Question
One yard is approximately 0.91 meter. Write a direct variation equation that relates x yards to y meters.
Answers
Answer
A company that makes trail mix uses 1 3/8 pounds of peanuts for every 2/3 pound of raisins. They make 245 pounds of trail mix using only peanuts and raisins. How many pounds of peanuts do they use?
Answered by
GPT-4o mini
In direct variation, there is a constant ratio between the two variables. In this case, you are given that 1 yard is approximately 0.91 meters. Therefore, you can express the relationship between yards (x) and meters (y) as:
\[
y = kx
\]
where \( k \) is the constant of variation. Here, \( k \) will be equal to 0.91 because for every yard, there are approximately 0.91 meters.
Thus, the direct variation equation relating \( x \) yards to \( y \) meters is:
\[
y = 0.91x
\]
\[
y = kx
\]
where \( k \) is the constant of variation. Here, \( k \) will be equal to 0.91 because for every yard, there are approximately 0.91 meters.
Thus, the direct variation equation relating \( x \) yards to \( y \) meters is:
\[
y = 0.91x
\]
Answered by
GPT-4o mini
To solve this problem, we first need to establish the ratio of peanuts to raisins.
The problem states that the company uses \(1 \frac{3}{8}\) pounds of peanuts for every \(\frac{2}{3}\) pound of raisins.
1. **Convert \(1 \frac{3}{8}\) to an improper fraction:**
\[
1 \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}
\]
So, the ratio of peanuts to raisins is:
\[
\frac{11/8}{2/3}
\]
2. **Simplifying this ratio:**
To simplify \(\frac{11/8}{2/3}\), we multiply by the reciprocal of the denominator:
\[
= \frac{11}{8} \times \frac{3}{2} = \frac{11 \times 3}{8 \times 2} = \frac{33}{16}
\]
This means for every 33 parts of peanuts, there are 16 parts of raisins.
3. **Calculating total parts:**
The total parts of the mixture (peanuts + raisins) is:
\[
33 + 16 = 49 \text{ parts}
\]
4. **Calculating the weight of each part:**
If they make a total of 245 pounds of trail mix, we can find the weight of one part:
\[
\text{Weight per part} = \frac{245 \text{ pounds}}{49 \text{ parts}} = 5 \text{ pounds per part}
\]
5. **Calculating the pounds of peanuts used:**
Since there are 33 parts of peanuts, the total weight of peanuts used is:
\[
\text{Pounds of peanuts} = 33 \text{ parts} \times 5 \text{ pounds/part} = 165 \text{ pounds}
\]
Therefore, the company uses **165 pounds of peanuts** in the trail mix.
The problem states that the company uses \(1 \frac{3}{8}\) pounds of peanuts for every \(\frac{2}{3}\) pound of raisins.
1. **Convert \(1 \frac{3}{8}\) to an improper fraction:**
\[
1 \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}
\]
So, the ratio of peanuts to raisins is:
\[
\frac{11/8}{2/3}
\]
2. **Simplifying this ratio:**
To simplify \(\frac{11/8}{2/3}\), we multiply by the reciprocal of the denominator:
\[
= \frac{11}{8} \times \frac{3}{2} = \frac{11 \times 3}{8 \times 2} = \frac{33}{16}
\]
This means for every 33 parts of peanuts, there are 16 parts of raisins.
3. **Calculating total parts:**
The total parts of the mixture (peanuts + raisins) is:
\[
33 + 16 = 49 \text{ parts}
\]
4. **Calculating the weight of each part:**
If they make a total of 245 pounds of trail mix, we can find the weight of one part:
\[
\text{Weight per part} = \frac{245 \text{ pounds}}{49 \text{ parts}} = 5 \text{ pounds per part}
\]
5. **Calculating the pounds of peanuts used:**
Since there are 33 parts of peanuts, the total weight of peanuts used is:
\[
\text{Pounds of peanuts} = 33 \text{ parts} \times 5 \text{ pounds/part} = 165 \text{ pounds}
\]
Therefore, the company uses **165 pounds of peanuts** in the trail mix.
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