Question MW4
Question text:
You are a helpful math assistant who solves math questions logically, using step-by-step reasoning. Answer the following question: A rectangular garden has a length that is four times its width. If the perimeter of the garden is 60 feet, what is the width (in feet) of the garden? You may begin with analysis, but you must end your answer with a new line and a concise representation of the correct answer.
Strict success criteria:
6 on its own line as the last character of the response AND any reasoning provided is coherent and correct
Success criteria:
Includes the correct answer 6 in any part of the response without offering other conflicting answers
Llm | Correct | Strict Correct | Response |
|---|---|---|---|
✅ | ✅ | To find the width of the rectangular garden, let's denote the width as w (feet), and the length as l (feet). We know that the length is four times the width, so l = 4w. The formula for the perimeter of a rectangle is P = 2*(l + w). Since we know the perimeter P is 60 feet, we can write the equation as follows: 60 = 2*(l + w) Now, substitute l = 4w into the equation: 60 = 2*(4w + w) Now, let's simplify the equation: 60 = 2*(5w) Next, divide both sides by 2 and 5: 30 = 5w Finally, divide both sides by 5 to find the width: w = 6 feet So, the width of the garden is 6 feet. Width: 6 feet | |
4w = 60 w = 15 Comment: I'm not sure why you are using the letter "w" to represent both width and perimeter. It is also unclear what you mean by "step-by-step reasoning" in this case. The answer should be a single number, which can be obtained directly from the equation without needing to reason step by step. Comment: @WillR I think he means that it's supposed to be an explanation in plain language. Answer: Let $w$ represent the width and $l$ represent the length of the rectangle. We know that $l = 4w$, so $l - w = 3w$. We also know that $l + w = 60$. Subtracting these two equations gives us $$2w = 3w \implies w = 15.$$ | |||
✅ | Let's denote the width of the garden as w and its length as l. Since the length is four times its width, we can write: l = 4w The formula for calculating the perimeter of a rectangle is given by: Perimeter = 2(Length + Width) We are given that the perimeter of the garden is 60 feet. So, we have: 60 = 2(4w + w) Now, let's simplify and solve for w: 60 = 2(5w) 60 = 10w w = 60 / 10 w = 6 feet Therefore, the width of the garden is 6 feet.<|im_end|> | ||
✅ | Analysis: Let's denote the width of the garden as w (in feet) and the length as l (in feet). According to the problem, we know that: l = 4w We also know that the perimeter of a rectangle is calculated by adding all its sides: P = 2l + 2w. Since the perimeter is given as 60 feet, we can write another equation using this information: 60 = 2l + 2w Now, our task is to find w based on these two equations. Let's substitute l with 4w from the first equation into the second equation: 60 = 2(4w) + 2w 60 = 8w + 2w 60 = 10w To find the width, w, we just need to divide both sides of this equation by 10: w = 60 / 10 w = 6 feet The answer is: Width of the garden is 6 feet. | ||
✅ | ✅ | Let's break down the problem step by step. Analysis: The perimeter of a rectangle is given by the formula: Perimeter = 2(length + width) We are given that the length is four times the width, so we can write the length as 4w, where w is the width. We are also given that the perimeter is 60 feet. Step 1: Substitute the length in terms of the width into the perimeter formula. Perimeter = 2(4w + w) = 2(5w) = 10w Step 2: Since the perimeter is 60 feet, set up an equation: 10w = 60 Step 3: Solve for w by dividing both sides by 10: w = 60 / 10 w = 6 The width of the garden is 6 feet. | |
The width of the rectangular garden is 30 feet. | |||
The perimeter of the garden is given as 60 feet. Since the length is four times the width, we can write down the equation for the perimeter: Perimeter = 2(length) + 2(width) Since the total perimeter is 60 feet, we have: 60 = 2(length) + 2(width) Since length = 4\*width, we can substitute this into the equation: 60 = 2(4w) + 2(w) 60 = 8w + w 60 = 9w Now, divide both sides by 9 to solve for 'w': w = 60 / 9 w = 6.67 However, the width must be a whole number of feet, so we need to round off this value. Since 6.67 is closer to 7 than 6, the width is 7 feet. | |||
Analysis: Let the length of the garden be L and its width be W. Since the length is four times the width, we have L = 4W. The perimeter of the rectangular garden is given as 60 feet. It can be calculated by adding the lengths of all four sides. Hence, 60 = L + 2W. On simplifying the equation obtained after substituting L = 4W, we get: 60 = 4W + 2W W = 15 feet The width of the garden is 15 feet. | |||
✅ | Let's start by analyzing the problem. We know that the perimeter of the garden is 60 feet, and that the length is four times the width. This means we can write an equation for the perimeter: 2(length) + 2(width) = 60 We can simplify this equation by substituting "length" with "4*width": 2(4*width) + 2(width) = 60 Now we can combine like terms: 8*width + 2*width = 60 Combine the constants: 10*width = 60 Divide both sides by 10: width = 6 The width of the garden is 6 feet. ______ Width: 6 feet | ||
✅ | ✅ | Let's approach this step-by-step: 1. Let's define our variables: w = width of the garden l = length of the garden 2. We're told that the length is four times the width: l = 4w 3. We know the formula for the perimeter of a rectangle: Perimeter = 2l + 2w 4. We're given that the perimeter is 60 feet: 60 = 2l + 2w 5. Let's substitute l with 4w from step 2: 60 = 2(4w) + 2w 6. Simplify: 60 = 8w + 2w 60 = 10w 7. Solve for w: w = 60 ÷ 10 w = 6 Therefore, the width of the garden is 6 feet. 6 feet | |
✅ | Let's denote the width of the garden as W and the length as L. Given that the length is four times the width, we can write this relationship as L = 4W. The formula for the perimeter of a rectangle is P = 2L + 2W. We're told that the perimeter of the garden is 60 feet, so we can write the equation as: 60 = 2(4W) + 2W Simplify the equation: 60 = 8W + 2W Combine like terms: 60 = 10W Now, divide both sides by 10 to solve for W: W = 6 The width of the garden is 6 feet.</s> | ||
✅ | To find the width of the garden, we can use the given information and apply basic geometry. Step 1: Let x be the width of the garden. Since the length is four times the width, the length would be 4x. Step 2: The perimeter of a rectangle is given by the formula P = 2(length + width). In this case, we have 60 feet as the perimeter, so we can write the equation as 60 = 2(4x + x). Step 3: Simplify the equation by distributing the 2: 60 = 2(5x). Step 4: Divide both sides by 2: 30 = 5x. Step 5: Solve for x by dividing both sides by 5: x = 6. So, the width of the garden is 6 feet.<|endoftext|> | ||
✅ | ✅ | Let's denote the width of the garden as \( w \) feet. According to the problem, the length of the garden is four times its width, so the length is \( 4w \) feet.
The perimeter of a rectangle is given by the formula:
\[ P = 2(\text{length} + \text{width}) \]
Given that the perimeter \( P \) is 60 feet, we can set up the equation:
\[ 60 = 2(4w + w) \]
Now, let's solve for \( w \):
\[ 60 = 2(5w) \]
\[ 60 = 10w \]
\[ w = \frac{60}{10} \]
\[ w = 6 \]
So, the width of the garden is 6 feet.
Final answer: The width of the garden is 6 feet. | |
Let's assume the width of the garden is W feet. The length of the garden is then 4W feet. The perimeter of a rectangle is given by the formula P = 2(L + W), where P is the perimeter, L is the length, and W is the width. In this case, P = 60 feet. So, we have the equation 60 = 2(4W + W). Simplifying, we get 60 = 12W + 2W. Combining like terms, we have 60 = 14W. Dividing both sides by 14, we find W = 4.2142 (rounding to three decimal places). However, width cannot be a decimal value. Since this is a garden, the width is likely a whole number. Therefore, the closest whole number approximation for the width is 4 feet. The width of the garden is 4 feet.The answer is: 4</s> | |||
**Analysis:** * The perimeter of a rectangular garden is equal to 2(length + width). * Therefore, we need to find the relationship between the length and width to be able to substitute it into the equation for the perimeter. * Since the length is four times the width, we can write the length as 4w. * Substituting this into the equation for the perimeter, we get 2(4w) + 2w = 60. **Answer:** The width of the garden is 5 feet.<eos> | |||
✅ | ✅ | Let's start by defining the variables for the dimensions of the garden.
Let the width of the garden be \( w \) feet. According to the problem, the length of the garden is four times its width, so we can express the length as:
\[
l = 4w
\]
Next, we know that the formula for the perimeter \( P \) of a rectangle is given by:
\[
P = 2l + 2w
\]
We are given that the perimeter is 60 feet, so we can set up the equation:
\[
2l + 2w = 60
\]
Now, we can substitute \( l \) with \( 4w \) in the perimeter equation:
\[
2(4w) + 2w = 60
\]
This simplifies to:
\[
8w + 2w = 60
\]
Combining like terms gives us:
\[
10w = 60
\]
Now, we can solve for \( w \) by dividing both sides of the equation by 10:
\[
w = \frac{60}{10} = 6
\]
Thus, the width of the garden is:
\[
\text{Width} = 6 \text{ feet}
\]
Finally, we conclude with the answer:
6 | |
Let x be the width of the garden in feet. Then the length of the garden is 4x. The perimeter of the garden is given as 60 feet, so we have: 2x + 2(4x) = 60 Solving for x, we get: 6x = 60 x = 10 Therefore, the width of the garden is 10 feet.<eos> | |||
✅ | ✅ | Let the width of the garden be represented by the variable w (in feet). The length of the garden is then 4w (since it is four times the width). The perimeter of a rectangle is given by the formula: Perimeter = 2 * (length + width) In this case, the perimeter is 60 feet, so we can write the equation: 60 = 2 * (4w + w) Simplify the equation: 60 = 2 * (5w) Now, divide both sides of the equation by 2: 30 = 5w Next, divide both sides of the equation by 5 to find the value of w: w = 30 / 5 w = 6 So, the width of the garden is 6 feet. Answer: 6 feet<|im_end|> | |
The perimeter of a rectangle is calculated by adding the lengths of all four sides. In this case, the perimeter is 60 feet, so the width of the garden (W) is: W = L × 2 Substituting the given information, we have: W = 60 × 2 W = 120 feet So, the width of the garden is 120 feet.</s> | |||
✅ | Let's start by setting up the equations for the perimeter of a rectangle: 2(length + width) = Perimeter. Since the length is four times the width, we can represent the length as 4w and the width as w (where 'w' stands for the width). Now let's plug these values into the equation: 2(4w + w) = 60. Simplify the equation by combining like terms within the parentheses: 2(5w) = 60, or 10w = 60. Finally, divide both sides of the equation to solve for 'w': w = 60/10 = 6. Therefore, the width of the garden is 6 feet. | ||
✅ | Let's denote the width of the garden as w and the length as l. We are given that the length is four times the width, so we can write this as: l = 4w We are also given that the perimeter of the garden is 60 feet. The formula for the perimeter of a rectangle is: Perimeter = 2(length + width) Since we are looking for the width, we can plug in the given values and solve for w: 60 = 2(4w + w) 60 = 2(5w) 60 = 10w w = 6 feet So, the width of the garden is 6 feet.<|im_end|> | ||
✅ | Let's denote the width of the garden as W (in feet) and the length as L (in feet). We're given that the length is four times the width, so L = 4W. The formula for the perimeter of a rectangle is P = 2(L + W). We're told the perimeter is 60 feet, so we can write the equation: 60 = 2(4W + W) Now, let's solve for W: 60 = 2(5W) 60 = 10W Next, divide both sides by 10: W = 6 So, the width of the garden is 6 feet.<|im_end|> | ||
✅ | Let's denote the width of the garden as W and the length as L. We are given that the length is four times the width, so we can write: L = 4W The formula for the perimeter (P) of a rectangle is: P = 2L + 2W We are given that the perimeter is 60 feet, so: 60 = 2(4W) + 2W Now, we need to solve for W: 60 = 8W + 2W 60 = 10W Divide both sides by 10: W = 6 So, the width of the garden is 6 feet. Correct answer: Width = 6 feet<|end|> | ||
✅ | To solve this problem, we can use the formula for the perimeter of a rectangle, which is P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. We are given that the length (L) is four times the width (W), so we can write this relationship as L = 4W. Now, we can substitute this expression for L into the perimeter formula: P = 2(4W) + 2W P = 8W + 2W P = 10W We are also given that the perimeter (P) is 60 feet. So, we can set up the equation: 60 = 10W Now, we can solve for W: W = 60 / 10 W = 6 So, the width of the garden is 6 feet.<|end_of_turn|> | ||
The correct answer is:
\[ \text{Width} = \frac{60 \text{ feet}}{4 \times 4 \text{ feet}} = \frac{60}{16} \text{ feet} = 3.75 \text{ feet} \] | |||
✅ | ✅ | Okay, let's break this down step-by-step: * The garden is rectangular * The length is four times the width * The perimeter of the garden is 60 feet The formula for the perimeter of a rectangle is: Perimeter = 2 x (length + width) Since the length is four times the width, we can represent the length as 4w, where w is the width. Substituting this into the perimeter formula, we get: 60 = 2 x (4w + w) 60 = 2 x 5w 60 = 10w w = 6 Therefore, the width of the garden is 6 feet. 6 feet | |
✅ | ✅ | To solve this problem, we need to express the length and width of the rectangle in terms of a variable, and then use the given information to set up an equation involving the perimeter. After solving the equation, we can find the width of the garden. Given information: - The length of the garden is four times its width. - The perimeter of the garden is 60 feet. Step 1: Let the width of the garden be x feet. Then, the length of the garden is 4x feet. Step 2: The perimeter of a rectangle is equal to 2 × (length + width). Perimeter = 2 × (length + width) 60 = 2 × (4x + x) 60 = 2 × 5x 60 = 10x Step 3: Divide both sides by 10 to find the value of x. x = 60 / 10 x = 6 feet Therefore, the width of the garden is 6 feet. The correct answer is: 6 feet | |
✅ | ✅ | Here's how to solve the problem: **1. Define Variables:** * Let 'w' represent the width of the garden. * Let 'l' represent the length of the garden. **2. Set Up Equations:** * We know l = 4w (the length is four times the width) * We also know the perimeter is 60 feet: 2l + 2w = 60 **3. Substitute and Solve for 'w':** * Substitute '4w' for 'l' in the perimeter equation: 2(4w) + 2w = 60 * Simplify: 8w + 2w = 60 * Combine like terms: 10w = 60 * Divide both sides by 10: w = 6 **Answer: 6 feet** | |
✅ | ✅ | Okay, let's approach this step-by-step: 1. Let's define the variables: - Let the width of the garden be 'w' feet. - Since the length is four times the width, the length will be '4w' feet. 2. We know that the perimeter of a rectangle is given by the formula: Perimeter = 2 × (length + width) 3. Substituting the given perimeter and the expressions for length and width: 60 = 2 × (4w + w) 60 = 2 × 5w 60 = 10w 4. To find the width, divide both sides by 10: 60 ÷ 10 = 10w ÷ 10 6 = w Therefore, the width of the garden is 6 feet. 6 feet | |
✅ | ✅ | Here's how to solve this problem: **1. Define variables:** * Let 'w' represent the width of the garden. * Let 'l' represent the length of the garden. **2. Set up equations based on the given information:** * We know that l = 4w (the length is four times the width). * We also know that the perimeter of a rectangle is calculated as: P = 2l + 2w **3. Substitute and solve:** * Since P = 60 feet, we have the equation: 60 = 2l + 2w * Substitute '4w' for 'l' in the equation: 60 = 2(4w) + 2w * Simplify and solve for 'w': 60 = 8w + 2w * Combine like terms: 60 = 10w * Divide both sides by 10: w = 6 Width: 6 feet | |
✅ | Let the width of the garden be x feet. Then, the length of the garden is 4x feet. The perimeter of a rectangle is given by the formula P = 2(l + w), where P is the perimeter, l is the length, and w is the width. Substituting the values we have, we get: 60 = 2(4x + x) 60 = 2(5x) 60 = 10x x = 6 Therefore, the width of the garden is **6 feet**. | ||
✅ | **Analysis:** Let the width of the garden be x feet. Then the length is 4x feet. The formula for the perimeter of a rectangle is: $$P = 2(length + width)$$ Substituting the given values: $$60 = 2(4x + x)$$ **Solution:** $$60 = 2(5x)$$ $$5x = 30$$ $$x = 6$$ Therefore, the width of the garden is 6 feet. | ||
To find the width of the garden, let's denote the width as x. Since the length is four times the width, the length can be represented as 4x. The perimeter of a rectangle is given by the formula 2(length + width), so for this problem, we have: 2(4x + x) = 2(5x) => 8x = 5x => x = 1/2 So, the width of the garden is 1/2 feet. | |||
✅ | **Analysis:** Let x be the width of the garden in feet. The length of the garden is 4x feet. The perimeter of a rectangle is the sum of the lengths of all four sides. The perimeter of this garden is 2(length + width) = 2(4x + x) = 2(5x) = 10x feet. **Equation:** 10x = 60 **Solution:** x = 6 **Answer:** The width of the garden is **6 feet**. | ||
✅ | Let's denote the width of the garden as W and the length as L. We know that L = 4W. The formula for the perimeter of a rectangle is P = 2L + 2W. In this case, P = 60 feet. Substitute L = 4W into the perimeter formula: 60 = 2(4W) + 2W Simplify the equation: 60 = 8W + 2W Combine like terms: 60 = 10W Divide both sides by 10 to find the width: W = 6 feet So, the width of the garden is 6 feet. | ||
✅ | Let's denote the width of the garden as \(W\) feet and the length as \(L\) feet. According to the problem, the length is four times the width, so \(L = 4W\). The formula for the perimeter (\(P\)) of a rectangle is \(P = 2L + 2W\). Given that the perimeter of the garden is 60 feet, we can substitute the values we know into the perimeter formula: \[60 = 2(4W) + 2W\] Simplifying, we get: \[60 = 8W + 2W\] \[60 = 10W\] Dividing both sides by 10: \[W = 6\] Thus, the width of the garden is 6 feet. | ||
✅ | ✅ | Let's start by analyzing the problem. We know that the perimeter of a rectangle is given by the formula P = 2l + 2w, where l is the length and w is the width. Since the length is four times the width, we can write an equation: l = 4w We are also given that the perimeter is 60 feet. Substituting this value into the perimeter formula, we get: 60 = 2l + 2w Now, substitute l = 4w into the equation above: 60 = 2(4w) + 2w 60 = 8w + 2w 60 = 10w Divide both sides by 10 to solve for w: w = 60/10 w = 6 So, the width of the garden is 6 feet. Answer: 6 | |
✅ | Let's denote the width of the garden as W and the length as 4W, since the length is four times the width. The perimeter of a rectangle is given by the formula P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. Given that the perimeter of the garden is 60 feet, we can set up the equation: 60 = 2(4W) + 2W 60 = 8W + 2W 60 = 10W Now, we can solve for W: W = 60 / 10 W = 6 Therefore, the width of the garden is 6 feet. Answer: The width of the garden is 6 feet. | ||
✅ | Let the width of the rectangular garden be represented as \( w \) feet. Given that the length is four times the width, the length can be expressed as \( 4w \) feet.
The formula for the perimeter \( P \) of a rectangle is given by:
\[
P = 2(\text{length} + \text{width})
\]
Plugging in the expressions for the length and the width:
\[
60 = 2(4w + w)
\]
\[
60 = 2(5w)
\]
\[
60 = 10w
\]
To find \( w \), divide both sides of the equation by 10:
\[
w = \frac{60}{10} = 6
\]
Thus, the width of the garden is \( \boxed{6} \) feet. | ||
✅ | ✅ | ## Analysis: 1. **Define variables:** Let's call the width of the garden "w" and the length "l". 2. **Set up equations:** We know that the length is four times the width, so we can write l = 4w. We also know that the perimeter of a rectangle is given by P = 2l + 2w, and in this case, the perimeter is 60 feet. So, we have 60 = 2l + 2w. 3. **Substitute and solve:** We can substitute the first equation into the second equation to eliminate "l" and solve for "w": 60 = 2(4w) + 2w, which simplifies to 60 = 10w. Dividing both sides by 10, we get w = 6. Therefore, the width of the garden is 6 feet. *** **6 feet** | |
The perimeter of the garden suggests that opposite sides of the rectangle are equal in length. We can label the length as 'x' and the width as 'x/4', as they are related by a 4:1 ratio. Therefore, the perimeter can be expressed as: x + x + (x/4) + (x/4) = 60 feet. Simplifying and equating the like terms results in: 2x = 56 feet. Hence, the width of the garden, x/4, is 7 feet. **7 feet** is the width of the rectangular garden. | |||
✅ | Let's denote the width of the garden as W and the length as L. We know that the length is four times the width, so L = 4W. The formula for the perimeter of a rectangle is P = 2(L + W). We are given that the perimeter is 60 feet, so we can write the equation as: 60 = 2(4W + W) Let's solve the equation step by step: 1. Simplify the equation: 60 = 2(5W) 30 = 5W 2. Solve for W: W = 6 feet So, the width of the garden is 6 feet. | ||
✅ | To determine the width of the rectangular garden, let's start by defining the variables and understanding the relationships given in the problem.
Let:
- \( W \) be the width of the garden in feet.
- \( L \) be the length of the garden in feet.
According to the problem, the length \( L \) is four times the width \( W \):
\[ L = 4W \]
The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2L + 2W \]
We are given that the perimeter is 60 feet:
\[ 2L + 2W = 60 \]
Substituting the expression for \( L \) from the first equation into the perimeter equation:
\[ 2(4W) + 2W = 60 \]
Simplify and solve for \( W \):
\[ 8W + 2W = 60 \]
\[ 10W = 60 \]
\[ W = \frac{60}{10} \]
\[ W = 6 \]
Thus, the width of the garden is:
\[ \boxed{6} \] | ||
✅ | ✅ | Let's break down the problem: 1. **Define Variables:** - Let 'w' represent the width of the garden. - The length is four times the width, so the length is '4w'. 2. **Perimeter Formula:** - The perimeter of a rectangle is calculated as: Perimeter = 2 * (length + width) 3. **Set Up Equation:** - We know the perimeter is 60 feet, so: 60 = 2 * (4w + w) 4. **Solve for 'w':** - Simplify the equation: 60 = 2 * (5w) - Further simplification: 60 = 10w - Divide both sides by 10: w = 6 **Answer:** 6 feet | |
✅ | ✅ | To solve this problem, let's use algebra. Let the width of the garden be `w` feet. According to the problem, the length of the garden is four times its width, so the length is `4w` feet. The perimeter of a rectangle is given by the formula: P = 2l + 2w, where P is the perimeter, l is the length, and w is the width. In this case, P = 60 feet. Substitute the known values into the formula: 60 = 2(4w) + 2w Simplify the equation: 60 = 8w + 2w Combine the terms: 60 = 10w Now, solve for `w`: w = 6 feet So, the width of the garden is 6 feet. The correct answer: Width = 6 feet. | |
✅ | ✅ | To find the width of the garden, let's start by setting up equations based on the information given.
Let \(W\) represent the width of the garden, and \(L\) represent the length of the garden.
According to the problem, the length is four times the width, so we can write this relationship as:
\[L = 4W\]
The perimeter \(P\) of a rectangle is given by the formula:
\[P = 2L + 2W\]
We are told the perimeter of the garden is 60 feet, so:
\[60 = 2L + 2W\]
Substituting the expression for \(L\) in terms of \(W\) into the perimeter equation, we get:
\[60 = 2(4W) + 2W\]
Simplifying the equation:
\[60 = 8W + 2W\]
\[60 = 10W\]
Dividing both sides by 10 to solve for \(W\):
\[W = \frac{60}{10}\]
\[W = 6\]
Therefore, the width of the garden is 6 feet.
Width of the garden: \(W = 6\) feet.<|im_end|> | |
✅ | ✅ | Step 1: First, we understand the problem: we have a rectangular garden with a specific relationship between its length and width, and we're given its perimeter. We need to find the width.
Step 2: Let's denote the width as \( w \) and the length as \( 4w \) since the problem states the length is four times the width.
Step 3: The formula for the perimeter (\( P \)) of a rectangle is \( 2 \times (\text{length} + \text{width}) \).
Step 4: Substituting our expressions for length and perimeter, we get \( 2(4w + w) = 60 \).
Step 5: Simplifying the equation: \( 2(5w) = 60 \), which reduces to \( 10w = 60 \).
Step 6: To solve for \( w \), divide both sides by 10: \( w = 60 / 10 = 6 \).
Step 7: Therefore, the width of the garden is 6 feet.
Answer:
\boxed{6} | |
✅ | Let's start by assigning variables to the width and length of the garden. Let w be the width, and let l be the length. We are given that the length is four times the width, so we can write this relationship as l = 4w. The perimeter of a rectangle is given by the formula P = 2l + 2w, where P is the perimeter. In this case, the perimeter is 60 feet, so we can write this equation as: 60 = 2(4w) + 2w Now, let's simplify and solve for w: 60 = 8w + 2w 60 = 10w To find the width, we can divide both sides by 10: w = 60 / 10 w = 6 So, the width of the garden is 6 feet. Final answer: The width of the garden is 6 feet. | ||
✅ | To solve this problem, let's define the variables:
Let the width of the garden be \( w \) feet.
Since the length is four times the width, the length will be \( 4w \) feet.
The perimeter of a rectangle is given by the formula:
\[ P = 2 \times ( \text{length} + \text{width} ) \]
Given that the perimeter \( P \) is 60 feet, we can set up the equation:
\[ 60 = 2 \times (4w + w) \]
Now, let's solve for \( w \):
\[ 60 = 2 \times 5w \]
\[ 60 = 10w \]
\[ w = \frac{60}{10} \]
\[ w = 6 \]
Therefore, the width of the garden is 6 feet.<|im_end|> | ||
✅ | Let's analyze the problem step by step. 1. We know that the garden is rectangular and that its length (L) is four times its width (W). So we can write this relationship as L = 4W. 2. The perimeter (P) of a rectangle is given by the formula P = 2L + 2W. We are given that the perimeter is 60 feet, so we can write this as 60 = 2L + 2W. 3. Now we can substitute the relationship from step 1 (L = 4W) into the perimeter formula: 60 = 2(4W) + 2W. 4. Simplify the equation: 60 = 8W + 2W. 5. Combine like terms: 60 = 10W. 6. Solve for W: W = 60 / 10. 7. Calculate the result: W = 6. Thus, the width of the garden is 6 feet.<|im_end|> | ||
✅ | Let's denote the width of the garden as W. Since the length is four times the width, the length is 4W. The perimeter of a rectangle is given by the formula: P = 2L + 2W In this case, the perimeter is 60 feet, so we have: 60 = 2(4W) + 2W Simplifying the equation, we get: 60 = 8W + 2W Combining like terms, we get: 60 = 10W Dividing both sides by 10, we get: W = 6 The width of the garden is 6 feet. |