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When starting a new unit, a teacher announces that each day after the first, the class must do twice the total number of problems that had been assigned on all previous days. the class works for 6 days and on the 7th day she says that this is our last day of the unit. on that last day, what fraction of the problems do they still have to complete.?

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the total number of problems is 1+2+6+18+54+162+486 = 729 486/729 = 2/3

Answer:

[tex]\text{Required fraction}= \frac{2}{3}[/tex]

Step-by-step explanation:

Let , The number of problem they did on first day = x

Then, the number of problems they did on second day is twice the number of  problem they did on first day = 2x

the number of problems they did on third day is twice the number of  problem they did on all previous days = 2(x+2x) = 6x

the number of problems they did on fourth day is twice the number of  problem they did on all previous days = 2(x+2x+6x) = 18x

the number of problems they did on fifth day is twice the number of  problem they did on all previous days = 2(x+2x+6x+18x) = 54x

the number of problems they did on sixth day is twice the number of  problem they did on all previous days = 2(x+2x+6x+18x+54x) = 162x

the number of problems they did on seventh day is twice the number of  problem they did on all previous days = 2(x+2x+6x+18x+54x+162x) = 486x

Total number of problem they did in seven days= (x+2x+6x+18x+54x+162x+486x) = 729x

[tex]\text{Required fraction}=\frac{\text{Number of questions to be done on seventh day}}{\text{Total number of questions}}[/tex]

[tex]\text{Required fraction}=\frac{486x}{729x} = \frac{486}{729}[/tex]  

Thus , [tex]\text{Required fraction}= \frac{2}{3}[/tex]