Gwen invested 2100 pounds for three years in a savings account, which pays compound interest. The interest rate for the first two years was five percent. The interest rate for the third year was 1.5 percent. How much did Gwen have in her savings account at the end of the three-year period?
Interest is an amount of money that can accrue from either borrowing or investing money. When money is invested, the bank pays the customer interest as a way of rewarding that customer for banking with them. Interest is usually paid yearly. And compound interest means that each year the interest received is added on to the amount in the account and the following year the interest is calculated based on that new amount.
To calculate the amount that Gwen will have at the end of the three years, we will first need to work out five percent of her investment. We’ll need to add that onto the original, then work out five percent of the new amount, add it on, and then work out 1.5 percent and add it on again.
Let’s see what this might look like and the two possible methods we might use. In the first method, we’re going to begin by finding the value of five percent of 2100. The original number is 100 percent. With that in mind, we can calculate 10 percent of the amount by dividing by 10, since 100 divided by 10 gives us 10.
2100 divided by 10 is 210. So 10 percent of 2100 is 210 pounds. We can then find five percent by dividing the value for 10 percent by two. 210 divided by two is 105. So we know that five percent of 2100 is 105. Let’s add this onto the original amount and that will tell us the amount that Gwen had at the end of the first year. That’s 2205 pounds.
Next, we’ll repeat this process for year two. This time, 100 percent is 2205 pounds. Once again, we can calculate 10 percent by dividing by 10. That gives us that 10 percent of 2205 pounds is 220 pounds and 50 pence. Let’s add a zero here because we’re working in pounds.
Once again, we find the value of five percent by dividing the value for 10 percent by two. That’s 110 pounds and 25. This time, we add that 110 pounds and 25 pence onto 2205, since that’s how much she had at the end of the first year. At the end of the second year, she had 2315 pounds and 25 pence.
Now, let’s consider year three. This time she earned 1.5 percent. We can find the value of one percent of a number by dividing by 100 since 100 divided by 100 is one. 2315.25 divided by 100 is 23.1525. Normally, at this stage, we try to round, but we’re not quite finished. So we will leave it in this form.
We want to find 1.5 percent. So we’re going to find 0.5 percent by dividing the value for one percent by two. That gives us 11.57625. Adding the values for one percent and 0.5 percent tells us that 1.5 percent is 34.72875. Once again, we won’t round just yet. We should then add this value to 2315 pounds and 25 pence to find the amount she had at the end of the third year. That’s 2349.97875.
Since we’ve now finished our calculations and we’re dealing with money, we should round this to two decimal places. The second number after the decimal place is the seven. The digit immediately to its right is an eight. And since eight is greater than five, that tells us we round our number up. At the end of the third year, she had 2349 pounds and 98 pence.
Now, whilst this is quite a long winding method, it will get you the marks. There is a slightly quicker way though and that involves finding the relevant decimal multiplier for each percentage increase.
Now, in the first two years, Gwen’s investment increased by five percent. We can say that the original amount was 100 percent. So increasing this by five percent, we’re adding five percent to the original. 100 plus five is 105 percent. At the end of the first year, Gwen had 105 percent of the original.
Now, since percent means out of 100, we can change this into a decimal by dividing by 100. And 105 percent is, therefore, equivalent to 1.05. In maths, of can usually be replaced with a multiplication symbol. So we can find 105 percent of 2100 by typing in 1.05 multiplied by 2100. Alternatively, we can actually use 1.05 multiplied by 2100 in our next calculation.
Once we know that at the end of year one, Gwen had 1.05 multiplied by 2100 pounds, we know that we’re again increasing it in year two by five percent. So we’re going to multiply that amount by 1.05 once again. That’s 1.05 multiplied by 1.05 multiplied by 2100. Remember when we multiply a number by itself, we’re squaring it. So this is the same as 1.05 squared multiplied by 2100.
Now, remember the percentage has changed in year three. It was an increase of 1.5 percent. We can, therefore, add 1.5 percent on to 100 percent and it gives us 101.5 percent. Once again, we divide 101.5 by 100 to get the decimal multiplier; that’s 1.015.
We know that at the end of year two, Gwen had 1.05 squared multiplied by 2100 pounds in her account. So to increase this by 1.5 percent, we can multiply that whole number by 1.015. Finally, if we type all of this into our calculator, we get 2349.97875 once again.
Rounding to two decimal places, we can see that either method tells us that Gwen had 2349 pounds and 98 pence in her savings account.