# How to Manually Calculate your Interest

Hi, and welcome to this overview of word problems that involve interest! In this video, we’re going to look at how interest accrues on money over time and how to solve word problems that deal with this process. Let’s get started!

Whether the money in question is a savings account balance or a credit card debt, the process is the same. Interest grows very rapidly, as the equation that is used to accrue interest is exponential in nature.

Let’s first review the steps and terminology that are involved in calculating a percentage increase with a simple example.

Suppose you have $100 in a savings account that returns an annual interest rate of 1.7%. This means that your bank promises to pay you 1.7% of your$100 balance at the end of the year. To determine that amount, you must:

First, convert the percentage to its decimal representation by moving the decimal point two places to the left,

Which gives us 1.7 or $1.70. $$100\times.017=1.70$$ This amount,$1.70, is the interest that is earned and added to your $100, which means that your new balance at the end of the year is$101.70.

These two steps can be combined for a more efficient process. If we start with a balance of $100 and add the calculated interest, .017times 100, the expression will look like this: $$100+(100\times.017)$$ We can now recognize a common factor of 100 in each of the two terms of the expression. Factoring out that initial balance, which is sometimes referred to as the principal balance, gives us 100 times 1 plus .017: $$100(1+.017)$$ Combining the terms in the parentheses results in a factor of 1.017, which when multiplied by$100 results in the new savings account balance at the end of the year, $101.70. Let’s practice a few more examples using the one-step process of adding interest to a given amount at the end of one year. Calculate the interest that is earned on a bank account balance of$1,235 with an annual percentage rate of 2.1%. Round your answer to the nearest cent.

Remember, we first need to convert 2.1% to its decimal. Then multiply this interest rate by the initial balance of $1,235. This gives us an interest amount of$25.94 earned at the end of the year.

Now that we know the interest amount, what would the ending balance be of the bank account after one year?

Remember, the initial balance plus the interest earned equals the ending balance after one year. So, in this case: 1,235 plus 25.94 is equal to $1,260.94. $$1,235 + 25.94= 1,260.94$$ Let’s think a step further. What would the account balance be at the end of two years, assuming that there were no banking fees and that you did not make any additional deposits or withdrawals? If you’d like, pause the video and see if you can come up with the answer yourself. To figure this out, all we have to do is take the balance at the end of year one,$1,260.94, and multiply it by a factor of 1.021. The ending balance at the end of the second year would, therefore, be equal to:

$$1,260.94(1.021)=1,287.41$$

The concept of time is very important in this type of problem, because the interest is added to the account balance at the end of each year. If we break things down a bit, it becomes very easy to see the pattern of the factors as each year passes.

Year Initial Ending
End Balance x Factor Balance
1 $1,235 x 1.021 =$1,260.94
2 $1,235 x 1.021 x 1.021 =$1,287.41
3 $1,23 x 1.021 x 1.021 x 1.021 =$1,314.45

At the end of the first year, our initial balance grows by 2.1%. At the end of year two, the annual percentage rate is compounded again on a larger value, and then again at the end of year three, and so on.

Now, what would your ending balance be at the end of year four?

Well, at the end of year four, the annual percentage rate, or APR, will have been compounded four times, which gives us a total of $1,342.05: $$1,235(1.021)^{4}=1,342.05$$ And that’s all there is to it! Now that we’ve gone through several problems together, you should be well on your way to mastering any interest-related word problem that’s thrown at you. Thanks for watching, and happy studying! ## Practice Questions Question #1: Calculate the interest that is earned in one year on a bank account with a balance of$3,640, with an annual interest rate of 1.4%. Round your answer to the nearest cent.

$51.23$46.53

$55.60$50.96

Interest can be calculated by turning the percentage rate into a decimal and then multiplying this by the account balance. For this example, 1.7% would be 0.017, and when multiplied by the account balance of $3,640, the result is$50.96. This means that over one year the account will earn $50.96 in interest. Question #2: What would the new account balance be after one year for an account that earns 2.9% interest annually, if the account starts with$50,000?

$51,000$51,450

$60,450$71,490

2.9% of $50,000 can be calculated by multiplying 0.029 by$50,000 which equals $1,450. This can be added back to the original account balance in order to determine the new balance. $$1{,}450+50{,}000=51{,}450$$ Question #3: Mike invests$3,000 into an account that offers 8% interest per year. How much money will Mike have in the account after three years if the interest is added to the account balance after each year?

$3,779.14$3,229.14

$4.779.14$3,700.14

8% of $3,000 can be calculated by multiplying 0.08 by$3,000, which equals $240. When this is added back to the original account balance, Mike now has$3,240. The second year of interest can be calculated by multiplying 0.08 by $3,240 which equals$259.20. When this second year of interest is added back to the account balance Mike now has $3,240 +$259.20, which equals $3,499.20. The third year of interest can be calculated by multiplying 0.08 by$3,499.20, which equals $279.94 (when rounded to the nearest cent). This third year of interest can be added back to Mike’s account balance which leaves him with$3,499.20 + $279.94, which equals$3,779.14. This is his account balance after three years of interest.

Question #4:

Eva opens a savings account and deposits $350 as principal. The account earns 4% interest, compounded annually. What will her balance be in one year?$435

$364$532

$402 Answer: 4% interest on$350 can be calculated by multiplying 0.04 by $350, which equals$14. This means that after one year, Eva’s balance will be $14 higher. Her account will grow from$350 to $364. Question #5: Joe invests$17,500 in Bank A at an interest rate of 3.5% compounded annually. Brian invests $18,900 in Bank B at an interest rate of 3.1% compounded annually. Who will earn more interest after one year? Joe will earn more interest. Brian will earn more interest. Joe and Brian will earn the same amount of interest. There is no way to compare the amount of interest earned because Joe and Brian have a different interest rate. Answer: Joe will invest$17,500 at a 3.5% interest rate. The interest can be calculated as $17,500 × 0.035, which equals$612.50 in interest.

Brian will invest $18,900 in Bank B at a 3.1% interest rate. The interest can be calculated as$18,900 × 0.031, which equals \$585.90 in interest.

After one year Joe will earn more interest than Brian.