Answer :
Certainly! Let's solve the problem step-by-step.
We are given that a supplier sells a mixture of mulch and gravel. Specifically, the supplier sells [tex]\(2 \frac{1}{4}\)[/tex] pounds of mulch for every [tex]\(1 \frac{1}{3}\)[/tex] pounds of gravel. The total weight of mulch and gravel combined is 172 pounds. Our task is to find out how many pounds of each item the supplier sells.
1. Convert Mixed Numbers to Improper Fractions:
- Mulch: [tex]\(2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}\)[/tex]
- Gravel: [tex]\(1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}\)[/tex]
2. Determine the Ratio:
Since we have [tex]\(\frac{9}{4}\)[/tex] pounds of mulch for every [tex]\(\frac{4}{3}\)[/tex] pounds of gravel, our goal is to work with these ratios.
3. Calculate Total Parts:
To handle this proportion, we need to find a common basis for the parts, which involves adding the two ratios:
First, find a common denominator to add the ratios together. For [tex]\(\frac{9}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex], the common denominator is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to twelfths: [tex]\( \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12} \)[/tex]
- Convert [tex]\(\frac{4}{3}\)[/tex] to twelfths: [tex]\( \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \)[/tex]
Adding these gives us the total parts:
[tex]\[
\frac{27}{12} + \frac{16}{12} = \frac{43}{12}
\][/tex]
4. Calculate the Weight of Each Item:
Now, determine how many parts out of the total 43 parts each item occupies and use the total weight (172 pounds) to find their weights.
- Weight of Mulch:
[tex]\[
\text{Mulch weight} = \left(\frac{\frac{27}{12}}{\frac{43}{12}}\right) \times 172 = \left(\frac{27}{43}\right) \times 172 = 108 \text{ pounds}
\][/tex]
- Weight of Gravel:
[tex]\[
\text{Gravel weight} = \left(\frac{\frac{16}{12}}{\frac{43}{12}}\right) \times 172 = \left(\frac{16}{43}\right) \times 172 = 64 \text{ pounds}
\][/tex]
Therefore, the supplier sells 108 pounds of mulch and 64 pounds of gravel.
We are given that a supplier sells a mixture of mulch and gravel. Specifically, the supplier sells [tex]\(2 \frac{1}{4}\)[/tex] pounds of mulch for every [tex]\(1 \frac{1}{3}\)[/tex] pounds of gravel. The total weight of mulch and gravel combined is 172 pounds. Our task is to find out how many pounds of each item the supplier sells.
1. Convert Mixed Numbers to Improper Fractions:
- Mulch: [tex]\(2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}\)[/tex]
- Gravel: [tex]\(1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}\)[/tex]
2. Determine the Ratio:
Since we have [tex]\(\frac{9}{4}\)[/tex] pounds of mulch for every [tex]\(\frac{4}{3}\)[/tex] pounds of gravel, our goal is to work with these ratios.
3. Calculate Total Parts:
To handle this proportion, we need to find a common basis for the parts, which involves adding the two ratios:
First, find a common denominator to add the ratios together. For [tex]\(\frac{9}{4}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex], the common denominator is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to twelfths: [tex]\( \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12} \)[/tex]
- Convert [tex]\(\frac{4}{3}\)[/tex] to twelfths: [tex]\( \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \)[/tex]
Adding these gives us the total parts:
[tex]\[
\frac{27}{12} + \frac{16}{12} = \frac{43}{12}
\][/tex]
4. Calculate the Weight of Each Item:
Now, determine how many parts out of the total 43 parts each item occupies and use the total weight (172 pounds) to find their weights.
- Weight of Mulch:
[tex]\[
\text{Mulch weight} = \left(\frac{\frac{27}{12}}{\frac{43}{12}}\right) \times 172 = \left(\frac{27}{43}\right) \times 172 = 108 \text{ pounds}
\][/tex]
- Weight of Gravel:
[tex]\[
\text{Gravel weight} = \left(\frac{\frac{16}{12}}{\frac{43}{12}}\right) \times 172 = \left(\frac{16}{43}\right) \times 172 = 64 \text{ pounds}
\][/tex]
Therefore, the supplier sells 108 pounds of mulch and 64 pounds of gravel.
