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Sagot :
Let's solve these problems step by step.
### Problem 6:
Given:
- The initial amount of water in the tank is [tex]\(\frac{15}{11}\)[/tex] liters.
- The leaking tap adds [tex]\(\frac{3}{4}\)[/tex] liters of water per hour.
- The tank is placed below the leaking tap for [tex]\(\frac{21}{2}\)[/tex] hours.
Step 1: Identify the water addition rate.
- The leaking tap adds [tex]\(\frac{3}{4}\)[/tex] liters per hour.
Step 2: Determine the duration for which the tap is leaking.
- The duration is [tex]\(\frac{21}{2}\)[/tex] hours.
Step 3: Calculate the total amount of water added by the tap.
[tex]\[ \text{Water added} = \left(\frac{3}{4}\right) \times \left(\frac{21}{2}\right) = 7.875 \text{ liters} \][/tex]
Step 4: Calculate the total amount of water in the tank after [tex]\(\frac{21}{2}\)[/tex] hours.
[tex]\[ \text{Total water in tank} = \text{Initial water} + \text{Water added} \][/tex]
[tex]\[ \text{Total water in tank} = \frac{15}{11} + 7.875 \approx 9.239 \text{ liters} \][/tex]
Therefore, the tank will have approximately 9.239 liters of water after [tex]\(\frac{21}{2}\)[/tex] hours.
---
### Problem 7:
Given:
- Anil spends [tex]\(\frac{1}{4}\)[/tex] of his monthly income on food.
- He spends [tex]\(\frac{2}{5}\)[/tex] on other expenses.
- The rest is donated.
Step 1: Let Anil's monthly income be denoted as [tex]\(I\)[/tex].
Step 2: Calculate the fraction of income used on food.
[tex]\[ \text{Food expenses} = \frac{1}{4}I \][/tex]
Step 3: Calculate the fraction of income used on other expenses.
[tex]\[ \text{Other expenses} = \frac{2}{5}I \][/tex]
Step 4: Determine the total expenditure on food and other expenses.
[tex]\[ \text{Total expenses} = \frac{1}{4}I + \frac{2}{5}I \][/tex]
Step 5: To add these fractions, we find a common denominator.
[tex]\[ \text{Total expenses} = \frac{5}{20}I + \frac{8}{20}I = \frac{13}{20}I \][/tex]
Step 6: Calculate the remaining amount, which is donated.
[tex]\[ \text{Donated amount} = I - \frac{13}{20}I \][/tex]
[tex]\[ \text{Donated amount} = \frac{20}{20}I - \frac{13}{20}I \][/tex]
[tex]\[ \text{Donated amount} = \frac{7}{20}I \][/tex]
Therefore:
- Anil spends [tex]\(\frac{1}{4}\)[/tex] of his income on food.
- He spends [tex]\(\frac{2}{5}\)[/tex] on other expenses.
- The remaining [tex]\(\frac{7}{20}\)[/tex] of his monthly income is donated.
### Problem 6:
Given:
- The initial amount of water in the tank is [tex]\(\frac{15}{11}\)[/tex] liters.
- The leaking tap adds [tex]\(\frac{3}{4}\)[/tex] liters of water per hour.
- The tank is placed below the leaking tap for [tex]\(\frac{21}{2}\)[/tex] hours.
Step 1: Identify the water addition rate.
- The leaking tap adds [tex]\(\frac{3}{4}\)[/tex] liters per hour.
Step 2: Determine the duration for which the tap is leaking.
- The duration is [tex]\(\frac{21}{2}\)[/tex] hours.
Step 3: Calculate the total amount of water added by the tap.
[tex]\[ \text{Water added} = \left(\frac{3}{4}\right) \times \left(\frac{21}{2}\right) = 7.875 \text{ liters} \][/tex]
Step 4: Calculate the total amount of water in the tank after [tex]\(\frac{21}{2}\)[/tex] hours.
[tex]\[ \text{Total water in tank} = \text{Initial water} + \text{Water added} \][/tex]
[tex]\[ \text{Total water in tank} = \frac{15}{11} + 7.875 \approx 9.239 \text{ liters} \][/tex]
Therefore, the tank will have approximately 9.239 liters of water after [tex]\(\frac{21}{2}\)[/tex] hours.
---
### Problem 7:
Given:
- Anil spends [tex]\(\frac{1}{4}\)[/tex] of his monthly income on food.
- He spends [tex]\(\frac{2}{5}\)[/tex] on other expenses.
- The rest is donated.
Step 1: Let Anil's monthly income be denoted as [tex]\(I\)[/tex].
Step 2: Calculate the fraction of income used on food.
[tex]\[ \text{Food expenses} = \frac{1}{4}I \][/tex]
Step 3: Calculate the fraction of income used on other expenses.
[tex]\[ \text{Other expenses} = \frac{2}{5}I \][/tex]
Step 4: Determine the total expenditure on food and other expenses.
[tex]\[ \text{Total expenses} = \frac{1}{4}I + \frac{2}{5}I \][/tex]
Step 5: To add these fractions, we find a common denominator.
[tex]\[ \text{Total expenses} = \frac{5}{20}I + \frac{8}{20}I = \frac{13}{20}I \][/tex]
Step 6: Calculate the remaining amount, which is donated.
[tex]\[ \text{Donated amount} = I - \frac{13}{20}I \][/tex]
[tex]\[ \text{Donated amount} = \frac{20}{20}I - \frac{13}{20}I \][/tex]
[tex]\[ \text{Donated amount} = \frac{7}{20}I \][/tex]
Therefore:
- Anil spends [tex]\(\frac{1}{4}\)[/tex] of his income on food.
- He spends [tex]\(\frac{2}{5}\)[/tex] on other expenses.
- The remaining [tex]\(\frac{7}{20}\)[/tex] of his monthly income is donated.
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