### Video Transcript

Scarlett invests 1000 pounds into an individual savings account at the beginning of the year. An interest rate of 2.5 percent is paid monthly on the balance in the account. How much money does Scarlett have in her individual savings account at the end of the year if no other deposits or withdrawals are made and the interest rate stays constant?

Weβre told that Scarlett invests 1000 pounds. The bank has an interest rate of 2.5 percent. And this is paid monthly on the balance in the account. To calculate the amount of money in the account at the end of the year, we could work out the amount after month one, month two, month three, and so on, all the way up to month 12. We would do this by working out 2.5 percent of the previous balance.

Whilst there is nothing wrong with this method, it is very time consuming in an exam. As a result, it is far easier for us to remember and use the following formula, π΄ is equal to π multiplied by one plus π to the power of π. The π is the original amount invested, sometimes known as the principal. The π is the rate of interest written as a decimal. And finally, the π is the number of payments.

Substituting in the appropriate numbers into this formula gives us the value of π΄, which is the new amount. In our question, π is equal to 1000 pounds, as this is the amount that Scarlett is investing. The interest rate was 2.5 percent. To turn this into a decimal, we need to divide by 100, as percentages are out of 100. 2.5 divided by 100 is equal to 0.025. This means that our value for π, the interest rate, written as a decimal is 0.025. π is equal to 12, as the interest was paid monthly and there were 12 months in a year.

Substituting these values into the formula gives us 1000 multiplied by one plus 0.025 to the power of 12. One plus 0.025 is equal to 1.025. Therefore, the new amount is equal to 1000 multiplied by 1.025 to the power of 12. Typing this into the calculator gives us an answer of 1344.888824. As weβre dealing with money, we need to round our answer to two decimal places. The third decimal place is an eight, so we need to round up. Rounding up gives us 1344.89.

The amount of money that Scarlett has in her account at the end of the year is 1344 pounds and 89 pence. Her initial investment has increased by 344 pounds and 89 pence. This is the interest that she has earned.