Answer :
To solve this problem, we need to determine how many pounds of mulch and gravel the supplier sells, given specific conditions about their combined weight and ratio.
Step 1: Understand the Problem
- We know that for every [tex]\(2 \frac{1}{4}\)[/tex] pounds of mulch, there are [tex]\(1 \frac{1}{3}\)[/tex] pounds of gravel sold.
- The total combined weight of mulch and gravel is 172 pounds.
Step 2: Express the Given Quantities as Improper Fractions
- Convert [tex]\(2 \frac{1}{4}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{1}{4} = \frac{9}{4}
\][/tex]
- Convert [tex]\(1 \frac{1}{3}\)[/tex] to an improper fraction:
[tex]\[
1 \frac{1}{3} = \frac{4}{3}
\][/tex]
Step 3: Set Up the Equation
- Let [tex]\(x\)[/tex] be the multiplier for the ratio.
- The equation for the total weight is:
[tex]\[
\left(\frac{9}{4}\times x\right) + \left(\frac{4}{3}\times x\right) = 172
\][/tex]
Step 4: Find a Common Denominator and Combine Terms
- The common denominator for 4 and 3 is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to twelfths:
[tex]\[
\frac{9}{4} = \frac{27}{12}
\][/tex]
- Convert [tex]\(\frac{4}{3}\)[/tex] to twelfths:
[tex]\[
\frac{4}{3} = \frac{16}{12}
\][/tex]
- Combine these ratios:
[tex]\[
\frac{27}{12}x + \frac{16}{12}x = \frac{43}{12}x
\][/tex]
Step 5: Solve for [tex]\(x\)[/tex]
- The equation becomes:
[tex]\[
\frac{43}{12}x = 172
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{172 \times 12}{43} \approx 48
\][/tex]
Step 6: Calculate the Amount of Each Item
1. Multiply the mulch ratio by [tex]\(x\)[/tex]:
[tex]\[
\frac{9}{4} \times x = \frac{9}{4} \times 48 \approx 108 \text{ pounds of mulch}
\][/tex]
2. Multiply the gravel ratio by [tex]\(x\)[/tex]:
[tex]\[
\frac{4}{3} \times x = \frac{4}{3} \times 48 \approx 64 \text{ pounds of gravel}
\][/tex]
Conclusion
- The supplier sells approximately 108 pounds of mulch and 64 pounds of gravel, keeping the total weight around 172 pounds, which justifies our solution with the given conditions.
Step 1: Understand the Problem
- We know that for every [tex]\(2 \frac{1}{4}\)[/tex] pounds of mulch, there are [tex]\(1 \frac{1}{3}\)[/tex] pounds of gravel sold.
- The total combined weight of mulch and gravel is 172 pounds.
Step 2: Express the Given Quantities as Improper Fractions
- Convert [tex]\(2 \frac{1}{4}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{1}{4} = \frac{9}{4}
\][/tex]
- Convert [tex]\(1 \frac{1}{3}\)[/tex] to an improper fraction:
[tex]\[
1 \frac{1}{3} = \frac{4}{3}
\][/tex]
Step 3: Set Up the Equation
- Let [tex]\(x\)[/tex] be the multiplier for the ratio.
- The equation for the total weight is:
[tex]\[
\left(\frac{9}{4}\times x\right) + \left(\frac{4}{3}\times x\right) = 172
\][/tex]
Step 4: Find a Common Denominator and Combine Terms
- The common denominator for 4 and 3 is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to twelfths:
[tex]\[
\frac{9}{4} = \frac{27}{12}
\][/tex]
- Convert [tex]\(\frac{4}{3}\)[/tex] to twelfths:
[tex]\[
\frac{4}{3} = \frac{16}{12}
\][/tex]
- Combine these ratios:
[tex]\[
\frac{27}{12}x + \frac{16}{12}x = \frac{43}{12}x
\][/tex]
Step 5: Solve for [tex]\(x\)[/tex]
- The equation becomes:
[tex]\[
\frac{43}{12}x = 172
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{172 \times 12}{43} \approx 48
\][/tex]
Step 6: Calculate the Amount of Each Item
1. Multiply the mulch ratio by [tex]\(x\)[/tex]:
[tex]\[
\frac{9}{4} \times x = \frac{9}{4} \times 48 \approx 108 \text{ pounds of mulch}
\][/tex]
2. Multiply the gravel ratio by [tex]\(x\)[/tex]:
[tex]\[
\frac{4}{3} \times x = \frac{4}{3} \times 48 \approx 64 \text{ pounds of gravel}
\][/tex]
Conclusion
- The supplier sells approximately 108 pounds of mulch and 64 pounds of gravel, keeping the total weight around 172 pounds, which justifies our solution with the given conditions.
