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Sagot :
Sure, let's go through each part of the question step by step:
### Part 9: Verification
#### (a)
Verify that:
[tex]\[ 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\[ 18 \times [7 + (-3)] = 18 \times 4 = 72 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\[ [18 \times 7] + [18 \times (-3)] = 126 + (-54) = 72 \][/tex]
Since both LHS and RHS equal 72, the equation is verified to be true.
[tex]\[ 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \][/tex]
#### (b)
Verify that:
[tex]\[ (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\[ (-21) \times [(-4) + (-6)] = (-21) \times (-10) = 210 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\[ [(-21) \times (-4)] + [(-21) \times (-6)] = 84 + 126 = 210 \][/tex]
Since both LHS and RHS equal 210, the equation is verified to be true.
[tex]\[ (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \][/tex]
### Part 10: Cement Company Profit and Loss
#### (a)
Calculate the company's profit or loss when it sells 3,000 bags of white cement and 5,000 bags of grey cement in a month.
1. Profit per bag of white cement = Rs 8
2. Loss per bag of grey cement = Rs 5
3. Calculate total profit from white cement:
[tex]\[ \text{Total profit} = 3000 \times 8 = 24000 \, \text{Rs} \][/tex]
4. Calculate total loss from grey cement:
[tex]\[ \text{Total loss} = 5000 \times 5 = 25000 \, \text{Rs} \][/tex]
5. Calculate net profit or loss:
[tex]\[ \text{Net profit or loss} = 24000 - 25000 = -1000 \, \text{Rs} \][/tex]
So, the company has a loss of Rs 1000.
#### (b)
Determine the number of white cement bags to be sold to have neither profit nor loss if the company sells 6,400 grey bags.
1. Loss from grey cement bags:
[tex]\[ \text{Total loss} = 6400 \times 5 = 32000 \, \text{Rs} \][/tex]
2. To break even (neither profit nor loss), the profit from white cement must equal the total loss from grey cement:
[tex]\[ \text{Total profit} = 32000 \, \text{Rs} \][/tex]
3. Let \( x \) be the number of white cement bags needed. Calculate \( x \) based on the required profit:
[tex]\[ 8x = 32000 \][/tex]
4. Solve for \( x \):
[tex]\[ x = \frac{32000}{8} = 4000 \][/tex]
Hence, the company must sell 4,000 bags of white cement to have neither profit nor loss.
### Part 9: Verification
#### (a)
Verify that:
[tex]\[ 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\[ 18 \times [7 + (-3)] = 18 \times 4 = 72 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\[ [18 \times 7] + [18 \times (-3)] = 126 + (-54) = 72 \][/tex]
Since both LHS and RHS equal 72, the equation is verified to be true.
[tex]\[ 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \][/tex]
#### (b)
Verify that:
[tex]\[ (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\[ (-21) \times [(-4) + (-6)] = (-21) \times (-10) = 210 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\[ [(-21) \times (-4)] + [(-21) \times (-6)] = 84 + 126 = 210 \][/tex]
Since both LHS and RHS equal 210, the equation is verified to be true.
[tex]\[ (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \][/tex]
### Part 10: Cement Company Profit and Loss
#### (a)
Calculate the company's profit or loss when it sells 3,000 bags of white cement and 5,000 bags of grey cement in a month.
1. Profit per bag of white cement = Rs 8
2. Loss per bag of grey cement = Rs 5
3. Calculate total profit from white cement:
[tex]\[ \text{Total profit} = 3000 \times 8 = 24000 \, \text{Rs} \][/tex]
4. Calculate total loss from grey cement:
[tex]\[ \text{Total loss} = 5000 \times 5 = 25000 \, \text{Rs} \][/tex]
5. Calculate net profit or loss:
[tex]\[ \text{Net profit or loss} = 24000 - 25000 = -1000 \, \text{Rs} \][/tex]
So, the company has a loss of Rs 1000.
#### (b)
Determine the number of white cement bags to be sold to have neither profit nor loss if the company sells 6,400 grey bags.
1. Loss from grey cement bags:
[tex]\[ \text{Total loss} = 6400 \times 5 = 32000 \, \text{Rs} \][/tex]
2. To break even (neither profit nor loss), the profit from white cement must equal the total loss from grey cement:
[tex]\[ \text{Total profit} = 32000 \, \text{Rs} \][/tex]
3. Let \( x \) be the number of white cement bags needed. Calculate \( x \) based on the required profit:
[tex]\[ 8x = 32000 \][/tex]
4. Solve for \( x \):
[tex]\[ x = \frac{32000}{8} = 4000 \][/tex]
Hence, the company must sell 4,000 bags of white cement to have neither profit nor loss.
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