Savings account math question

Question

You are going to start with $1 in a savings account and it will double everyday. Your friend will start with $100,000 and add $10,000 each day. Will you both ever have close to the same amount of money in the account at the same time? If so, what is that dollar amount and how many days will it take ?

Answer 1

If you start off with $1 in your account and it doubles every day, it goes up to $2 one day later, then $4 the day after, then $8, then $16, then so on. Essentially, every successive day, it gets multiplied by 2.

There is a mathematical operator that can represent multiplying a number multiple times: the exponent operator, or a^b. Essentially, a^b means "multiply a by itself, b times".

We can represent the value in your bank account like this: if x days have passed, then the balance in your account is 2^x, or 2 multiplied by itself, x times.

Your friend's bank account, on the other hand, starts off at $100,000, then increases to $10,000 every day. Their balance after x days can thus be represented as 10,000x + 100,000.

We can then set up an equation to see when both amounts will be equal: 2^x = 10,000x + 100,000.

To solve this equation, we can try plugging different values for x, and see which one gives us the closest result on both sides. Let's try an initial guess, 10 days: 2¹⁰ = 1024, and 10,000(10) + 100,000 = 200,000. Far off, and probably not the right answer. Let's try again, 15 days. 2¹⁵ = 32,768, and 10,000(15) = 250,000. Closer, but still pretty far. Let's try 20: 2²⁰ = 1,048,576, and 10,000(20) + 100,000 = 300,000. Overshot it slightly.

Let's try, I don't know, 18: 2¹⁸ = 262,144, and 10,000(18) + 100,000 = 280,000. That seems like the closest match, but to verify, we'll check 17 and 19, and if you check those yourself using a calculator, you'll see that those are much further off, so 18 is the closest.

The question asks for not only the number of days when they'll be equal, but also what the dollar amounts are; as we've plugged our x into both sides of the equation, we have the answers to those as well. So, our final answer is: the bank accounts will have close to the same amount of money after 18 days, at which point your account will have $262,144, and your friend's will have $280,000.

Answer 2

Since your money doubles every day this could be written in the form.

1 * 2ⁿ

If it has been 5 days you would have 2⁵ or 32 dollars.

Your friend has 100,000 + 10,000n dollars after n days..

So after 5 days your friend will have 150,000 dollars.

Your friend starts with a lot of money, but it increases slowly.

You start with very little money but it increases very quickly over some time.

First lets use the calculator to find when you will have 100,000 dollars.

So use the calculator and start with any number of days.

Let's start with 10 days.

After 10 days you will have 2¹⁰ or 1024 dollars.

This is not enough.

Lets try 15 days.

After 15 days you will have 2¹⁵ or 32,768 dollars. this is really close.

Lets check the next few days.

2¹⁶ = 65,536 , 2¹⁷ = 131,072 dollars.

This is when you will exceed your friends starting amount.

You could have found this by using logarithms by saying

2ⁿ = 100,000

log(2ⁿ) = log(100,000)

n*log(2) = 5

n = 5/log(2) = 16.6 which rounds up to 17

Now let's find how much your friend has after 17 days.

100,000 + 10,000*17 = 270,000

270,000 is more than 131,072

So you will need to wait another day to catch up.

Day 18

you will have 262,144 dollars

your friend will have 280,000 dollars.

This is as close as you will get.

If you wait another day your money will be nearly twice your friends and that is not close at all.

So it will take.

18 days.

You will have 262,144 dollars

Your friend will have 280,000 dollars.