Question
Chaz draws a square with sides equal to 4 units, then he dilates the square so the sides are equal to 6 units. Chaz writes a fraction in lowest terms to express the relationship between the area of the original square and the dilated square, with the numerator expressing the area of the dilated square. What is that fraction? (1 point) Responses 49 Start Fraction 4 over 9 End Fraction 214 2 Start Fraction 1 over 4 End Fraction 1636 16 over 36 23
Answers
GPT-4o mini
To find the fraction that expresses the relationship between the area of the original square and the dilated square, we first need to calculate the areas of both squares.
1. **Area of the original square**:
\[
\text{Area}_{\text{original}} = \text{side}^2 = 4^2 = 16 \text{ square units}
\]
2. **Area of the dilated square**:
\[
\text{Area}_{\text{dilated}} = \text{side}^2 = 6^2 = 36 \text{ square units}
\]
3. **Relationship between the areas**:
We express the relationship using a fraction where the numerator is the area of the dilated square and the denominator is the area of the original square.
\[
\text{Fraction} = \frac{\text{Area}_{\text{dilated}}}{\text{Area}_{\text{original}}} = \frac{36}{16}
\]
4. **Simplifying the fraction**:
We simplify \( \frac{36}{16} \) by finding the greatest common divisor (GCD) of 36 and 16, which is 4:
\[
\frac{36 \div 4}{16 \div 4} = \frac{9}{4}
\]
Thus, the fraction that expresses the relationship between the area of the dilated square and the area of the original square in lowest terms is:
\[
\frac{9}{4}
\]
This fraction represents that the area of the dilated square is 9/4 times the area of the original square.
1. **Area of the original square**:
\[
\text{Area}_{\text{original}} = \text{side}^2 = 4^2 = 16 \text{ square units}
\]
2. **Area of the dilated square**:
\[
\text{Area}_{\text{dilated}} = \text{side}^2 = 6^2 = 36 \text{ square units}
\]
3. **Relationship between the areas**:
We express the relationship using a fraction where the numerator is the area of the dilated square and the denominator is the area of the original square.
\[
\text{Fraction} = \frac{\text{Area}_{\text{dilated}}}{\text{Area}_{\text{original}}} = \frac{36}{16}
\]
4. **Simplifying the fraction**:
We simplify \( \frac{36}{16} \) by finding the greatest common divisor (GCD) of 36 and 16, which is 4:
\[
\frac{36 \div 4}{16 \div 4} = \frac{9}{4}
\]
Thus, the fraction that expresses the relationship between the area of the dilated square and the area of the original square in lowest terms is:
\[
\frac{9}{4}
\]
This fraction represents that the area of the dilated square is 9/4 times the area of the original square.