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9. Verify the following:
(a) [tex]18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)][/tex]
(b) [tex](-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)][/tex]

10. A cement company earns a profit of Rs 8 per bag of white cement sold and a loss of Rs 5 per bag of grey cement sold.
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags?

Sagot :

GinnyAnswer
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.
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