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Answer :
To solve this problem, we need to distribute the 172 pounds between mulch and gravel according to their given ratio. Here's a step-by-step guide to finding the solution:
1. Understanding the Ratio:
- Mulch is sold at a rate of [tex]\(2 \frac{1}{4}\)[/tex] pounds.
- Gravel is sold at a rate of [tex]\(1 \frac{1}{3}\)[/tex] pounds.
2. Converting Mixed Numbers to Improper Fractions:
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
- [tex]\(1 \frac{1}{3} = \frac{4}{3}\)[/tex]
3. Finding the Total Ratio Part:
- The total ratio of mulch to gravel is [tex]\(\frac{9}{4} + \frac{4}{3}\)[/tex].
- To add these fractions, find a common denominator, which is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to [tex]\(\frac{27}{12}\)[/tex].
- Convert [tex]\(\frac{4}{3}\)[/tex] to [tex]\(\frac{16}{12}\)[/tex].
4. Adding the Ratios:
- [tex]\(\frac{27}{12} + \frac{16}{12} = \frac{43}{12}\)[/tex]
5. Setting Up the Equation:
- The total combined weight of mulch and gravel is 172 pounds.
- Let [tex]\(x\)[/tex] represent how many times this full ratio fits into 172 pounds.
- So, [tex]\(\frac{43}{12}x = 172\)[/tex].
6. Solving for [tex]\(x\)[/tex]:
- Multiply both sides by 12 to clear the fraction: [tex]\(43x = 172 \times 12\)[/tex].
- Calculate [tex]\(172 \times 12 = 2064\)[/tex].
- Now solve for [tex]\(x\)[/tex]: [tex]\(x = \frac{2064}{43}\)[/tex].
- [tex]\(x = 48\)[/tex].
7. Calculating the Pounds of Each Item:
- Since [tex]\(x = 48\)[/tex], multiply the individual ratios by 48 to find the actual pounds.
- Pounds of Mulch:
- Multiply the mulch ratio by 48: [tex]\(\frac{9}{4} \times 48 = 9 \times 12 = 108\)[/tex].
- Pounds of Gravel:
- Multiply the gravel ratio by 48: [tex]\(\frac{4}{3} \times 48 = 4 \times 16 = 64\)[/tex].
8. Conclusion:
- The supplier sells 108 pounds of mulch and 64 pounds of gravel.
These calculations show how the mulch and gravel quantities are determined using the given ratio and total weight.
1. Understanding the Ratio:
- Mulch is sold at a rate of [tex]\(2 \frac{1}{4}\)[/tex] pounds.
- Gravel is sold at a rate of [tex]\(1 \frac{1}{3}\)[/tex] pounds.
2. Converting Mixed Numbers to Improper Fractions:
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
- [tex]\(1 \frac{1}{3} = \frac{4}{3}\)[/tex]
3. Finding the Total Ratio Part:
- The total ratio of mulch to gravel is [tex]\(\frac{9}{4} + \frac{4}{3}\)[/tex].
- To add these fractions, find a common denominator, which is 12.
- Convert [tex]\(\frac{9}{4}\)[/tex] to [tex]\(\frac{27}{12}\)[/tex].
- Convert [tex]\(\frac{4}{3}\)[/tex] to [tex]\(\frac{16}{12}\)[/tex].
4. Adding the Ratios:
- [tex]\(\frac{27}{12} + \frac{16}{12} = \frac{43}{12}\)[/tex]
5. Setting Up the Equation:
- The total combined weight of mulch and gravel is 172 pounds.
- Let [tex]\(x\)[/tex] represent how many times this full ratio fits into 172 pounds.
- So, [tex]\(\frac{43}{12}x = 172\)[/tex].
6. Solving for [tex]\(x\)[/tex]:
- Multiply both sides by 12 to clear the fraction: [tex]\(43x = 172 \times 12\)[/tex].
- Calculate [tex]\(172 \times 12 = 2064\)[/tex].
- Now solve for [tex]\(x\)[/tex]: [tex]\(x = \frac{2064}{43}\)[/tex].
- [tex]\(x = 48\)[/tex].
7. Calculating the Pounds of Each Item:
- Since [tex]\(x = 48\)[/tex], multiply the individual ratios by 48 to find the actual pounds.
- Pounds of Mulch:
- Multiply the mulch ratio by 48: [tex]\(\frac{9}{4} \times 48 = 9 \times 12 = 108\)[/tex].
- Pounds of Gravel:
- Multiply the gravel ratio by 48: [tex]\(\frac{4}{3} \times 48 = 4 \times 16 = 64\)[/tex].
8. Conclusion:
- The supplier sells 108 pounds of mulch and 64 pounds of gravel.
These calculations show how the mulch and gravel quantities are determined using the given ratio and total weight.
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