Solve {l}{(4*1/10)+(60*1/100)+(100*1/1000)}{(4*100)+(6*10)+(1*1)}{(4*1)+(6*0.1)+(1*0.01)}{(4*0.1)+(6*0.01)+(1*0.001)}

$$\left. \begin{array} { l } { ( 4 \times \frac { 1 } { 10 } ) + ( 60 \times \frac { 1 } { 100 } ) + ( 100 \times \frac { 1 } { 1000 } ) } \\ { ( 4 \times 100 ) + ( 6 \times 10 ) + ( 1 \times 1 ) } \\ { ( 4 \times 1 ) + ( 6 \times 0.1 ) + ( 1 \times 0.01 ) } \\ { ( 4 \times 0.1 ) + ( 6 \times 0.01 ) + ( 1 \times 0.001 ) } \end{array} \right.$$

Sort

$$0.461,1.1,4.61,461$$

Solution Steps

Multiply 4 and \frac{1}{10} to get \frac{4}{10}.

$$sort(\frac{4}{10}+60\times \frac{1}{100}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Reduce the fraction \frac{4}{10} to lowest terms by extracting and canceling out 2.

$$sort(\frac{2}{5}+60\times \frac{1}{100}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 60 and \frac{1}{100} to get \frac{60}{100}.

$$sort(\frac{2}{5}+\frac{60}{100}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Reduce the fraction \frac{60}{100} to lowest terms by extracting and canceling out 20.

$$sort(\frac{2}{5}+\frac{3}{5}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Since \frac{2}{5} and \frac{3}{5} have the same denominator, add them by adding their numerators.

$$sort(\frac{2+3}{5}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 2 and 3 to get 5.

$$sort(\frac{5}{5}+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Divide 5 by 5 to get 1.

$$sort(1+100\times \frac{1}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 100 and \frac{1}{1000} to get \frac{100}{1000}.

$$sort(1+\frac{100}{1000},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Reduce the fraction \frac{100}{1000} to lowest terms by extracting and canceling out 100.

$$sort(1+\frac{1}{10},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Convert 1 to fraction \frac{10}{10}.

$$sort(\frac{10}{10}+\frac{1}{10},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Since \frac{10}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$sort(\frac{10+1}{10},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 10 and 1 to get 11.

$$sort(\frac{11}{10},4\times 100+6\times 10+1\times 1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 4 and 100 to get 400. Multiply 6 and 10 to get 60. Multiply 1 and 1 to get 1.

$$sort(\frac{11}{10},400+60+1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 400 and 60 to get 460.

$$sort(\frac{11}{10},460+1,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 460 and 1 to get 461.

$$sort(\frac{11}{10},461,4\times 1+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 4 and 1 to get 4.

$$sort(\frac{11}{10},461,4+6\times 0.1+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 6 and 0.1 to get 0.6.

$$sort(\frac{11}{10},461,4+0.6+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 4 and 0.6 to get 4.6.

$$sort(\frac{11}{10},461,4.6+1\times 0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 1 and 0.01 to get 0.01.

$$sort(\frac{11}{10},461,4.6+0.01,4\times 0.1+6\times 0.01+1\times 0.001)$$

Add 4.6 and 0.01 to get 4.61.

$$sort(\frac{11}{10},461,4.61,4\times 0.1+6\times 0.01+1\times 0.001)$$

Multiply 4 and 0.1 to get 0.4.

$$sort(\frac{11}{10},461,4.61,0.4+6\times 0.01+1\times 0.001)$$

Multiply 6 and 0.01 to get 0.06.

$$sort(\frac{11}{10},461,4.61,0.4+0.06+1\times 0.001)$$

Add 0.4 and 0.06 to get 0.46.

$$sort(\frac{11}{10},461,4.61,0.46+1\times 0.001)$$

Multiply 1 and 0.001 to get 0.001.

$$sort(\frac{11}{10},461,4.61,0.46+0.001)$$

Add 0.46 and 0.001 to get 0.461.

$$sort(\frac{11}{10},461,4.61,0.461)$$

Convert decimal numbers in the list \frac{11}{10},461,4.61,0.461 to fractions.

$$\frac{11}{10},461,\frac{461}{100},\frac{461}{1000}$$

Least common denominator of the numbers in the list \frac{11}{10},461,\frac{461}{100},\frac{461}{1000} is 1000. Convert numbers in the list to fractions with denominator 1000.

$$\frac{1100}{1000},\frac{461000}{1000},\frac{4610}{1000},\frac{461}{1000}$$

To sort the list, start from a single element \frac{1100}{1000}.

$$\frac{1100}{1000}$$

Insert \frac{461000}{1000} to the appropriate location in the new list.

$$\frac{1100}{1000},\frac{461000}{1000}$$

Insert \frac{4610}{1000} to the appropriate location in the new list.

$$\frac{1100}{1000},\frac{4610}{1000},\frac{461000}{1000}$$

Insert \frac{461}{1000} to the appropriate location in the new list.

$$\frac{461}{1000},\frac{1100}{1000},\frac{4610}{1000},\frac{461000}{1000}$$

Replace the obtained fractions with the initial values.

$$0.461,\frac{11}{10},4.61,461$$

Evaluate

$$1.1,\ 461,\ 4.61,\ 0.461$$