Solve -250+100/1.1+100/1.21+75/1.1^3+75/1.1^4+frac{50}{1.1^5}+frac{25}{1.1}

$$-250+ \frac{ 100 }{ 1.1 } + \frac{ 100 }{ 1.21 } + \frac{ 75 }{ { 1.1 }^{ 3 } } + \frac{ 75 }{ { 1.1 }^{ 4 } } + \frac{ 50 }{ { 1.1 }^{ 5 } } + \frac{ 25 }{ 1.1 }$$

Evaluate

$$\frac{13673500}{161051}\approx 84.901677108$$

Solution Steps

Expand \frac{100}{1.1} by multiplying both numerator and the denominator by 10.

$$-250+\frac{1000}{11}+\frac{100}{1.21}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Convert -250 to fraction -\frac{2750}{11}.

$$-\frac{2750}{11}+\frac{1000}{11}+\frac{100}{1.21}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Since -\frac{2750}{11} and \frac{1000}{11} have the same denominator, add them by adding their numerators.

$$\frac{-2750+1000}{11}+\frac{100}{1.21}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Add -2750 and 1000 to get -1750.

$$-\frac{1750}{11}+\frac{100}{1.21}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Expand \frac{100}{1.21} by multiplying both numerator and the denominator by 100.

$$-\frac{1750}{11}+\frac{10000}{121}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Least common multiple of 11 and 121 is 121. Convert -\frac{1750}{11} and \frac{10000}{121} to fractions with denominator 121.

$$-\frac{19250}{121}+\frac{10000}{121}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Since -\frac{19250}{121} and \frac{10000}{121} have the same denominator, add them by adding their numerators.

$$\frac{-19250+10000}{121}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Add -19250 and 10000 to get -9250.

$$-\frac{9250}{121}+\frac{75}{1.1^{3}}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Calculate 1.1 to the power of 3 and get 1.331.

$$-\frac{9250}{121}+\frac{75}{1.331}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Expand \frac{75}{1.331} by multiplying both numerator and the denominator by 1000.

$$-\frac{9250}{121}+\frac{75000}{1331}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Least common multiple of 121 and 1331 is 1331. Convert -\frac{9250}{121} and \frac{75000}{1331} to fractions with denominator 1331.

$$-\frac{101750}{1331}+\frac{75000}{1331}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Since -\frac{101750}{1331} and \frac{75000}{1331} have the same denominator, add them by adding their numerators.

$$\frac{-101750+75000}{1331}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Add -101750 and 75000 to get -26750.

$$-\frac{26750}{1331}+\frac{75}{1.1^{4}}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Calculate 1.1 to the power of 4 and get 1.4641.

$$-\frac{26750}{1331}+\frac{75}{1.4641}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Expand \frac{75}{1.4641} by multiplying both numerator and the denominator by 10000.

$$-\frac{26750}{1331}+\frac{750000}{14641}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Least common multiple of 1331 and 14641 is 14641. Convert -\frac{26750}{1331} and \frac{750000}{14641} to fractions with denominator 14641.

$$-\frac{294250}{14641}+\frac{750000}{14641}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Since -\frac{294250}{14641} and \frac{750000}{14641} have the same denominator, add them by adding their numerators.

$$\frac{-294250+750000}{14641}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Add -294250 and 750000 to get 455750.

$$\frac{455750}{14641}+\frac{50}{1.1^{5}}+\frac{25}{1.1}$$

Calculate 1.1 to the power of 5 and get 1.61051.

$$\frac{455750}{14641}+\frac{50}{1.61051}+\frac{25}{1.1}$$

Expand \frac{50}{1.61051} by multiplying both numerator and the denominator by 100000.

$$\frac{455750}{14641}+\frac{5000000}{161051}+\frac{25}{1.1}$$

Least common multiple of 14641 and 161051 is 161051. Convert \frac{455750}{14641} and \frac{5000000}{161051} to fractions with denominator 161051.

$$\frac{5013250}{161051}+\frac{5000000}{161051}+\frac{25}{1.1}$$

Since \frac{5013250}{161051} and \frac{5000000}{161051} have the same denominator, add them by adding their numerators.

$$\frac{5013250+5000000}{161051}+\frac{25}{1.1}$$

Add 5013250 and 5000000 to get 10013250.

$$\frac{10013250}{161051}+\frac{25}{1.1}$$

Expand \frac{25}{1.1} by multiplying both numerator and the denominator by 10.

$$\frac{10013250}{161051}+\frac{250}{11}$$

Least common multiple of 161051 and 11 is 161051. Convert \frac{10013250}{161051} and \frac{250}{11} to fractions with denominator 161051.

$$\frac{10013250}{161051}+\frac{3660250}{161051}$$

Since \frac{10013250}{161051} and \frac{3660250}{161051} have the same denominator, add them by adding their numerators.

$$\frac{10013250+3660250}{161051}$$

Add 10013250 and 3660250 to get 13673500.

$$\frac{13673500}{161051}$$

Factor

$$\frac{23 \cdot 29 \cdot 41 \cdot 2 ^ {2} \cdot 5 ^ {3}}{11 ^ {5}} = 84\frac{145216}{161051} = 84.90167710849359$$