13mc019-2.jpg inches, which is incorrect.
What should be the correct dimensions of the card?
The scale factor is a reduction, and the new card should be 4 in x 2 in.
The scale factor is a reduction, and the new card should be 4 in x 6 in.
The scale factor is for an enlargement, and the print shop created the right size card.
The scale factor is for an enlargement, but the print shop should have made the card 9 in by 6 in.
Mark this and return   

To determine the correct dimensions of the card after applying the scale factor, we first need to clarify whether the scale factor is for a reduction or an enlargement.

Beth's original card dimensions are:
- Length: 6 inches
- Width: 3 inches

When a scale factor is applied, the new dimensions are obtained by multiplying the original dimensions by the scale factor. 

Since the dimensions given when Beth returns to the print shop are:
- Length: 9 inches
- Width: 13 inches (this value seems to be incomplete and not clear, but let's assume it's meant to represent an actual width)

Now let's analyze the options provided:

1. **The scale factor is a reduction, and the new card should be 4 in x 2 in.**
   - This would not be consistent with her original dimensions.

2. **The scale factor is a reduction, and the new card should be 4 in x 6 in.**
   - This also does not align properly with the original sizes.

3. **The scale factor is for an enlargement, and the print shop created the right size card.**
   - If we take the card's dimensions at face value (9 inches by 13 inches), it does not equal a consistent enlargement of 6 inches by 3 inches. 

4. **The scale factor is for an enlargement, but the print shop should have made the card 9 in by 6 in.**
   - This option suggests the dimensions still do not align.

To correctly identify the dimensions after finding the proper scale factor, let’s check the ratio between the desired card and the original size. 

If Beth intended to have an enlargement proportionate to her original size of 6 inches by 3 inches, the ratio between the original card sizes and the new card sizes should be checked. 

A common enlargement would involve multiplying both dimensions by the same factor. For the dimensions given (9 inches and 13 inches) could be none of these options (especially since one of them does not seem to align well).

**Correct Option**: If we had a scale factor to work with from her original 6 inches by 3 inches, and if we assumed a factor of 1.5 to create a proportion where both dimensions scale up consistently, you would get dimensions surrounding the values of 9 inches by approximately 4.5 inches.

Given all the analysis, none of the options appear fully correct, but a logical conclusion points toward a miscommunication regarding dimensions.
  
To summarize:

**The likely correct approach if given the dimensions properly should clarify that the scale factor is for an enlargement with consistent ratios, while print discrepancies could be noted with 9 inches by 4.5 inches making sense; hence, it is unclear largely provided the description here.** But such understanding of enlargement does apply correctly when analyzing dimensions and expecting a similar ratio factor on a proportional scale would apply even further—suggest continually monitoring exact specifications between both sizes initially provided for the full answer. 

If you need specific calculations or detailed feedback, please share the intended ratio or details directly.

The square represents a scale model that was created by using a factor of 4.

A square with sides lengths of 8 feet.

Which is true of the measures of the sides of the original square?
Each side of the original square is One-fourth the length of each side of the scale model.
Each side of the original square is 4 times the length of each side of the scale model.
Each side of the original square is the same length as each side of the scale model.
Each side of the original square is 4 feet less than the scale model.
Mark this and return   

To determine the relationship between the sides of the scale model and the original square, we need to analyze the given information:

- The scale factor is 4.
- The side length of the scale model is 8 feet.

1. **Understanding the Scale Factor**: A scale factor of 4 means that the original square's dimensions are larger than the model's dimensions. Specifically, the original square will have sides that are 4 times longer than the corresponding sides of the scale model.

2. **Calculating the Original Side Length**:
   \[
   \text{Original side length} = \text{Scale model side length} \times \text{Scale factor}
   \]
   \[
   \text{Original side length} = 8 \, \text{feet} \times 4 = 32 \, \text{feet}
   \]

Now, let's evaluate the statements:

1. **Each side of the original square is one-fourth the length of each side of the scale model.**
   - **Incorrect**: The original square is larger, not smaller.

2. **Each side of the original square is 4 times the length of each side of the scale model.**
   - **Correct**: The original square's dimensions (32 feet) are indeed 4 times the scale model's dimensions (8 feet).

3. **Each side of the original square is the same length as each side of the scale model.**
   - **Incorrect**: The original square is larger.

4. **Each side of the original square is 4 feet less than the scale model.**
   - **Incorrect**: The original square is significantly larger, not smaller.

Based on this analysis:

- The **correct statement** is: **Each side of the original square is 4 times the length of each side of the scale model.**

Alan solved the proportion StartFraction x over 200 EndFraction = StartFraction 8 over 25 EndFraction as shown.

StartFraction x over 200 EndFraction = StartFraction 8 over 25 EndFraction. (8) (x) = (25) (200). 8 x = 5,000. StartFraction 8 x over 8 EndFraction = StartFraction 5,000 over 8 EndFraction. X = 625.

What is Alan’s error?
He got the wrong product when he multiplied 25 by 200.
He got the wrong quotient when he divided 5,000 by 8.
He mixed up the positions of 8 and 25 in the equation (8) (x) = (25) (200).
He mixed up the positions of 8 and 200 in the equation (8) (x) = (25) (200).
Mark this and return

To analyze Alan's solution to the proportion \( \frac{x}{200} = \frac{8}{25} \), we can go through his steps to find the error.

1. **Starting with the proportion:**
   \[ 
   \frac{x}{200} = \frac{8}{25} 
   \]

2. **Cross-multiplying:**
   - Correctly, this should yield:
   \[
   25x = 8 \times 200 
   \]

3. **Calculating \(8 \times 200\):**
   - \(8 \times 200 = 1600\), not 5000. This shows that Alan made a mistake in his multiplication.

4. **Continuing with Alan's steps:**
   - He incorrectly calculated \( 25 \times 200\) as 5000. 
   - The equation he derived is:
   \[
   8x = 5000
   \]
   (which is incorrect)

5. **Solving for \(x\):**
   - Later in his solution, Alan divided by 8 to find \(x\), but since his product was wrong, his final answer is incorrect.

### Conclusion:
The error Alan made is:
**He got the wrong product when he multiplied 25 by 200.** 

So the correct choice is: **He got the wrong product when he multiplied 25 by 200.**

A rectangular magazine cover is photocopied by using a scale factor of One-third. The length of the photocopy of the magazine cover is
enlarged by a factor of 3.
enlarged by a factor of One-third.
reduced by a factor of 3.
reduced by a factor of One-third.
Mark this and return

When a rectangular magazine cover is photocopied using a scale factor of **One-third**, it means that the dimensions of the original cover are reduced to one-third of their original size.

Now let's analyze the options provided regarding the photocopy:

1. **Enlarged by a factor of 3.**
   - **Incorrect**: This would mean increasing the size, not reducing it.

2. **Enlarged by a factor of One-third.**
   - **Incorrect**: This statement suggests an enlargement, which contradicts the concept of scaling down the size.

3. **Reduced by a factor of 3.**
   - **Incorrect**: This phrasing is ambiguous; however, it is not representative of the scale factor as it suggests reducing by a factor of 3.

4. **Reduced by a factor of One-third.**
   - **Correct**: The scale factor of One-third indicates that the dimensions of the original magazine cover are reduced to one-third of their original length.

Thus, the correct answer is:

**Reduced by a factor of One-third.**

What proportion results in the equation 9 m = 10 n?
StartFraction 9 over n EndFraction = StartFraction 10 over m EndFraction
StartFraction 9 over m EndFraction = StartFraction 10 over n EndFraction
StartFraction 9 over 10 EndFraction = StartFraction m over n EndFraction
StartFraction 10 over 9 EndFraction = StartFraction n over m EndFraction
Mark this and return

To determine which proportion results in the equation \(9m = 10n\), we need to manipulate this equation to identify the correct proportions.

1. **Starting with the equation**:
   \[
   9m = 10n
   \]

2. **Rearranging the equation**:
   - Divide both sides by \(mn\) to set up a proportion:
   \[
   \frac{9}{n} = \frac{10}{m}
   \]

Now, let's look at the provided answer choices:

1. **\(\frac{9}{n} = \frac{10}{m}\)**
   - This is consistent with our rearrangement of \(9m = 10n\).
  
2. **\(\frac{9}{m} = \frac{10}{n}\)**
   - This is not equivalent to the original equation. 

3. **\(\frac{9}{10} = \frac{m}{n}\)**
   - This is not equivalent either; it's a different proportion.

4. **\(\frac{10}{9} = \frac{n}{m}\)**
   - This is also not equivalent to the original equation.

### Conclusion:
The correct choice, which accurately represents the proportion resulting from the equation \(9m = 10n\), is:

**\(\frac{9}{n} = \frac{10}{m}\)**.