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${:[1170.16=[C ×(1*(1)/((1+0.08)^(5)))/(0.08)]+[0.4 ×(1000+2C)×(1)/((1+0.08)^(5))]+],[[0.6 ×(C ×(1-(1)/((1+0.12)^(15)))/(0.12)+(1000)/((1+0.12)^(15)))×(1)/((1+0.08)^(5))]]:}$

$\begin{aligned} 1170.16= & {\left[C \times \frac{1 \cdot \frac{1}{(1+0.08)^{5}}}{0.08}\right]+\left[0.4 \times(1000+2 C) \times \frac{1}{(1+0.08)^{5}}\right]+} \\ & {\left[0.6 \times\left(C \times \frac{1-\frac{1}{(1+0.12)^{15}}}{0.12}+\frac{1000}{(1+0.12)^{15}}\right) \times \frac{1}{(1+0.08)^{5}}\right] }\end{aligned}$

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Math Problems

Grade 6

Properties of multiplication

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\begin{aligned} 1170.16= & {\left[C \times \frac{1 \cdot \frac{1}{(1+0.08)^{5}}}{0.08}\right]+\left[0.4 \times(1000+2 C) \times \frac{1}{(1+0.08)^{5}}\right]+} \\ & {\left[0.6 \times\left(C \times \frac{1-\frac{1}{(1+0.12)^{15}}}{0.12}+\frac{1000}{(1+0.12)^{15}}\right) \times \frac{1}{(1+0.08)^{5}}\right] }\end{aligned}

Identify Equation Structure: Identify the given equation and understand its structure. $\newline$ The equation is a financial equation that seems to involve present value calculations for different cash flows. It is structured as follows: $\newline$ $1170.16 = [C \times (1 \times (1) / ((1 + 0.08)^{(5)}) / (0.08)] + [0.4 \times (1000 + 2C) \times (1) / ((1 + 0.08)^{(5)})] + [0.6 \times (C \times (1 - (1) / ((1 + 0.12)^{(15)}) / (0.12) + (1000) / ((1 + 0.12)^{(15)})) \times (1) / ((1 + 0.08)^{(5)})]$ $\newline$ We need to solve for $C$ .
Calculate Present Value Factors: Simplify the equation by calculating the present value factors for the discount rates $8\%$ and $12\%$ for the respective periods. $\newline$ First, calculate the present value factor for a period of $5$ years at an $8\%$ discount rate: $\newline$ $(1 + 0.08)^{5}$
Substitute and Simplify Equation: Perform the calculation for the present value factor at $8\%$ for $5$ years. $\newline$ $(1 + 0.08)^{5} = (1.08)^{5} \approx 1.46933$
Combine and Solve for C: Calculate the present value factor for a period of $15$ years at a $12\%$ discount rate: $(1 + 0.12)^{15}$
Clear Denominators and Calculate: Perform the calculation for the present value factor at $12\%$ for $15$ years. $\newline$ $(1 + 0.12)^{15} = (1.12)^{15} \approx 5.47395$
Add Coefficients and Multiply Constants: Substitute the present value factors back into the equation and simplify. $\newline$ \(1170. $16$ = \left[C \times \left(\frac{ $1$ }{ $1$ . $46933$ }\right) / $0$ . $08$ \right] + \left[ $0$ . $4$ \times \left( $1000$ + $2$ C\right) / $1$ . $46933$ \right] + \left[ $0$ . $6$ \times \left(C \times \left( $1$ - \frac{ $1$ }{ $5$ . $47395$ }\right) / $0$ . $12$ \right) + \frac{ $1000$ }{ $5$ . $47395$ }\right] / $1$ . $46933$ $\newline$ Now, we need to simplify each term.
Combine C Terms and Subtract Constants: Simplify the first term of the equation: $\newline$ $C \times (1 / 1.46933) / 0.08 = C / (1.46933 \times 0.08) = C / 0.1175464$
Calculate Coefficients Sum: Simplify the second term of the equation: $\newline$ $0.4 \times (1000 + 2C) / 1.46933 = (400 + 0.8C) / 1.46933$
Divide to Solve for C: Simplify the third term of the equation: $\newline$ $0.6 \times (C \times (1 - 1 / 5.47395) / 0.12) + 1000 / 5.47395 = 0.6 \times (C \times (5.47395 - 1) / (5.47395 \times 0.12)) + 182.709$
Perform Division to Find C: Simplify the third term further: $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{0.8C}{1.46933} + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{0.8C}{1.46933} + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = C / 0.1175464 + (400 + 0.8C) / 1.46933 + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = C / 0.1175464 + 0.8C / 1.46933 + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{0.8C}{1.46933} + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.Add the coefficients of C together and multiply the constants: $\newline$ $1170.16 \times 0.172703 = C + 0.09403712C + 0.702073C + 182.709 \times 0.172703$ $\newline$ Now, calculate the products: $\newline$ $202.202 = C + 0.09403712C + 0.702073C + 31.556$ $\newline$ Combine the C terms and subtract the constant from both sides.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{0.8C}{1.46933} + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.Add the coefficients of C together and multiply the constants: $\newline$ $1170.16 \times 0.172703 = C + 0.09403712C + 0.702073C + 182.709 \times 0.172703$ $\newline$ Now, calculate the products: $\newline$ $202.202 = C + 0.09403712C + 0.702073C + 31.556$ $\newline$ Combine the C terms and subtract the constant from both sides.Combine the C terms and subtract the constant from both sides: $\newline$ $202.202 = (1 + 0.09403712 + 0.702073)C + 31.556$ $\newline$ $202.202 - 31.556 = (1 + 0.09403712 + 0.702073)C$ $\newline$ Now, calculate the sum of the coefficients and the difference of the constants.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{(400 + 0.8C)}{1.46933} + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = \frac{C}{0.1175464} + \frac{0.8C}{1.46933} + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.Add the coefficients of C together and multiply the constants: $\newline$ $1170.16 \times 0.172703 = C + 0.09403712C + 0.702073C + 182.709 \times 0.172703$ $\newline$ Now, calculate the products: $\newline$ $202.202 = C + 0.09403712C + 0.702073C + 31.556$ $\newline$ Combine the C terms and subtract the constant from both sides.Combine the C terms and subtract the constant from both sides: $\newline$ $202.202 = (1 + 0.09403712 + 0.702073)C + 31.556$ $\newline$ $202.202 - 31.556 = (1 + 0.09403712 + 0.702073)C$ $\newline$ Now, calculate the sum of the coefficients and the difference of the constants.Calculate the sum of the coefficients and the difference of the constants: $\newline$ $170.646 = (1.79611012)C$ $\newline$ Now, divide both sides by the sum of the coefficients to solve for C.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = C / 0.1175464 + (400 + 0.8C) / 1.46933 + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = C / 0.1175464 + 0.8C / 1.46933 + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.Add the coefficients of C together and multiply the constants: $\newline$ $1170.16 \times 0.172703 = C + 0.09403712C + 0.702073C + 182.709 \times 0.172703$ $\newline$ Now, calculate the products: $\newline$ $202.202 = C + 0.09403712C + 0.702073C + 31.556$ $\newline$ Combine the C terms and subtract the constant from both sides.Combine the C terms and subtract the constant from both sides: $\newline$ 202.202 = (1 + 0.09403712 + 0.702073)C + 31.556$\newline202.202 - 31.556 = (1 + 0.09403712 + 0.702073)C$ $\newline$ Now, calculate the sum of the coefficients and the difference of the constants.Calculate the sum of the coefficients and the difference of the constants: $\newline$ $170.646 = (1.79611012)C$ $\newline$ Now, divide both sides by the sum of the coefficients to solve for C.Divide both sides by the sum of the coefficients to solve for C: $\newline$ $C = 170.646 / 1.79611012$ $\newline$ Now, perform the division to find the value of C.
Perform Division to Find C: Simplify the third term further: $\newline$ $0.6 \times (C \times (4.47395) / (0.656874)) + 182.709 = 0.6 \times (C \times 6.81017) + 182.709 = 4.0861C + 182.709$ Now, we have the simplified equation: $\newline$ $1170.16 = C / 0.1175464 + (400 + 0.8C) / 1.46933 + 4.0861C + 182.709$ $\newline$ Combine like terms and solve for C.Combine the C terms and constant terms: $\newline$ $1170.16 = C / 0.1175464 + 0.8C / 1.46933 + 4.0861C + 182.709$ $\newline$ To combine the C terms, we need to find a common denominator or multiply through by the denominators to clear them.Multiply through by the denominators to clear them: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.8C \times (0.1175464) + 4.0861C \times (0.1175464 \times 1.46933) + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Calculate the constants and coefficients.Perform the calculations for the constants and coefficients: $\newline$ $1170.16 \times (0.1175464 \times 1.46933) = C + 0.09403712C + 0.702073C + 182.709 \times (0.1175464 \times 1.46933)$ $\newline$ Now, add the coefficients of C together and multiply the constants.Add the coefficients of C together and multiply the constants: $\newline$ $1170.16 \times 0.172703 = C + 0.09403712C + 0.702073C + 182.709 \times 0.172703$ $\newline$ Now, calculate the products: $\newline$ $202.202 = C + 0.09403712C + 0.702073C + 31.556$ $\newline$ Combine the C terms and subtract the constant from both sides.Combine the C terms and subtract the constant from both sides: $\newline$ 202.202 = (1 + 0.09403712 + 0.702073)C + 31.556$\newline202.202 - 31.556 = (1 + 0.09403712 + 0.702073)C$ $\newline$ Now, calculate the sum of the coefficients and the difference of the constants.Calculate the sum of the coefficients and the difference of the constants: $\newline$ $170.646 = (1.79611012)C$ $\newline$ Now, divide both sides by the sum of the coefficients to solve for C.Divide both sides by the sum of the coefficients to solve for C: $\newline$ $C = 170.646 / 1.79611012$ $\newline$ Now, perform the division to find the value of C.Perform the division to find the value of C: $\newline$ $1170.16 = C / 0.1175464 + (400 + 0.8C) / 1.46933 + 4.0861C + 182.709$ $0$ $\newline$ We have found the value of C that satisfies the given equation.

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