Question

Mohan ate half a pizza on Monday. He ate half of what was left on Tuesday and so on. He followed this pattern for one week. How much of the pizza would he have eaten during the week?
(A) $99.22\% $
(B) $95\% $
(C) $98.22\% $
(D) $100\% $

Answer 1

Hint: Initially, he has a whole pizza with him. He ate half of it on the first day (Monday), half of remaining on Tuesday and continued this for one week. we need to add the fraction of pizza eaten per day and multiply it by $100$ to get the percentage of pizza eaten by him during the week.

Complete Step by Step Solution:
Pizza he ate on Monday= $\dfrac{1}{2}$
Pizza left with him= $1 - $(pizza ate on Monday) =$1 - \dfrac{1}{2} = \dfrac{1}{2}$
Pizza he ate on Tuesday = $\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}$
Pizza left with him = $\dfrac{1}{2} - \dfrac{1}{4}$$ = \dfrac{1}{4}$
Pizza he ate on Wednesday = $\dfrac{1}{2} \times \dfrac{1}{4} = \dfrac{1}{8}$
Pizza left with him = $\dfrac{1}{4} - \dfrac{1}{8} = \dfrac{1}{8}$
Pizza he ate on Thursday = $\dfrac{1}{2} \times \dfrac{1}{8} = \dfrac{1}{{16}}$
Pizza left with him = $\dfrac{1}{8} - \dfrac{1}{{16}} = \dfrac{1}{{16}}$
Continuing this pattern, we can observe that pizza he ate on Friday, Saturday and Sunday is $\dfrac{1}{{32}},\dfrac{1}{{64}}$and $\dfrac{1}{{128}}$.
$\therefore $quantity of pizza he ate during the week = $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{{16}} + \dfrac{1}{{32}} + \dfrac{1}{{64}} + \dfrac{1}{{128}} = \dfrac{{64 + 32 + 16 + 8 + 4 + 2 + 1}}{{128}}$= $\dfrac{{127}}{{128}}$
Percentage of total quantity he ate during the week = $\dfrac{{127}}{{128}} \times 100\% = 99.22\% $

Therefore, the correct answer is option (A).

Note:
Instead of adding $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{{16}} + \dfrac{1}{{32}} + \dfrac{1}{{64}} + \dfrac{1}{{128}}$ , we can use a different approach. We can observe that the given series in a GP whose first term is $\dfrac{1}{2}$and the common ratio is also $\dfrac{1}{2}$.
Number of terms = $7$
Sum of $n$terms of a GP whose first term is $a$and common ratio is $r = $$\dfrac{{a({r^n} - 1)}}{{r - 1}}$, if $r \ne 1$and $an$if $r = 1$
Here, $r = \dfrac{1}{2}$and $a = \dfrac{1}{2}$
Sum of seven terms = $\dfrac{{\dfrac{1}{2}[{{(\dfrac{1}{2})}^7} - 1]}}{{(\dfrac{1}{2} - 1)}}$= $\dfrac{{127}}{{128}}$
Percentage of pizza ate during the week= $\dfrac{{127}}{{128}} \times 100\% = 99.22\% $
This formula is very important as you can use this to solve all the geometric series.