Math Concepts & Calculating Interest Flashcards
Let’s start with a very common symbol in real estate: the percent.
A percent: is one part of every hundred. Percent can also be symbolized by the math symbol %.
Example 1: My husband forgets to take out the trash 75% of the time.
Example 2: My client has been pre-approved for a $300,000 loan with a 4.125% interest rate.
Percent comes from the latin phrase “Per Centum” and centum means 100, so percent essentially means “per hundred.” If someone orders the lobster 80% of the time they go to a restaurant, they are saying that if they go to that restaurant 100 times, they would order lobster 80 times.
99% of my students make me happy; don’t be that 1%, Anthony.
We will be solving a number of problems in this level using percentages.
So, the following tidbit is critical: You can never solve a problem while a percentage is still in percentage form — it must be changed into its decimal form.
Percentages
Percentages can be turned into decimals. You’ll use this one a lot!
We’ll first start with double-digit percentages as they are a bit easier (but follow the same steps as single-digit percentages). Come with me as we convert 35% into a decimal.
Note: In order to solve problems that involve percentages, we will convert percentages into decimals very often.
Step 1: Eliminate the percent sign.
First, we eliminate the percent sign as our first step. Bye-bye, percent sign!
Result: 35
Step 2: Insert and move the decimal two spaces to the left.
For step two, we’ll add a decimal point to our number and move it to the left. The decimal will always start to the right of the whole number.
Final Answer: 0.35
35% is changed to a decimal by replacing the percentage sign with a decimal, then moving the decimal two spaces left.
Where Does the Decimal Come From?
Why does the decimal move two places? Recall that the denominator of a percent is 100. This means it is divided by 100. Our number system is base 10, meaning every place represents 10 times more than the place before it. One hundred has two powers of 10 (100 = 10 x 10). This is why the decimal place moves two times.
More Examples:
14% becomes 0.14
38% becomes 0.38
62% becomes 0.62
A table shows 14% equals 0.14, 38% equals 0.38, and 62% equals 0.62.
Converting Percentages to Decimal Form
Converting two-digit percentages into decimal form? No problem.
But what about converting single-digit percentages into decimals? Let’s see how we convert 7% into a decimal.
Step 1: Eliminate the percent sign.
First things first — let’s drop that percent sign.
Result: 7
Step 2: Add the decimal, move it to the left, add a zero.
Just like we did with double-digit percentages, we’ll start the decimal point to the right of the number farthest to the right (7). We’ll then move our decimal point over two spaces to the left.
We’ll need to drop a zero (0) into the empty space between our decimal point and seven. The result of that zero addition will be our answer.
Final Answer: 0.077% is changed to a decimal by replacing the percent sign with a decimal, moving it 2 spaces left, and adding zeros for 0.07.
More Examples:
2% becomes 0.02
4% becomes 0.04
6% becomes 0.06
A table shows 2% equals 0.02, 4% equals 0.04, and 6% equals 0.06.
Converting Single-Digit Percentages into Decimals
Percentage Problems
Percentage Problems
As you can see below, total, part, and percentage come together to create a nifty formula. The part divided by percentage will equal the total.
Part ÷ Percentage = Total
We can use our percentage formula to solve problems such as the one below.
Solving for the Sales Price
Mikayla, a licensed real estate sales agent, helped her client buy a property outside of Springfield. The listing agreement states that 3% of the sales price will go to the buyer’s agent as commission. Mikayla made $7,500 in commission in the purchase of the home.
For how much did the property sell?
Step 1: Insert the known variables.
First, we’ll plug in the numbers that we know from the story. We know the part is the $7,500 that Mikayla made in commission and the percentage is 3%.
Here’s what that looks like plugged in:
$7,500 ÷ 3% = Total
Step 2: Convert the percent and divide.
Before dividing, we’ll need to convert the percentage into a decimal. Once converted, you just need to divide 7,500 by 0.03.
Answer: $7,500 ÷ 0.03 = $250,000
Final Answer: The sales price of the home that Mikayla sold was $250,000.
Math Practice: Solving for Total
When solving for part, you’ll just need to multiply the total by percentage. Yep, it’s that easy!
Total x Percentage = Part
Let’s use this formula to solve for the problem below.
Solving for Mikayla’s Commission
Let’s use the same story, but now let’s pretend we don’t know how much Mikayla made in commission. What we do know is the sales price is $250,000 and the commission rate is 3%.
Step 1: Insert the known variables.
So we know the total is the total sales price of the house: $250,000. And we know the percent is the 3% commission that Mikayla made on the sale. Let’s put those numbers into the formula.
Answer: $250,000 x 3% = Part
Step 2: Convert the percentage to a decimal and multiply.
Once the variables are filled in with numbers and our percentage has been transformed into its decimal form, all you have to do is multiply the two numbers to find the part (the commission Mikayla made).
Answer: $250,000 x 0.03 = $7,500
Final Answer: Mikayla made $7,500 in commission.
Math Practice: Solving for Part
For our final formula of this formula family, let’s check out solving for percentage. As you can see from the formula below, the part divided by total will give you the percentage.
Part ÷ Total = Percentage
Let’s use this formula to solve for the problem below.
Solving for Commission Rate
This is the same story as before, but now we will use the commission payment and the sales price to figure out the commission rate. Remember, Mikayla made $7,500 on a $250,000 sale of a home.
Step 1: Insert the known variables.
We know that the total is the sales price of $250,000 and the part is the amount Mikalya received in commission ($7,500).
So, here’s what the formula looks like with those variables plugged in.
$7,500 ÷ $250,000 = Percentage
Step 2: Divide the part by the total.
0.03 = Percentage
Result: 0.03
Step 3: Convert the decimal to a percentage.
To convert 0.03 into a decimal, we’ll need to move the decimal point to the right two spaces. Then just drop the decimal point and slap on a percent sign.
Final Answer: The commission rate is 3%.
Math Practice: Solving for Percentage
Next, we will be covering a lot of common loan calculations. I will give examples and walk you through how to solve the problems.
For some problems you will be required to do a math workshop, which will give you an open-ended opportunity to solve the problem for yourself.
To start, let’s get down to the nitty gritty and take a look at every borrower’s least favorite part about borrowing: interest.
Non-Amortized Loans
All of the calculations in this chapter are for non-amortized loans. This means that the borrower will be paying interest only throughout the life of the loan. The principal will remain the same and will be due as a balloon payment at the end of the loan’s term.
These loans are more common in commercial and investment real estate. Most residential borrowers will probably take out amortized loans, meaning they are paying the interest and principal with each payment. The portion that goes to interest and the portion that goes to principal shifts throughout the life of the loan. We’ll solve for amortized loans later in this chapter.
Again, with non-amortized loans the payments stay the same, with the borrower paying only the interest. Often non-amortized loans take the form of straight loans.
Non-PITI
Additionally, the calculations we do in this level will not take into account the entire PITI payment (principal, interest, taxes, insurance). To make things simple, we are going to just focus on the PI part of the payment: principal and interest.
Interest
Okay, let’s start with the annual interest total. If it looks familiar, that’s because it is the same mathematical relationship for solving for part that you saw earlier.
Principal x Interest rate = Annual interest
Reminder: This formula for annual interest works for loans that are not amortized. This would not work to solve for amortized loans, as the amount paid towards interest changes each month due to the incremental reduction of the principal balance.
Solving for Annual Interest
Here’s a quick example of calculating the total annual interest:
Jamie took out a straight loan of $200,000 with an interest rate of 4.5%. How much will Jamie pay in annual interest?
Step 1: Insert known variables
We know the principal is $200,000 and the interest rate is 4.5%. Throw those into the equation and see how that looks.
Answer: $200,000 x 4.5% = Annual interest
Step 2: Convert the percentage to a decimal and then multiply the two numbers
After converting 4.5% to a decimal we get 0.045. Then just multiply those two numbers and you have the amount in annual interest.
$200,000 x 0.045 = $9,000
Final Answer: The annual interest that Jamie will pay is $9,000.
Math Practice: Solving for Annual Interest
Next, let’s solve for semi-annual interest. It’s pretty simple – you just divide the annual interest by 2. Easy!
Annual interest ÷ 2 = Semi-annual interest
Step 1: Insert known variables
Using Jamie’s annual interest from the previous example, we know that the annual interest is $9,000. So all we need to do is insert that variable into the formula.
Answer: $9,000 ÷ 2 = Semi-annual interest
Step 2: Divide the annual interest by 2
Easy as that.
$9,000 ÷ 2 = $4,500
Final Answer: The semi-annual interest for this loan equals $4,500.
Math Practice: Solving for Semi-Annual Interest
And now, check out the formula for calculating monthly interest. Not too difficult, either!
Annual interest ÷ 12 months = Monthly interest
Note 1: To identify the amount of interest owed in a month, we divide the annual interest by 12 since there are 12 months in a year.
Note 2: This formula would work to solve the monthly interest owed for one payment (either one month or several months at once). It would not work to solve the monthly interest for several payments, as the interest amount would change as the principal was reduced.
Solving for Monthly Interest
Let’s use our new equation to determine what Lauren owes in interest.
Lauren has a remaining balance of $300,000 on her house. Her interest rate is 4.15%. How much interest does Lauren owe for her next mortgage payment?
Step 1: Insert known variables
Here’s what we know:
Remaining balance: Lauren has a remaining principal of $300,000.
Interest rate: Lauren has an interest rate of 4.15%.
Annual interest: Unknown.
Answer: $300,000 x 4.15% = Annual interest
Step 2: Convert to decimal and multiply
First, let’s convert 4.15% to decimal form (0.0415), then multiply it by the principal ($300,000).
$300,000 x 0.0415 = $12,450
Answer: The annual interest is $12,450. But we still need to solve for monthly interest.
Step 3: Divide the annual interest by 12
To solve for monthly interest, we’ll divide our annual interest ($12,450) by 12 (since there are twelve months in a year).
$12,450 ÷ 12 = $1,037.50
Final Answer: For her mortgage payment next month, Lauren will owe her lender $1,037.50 in interest.
Math Practice: Solving for Monthly Interest
Solving for quarterly interest is the same as monthly interest, except instead of 12, you divide by 4. Here’s the last example solved for quarterly interest.
Step 1: Insert known variables
Since it’s the same example as before, the annual interest total is still $12,450. But instead of dividing by the 12 months of the year, we will divide by the quarters of the year: 4.
Annual interest ÷ 4 = Quarterly interest
Step 2: Divide the annual interest by 4
Easy as that.
$12,450 ÷ 4 = $3,112.50
Final Answer: The quarterly interest for this loan equals $3,112.50.
Other Interest Totals
At times, you may need to use both annual interest and monthly interest to solve a problem. For example, if you were asked to solve the interest for a single payment loan that covered 15 months, you would need to add annual interest (12 months) + monthly interest x 3 (3 months).
(Monthly interest x 3) + Annual interest = 15 month interest
Alternatively, you can also find the monthly interest and then multiply that number by the number of months you need to find the interest for.
Math Practice: Solving for Quarterly Interest
Okay, so you can solve for the interest. Let’s see if we can solve for the principal now.
You might remember that you can solve for total, part, and percent. We just found the part (the interest), but now we want to find the total (the principal).
Part ÷ Percentage = Total
Sit tight, I’ll walk you through how we find that pesky principal now.
The Story
Jeremy is paying $7,500 in annual interest and his loan has an interest rate of 3%.
Now, let’s figure out how much his loan principal is! It’ll be fun!
Step 1: Insert known variables
Total: The total is the principal and it is not given — it’s unknown and is the answer for which we are searching.
Part: Jeremy paid $7,500 in interest — so we’ll use the $7,500 for the “part” of the formula.
Percent: We know the interest rate is 3%. So we’ll use that rate to fill in the percentage part of our problem.
$7,500 ÷ 3% = Total
Result: $7,500 ÷ 3% = Total
Step 2: Convert to a decimal and divide
First we’ll need to convert the percentage into a decimal. And after we convert the percentage into a decimal, we’re now left with an easy equation to solve: $7,500 ÷ 0.03 = Total
$7,500 ÷ 0.03 = $250,000
Final Answer: The principal of the loan is $250,000.
So, to sum it up, Jeremy took out a loan with a 3% interest rate. He ended up paying $7,500 in interest on the loan. Knowing that, we were able to solve for the principal of the loan, which was $250,000.
Math Practice: Finding the Principal
Alright, now let’s complete the trio of formulas and solve for (percentage) interest rate. Here is the scenario:
The Story
Andie took out a loan of $340,000. Nice!
Andie paid $10,200 in interest over the year. What was Andie’s interest rate?
The Formula
To answer this question, we’ll need to solve for percentage. We’ll use the following equation to find the answer:
Part ÷ Total = Percentage
As you can see, this is just like the previous scenario, except with rearranged variables.
Step 1: Insert known variable
We know that the part is the $10,200 in interest that Andie paid. We also know that the total is the loan principal of $340,000. So, here is what the equation looks like with the variables inserted:
Answer: $10,200 ÷ $340,000 = Percentage
Step 2: Divide
We have all we need to finish this problem off. Divide 10,200 by 340,000.
$10,200 ÷ $340,000 = 3%
Final Answer: Andie’s loan has a 3% interest rate.
Math Practice: Finding the Interest Rate
Again, here are the formulas you should know for solving for the total, part, and percentage, as well as the corresponding formulas for finding principal, interest, and rate.
Part ÷ Percentage = Total
Annual interest ÷ Rate = Principal
Total x Percentage = Part
Principal x Interest rate = Annual interest
Part ÷ Total = Percentage
Annual interest ÷ Total interest = Rate
Formula Recap
