A new shopping mall is gaining in popularity. Every day since it opened, the number of shoppers is $20 \%$ more than the number of shoppers the day before. The total number of shoppers over the first $4$ days is $671$ . $\newline$ How many shoppers were at the mall on the first day? $\newline$ Round your final answer to the nearest integer. $\newline$ $\square$ shoppers

Full solution

Q. A new shopping mall is gaining in popularity. Every day since it opened, the number of shoppers is

20 \%

more than the number of shoppers the day before. The total number of shoppers over the first

4

days is

671

\newline

How many shoppers were at the mall on the first day?

\newline

Round your final answer to the nearest integer.

\newline

\square

shoppers

Denote Shoppers: Let's denote the number of shoppers on the first day as $S$ . According to the problem, each day the number of shoppers is $20$ % more than the previous day. This means that on the second day, there are $S \times 1.20$ shoppers, on the third day $S \times 1.20^2$ shoppers, and on the fourth day $S \times 1.20^3$ shoppers. The total number of shoppers over the first $4$ days is given as $671$ . We can set up the equation as follows: $\newline$ $S + S \times 1.20 + S \times 1.20^2 + S \times 1.20^3 = 671$
Set Up Equation: First, factor out $S$ from the left side of the equation: $\newline$ $S(1 + 1.20 + 1.20^2 + 1.20^3) = 671$ $\newline$ Now calculate the sum inside the parentheses: $\newline$ $1 + 1.20 + 1.20^2 + 1.20^3 = 1 + 1.20 + 1.44 + 1.728$ $\newline$ $= 5.368$ $\newline$ So the equation simplifies to: $\newline$ $S \times 5.368 = 671$
Factor Out S: Now, divide both sides of the equation by $5$ . $368$ to solve for $S$ : $\newline$ $S = \frac{671}{5.368}$ $\newline$ $S \approx 125.046$ $\newline$ Since the number of shoppers must be a whole number, we round to the nearest integer. $\newline$ $S \approx 125$