A student has a savings account earning 6% simple interest. She must pay $1650 for first-semester tuition by September 1 and $1650 for second-semester tuition by January 1. How much must she earn in the summer (by September 1) in order to pay the first-semester bill on time and still have the remainder of her summer earnings grow to $1650 between September 1 and January 1?
To solve this problem, let's break it down into steps:
Step 1: Calculate the interest earned on the first-semester bill from September 1 to January 1.
Since the first-semester bill is $1650 and she needs to grow it to $1650 by January 1, the interest earned is $1650 - $1650 = $0.
Step 2: Determine the amount of money the student needs to have on September 1 in her savings account to pay the first-semester bill.
Since the first-semester bill is $1650, she needs to have $1650 in her savings account on September 1.
Step 3: Calculate the amount of money the student must earn in the summer.
Since she needs to have $1650 on September 1 and the interest earned is $0, she must earn $1650.
Therefore, the student must earn $1650 in the summer in order to pay the first-semester bill on time and still have the remainder of her summer earnings grow to $1650 between September 1 and January 1.
To solve this problem, we need to break it down into two parts:
1. Calculate how much the student needs to pay for the first-semester tuition.
2. Calculate how much the remainder of her summer earnings needs to grow to reach $1650 by January 1.
Let's start with the first part, calculating the amount needed for the first-semester tuition:
Step 1: Multiply the tuition amount by 100 and divide it by the interest rate (6%) to calculate the principal amount needed. This is because 6% simple interest means the interest earned is 6% of the principal amount.
Principal amount = (Tuition amount x 100) / Interest rate
Principal amount = ($1650 x 100) / 6
Principal amount = $27500 / 6
Principal amount = $4583.33 (rounded to 2 decimal places)
Therefore, the student needs $4583.33 for the first-semester tuition.
Now, let's move on to the second part, calculating how much the remainder of her summer earnings needs to grow to reach $1650 by January 1:
Step 1: Subtract the amount needed for the first-semester tuition from $1650 to find out how much needs to grow.
Remaining amount to grow = $1650 - $4583.33
Remaining amount to grow = -$2933.33 (negative value indicates a deficit)
Since the remaining amount is negative, this means the student doesn't need to earn any additional money to reach $1650 by January 1. In fact, she has an excess of $2933.33.
So, the student doesn't need to earn any additional money in the summer to pay the first-semester bill on time and still have the remainder of her summer earnings grow to $1650 by January 1.