Question

Lucy's lemonade stand has increasing profits each week during the hot summer months. Lucy finds that each week during the summer her profits increase by 10% over what they were the previous week. In her first week of business, her profits were $50.How much money will she make in all over 4 weeks? Enter your answer, rounded to the nearest cent.

Answer 1

We are told that the first week Lucy earns $50. We want to calculate the second week's earnings. Since earnings increase 10% every week, we will calculate 10% of 50 and then add it to 50. To calculate 10% of a number, we simply multiply the number by 0.1.

Then, the 10% of 50 would be

`50\cdot0.1=5`

Now we add this quantity to 50. This would give the earnings of the second week. So, the earnings of the second week would be

`50+5=55=50+50\cdot0.1=50\cdot(1+(10)/(100))`

From here, it is important to note that to increase a quantity by a 10% we simply multiply the quantity by the number (1+10/100). So, to calculate the earnings of week 3, we simply multiply the earnings of week 2 by (1+10/100)

This would be

`50\cdot(1+(10)/(100))\cdot(1+(10)/(100))=50\cdot(1+(10)/(100))^2`

Finally, we repeat the process for week 4, This would give

`50\cdot(1+(10)/(100))^2\cdot(1+(10)/(100))=50\cdot(1+(10)/(100))^3`

So we have

`50\cdot(1+(10)/(100))^3=66.55`

To calculate the earnings of all 4 weeks, we simply add this numbers, so we have

`50+50\cdot(1+(10)/(100))+50\cdot(1+(10)/(100))^2+50\cdot(1+(10)/(100))^3=232.05`

So in 4 weeks, Lucy earns 232.05