Question

Douglas invests money in two simple interest accounts. He invests three times as much in an account paying 14% as he does in an account paying 5%. If he earns $152.75 in interest in one year from both accounts combined, how much did he invest altogether?

Answer 1

Answer:

Altogether, he invested $1300.

Step-by-step explanation:

This is a simple interest problem.

The simple interest formula is given by:

$E = P*I*t$

In which E are the earnings, P is the principal(the initial amount of money), I is the interest rate(yearly, as a decimal) and t is the time.

He invests three times as much in an account paying 14% as he does in an account paying 5%.

I am going to call the earnings from the account paying 14% $E_{1}$ and the earnings from the account paying 5% $E_{2}$. The principals are $P_1$ and $P_{2}$, in which $P_{1} = 3P_{2}$.

So

$E_{1} = P_{1}*0.14t$

$E_{2} = P_{2}*0.05t$

He earns $152.75 in interest in one year from both accounts combined.

This means that

$E_{1} + E_{2} = 152.75$

I am going to write $E_{1}$ as a function of $E_{2}$ and replace in the first equation, that of $E_{1}$.

So

$E_{1} = 152.75 - E_{2}$

$E_{1} = P_{1}*0.14t$

We also have that

$P_{1} = 3P_{2}$

So

$152.75 - E_{2} = 3*P_{2}*0.14t$

In which

$E_{2} = P_{2}*0.05t$

So

$152.75 - P_{2}*0.05t = 0.42P_{2}t$

His earnings are after 1 year, so $t = 1$

$152.75 - P_{2}*0.05 = 0.42P_{2}$

$0.42P_{2} + P_{2}*0.05 = 152.75$

$0.47P_{2} = 152.75$

$P_{2} = \frac{152.75}{0.47}$

$P_{2} = 325$

His smaller investment is 325.

$P_{1} = 3P_{2} = 3*325 = 975$

How much did he invest altogether?

This is $P_{1} + P_{2}$

$P_{1} + P_{2} = 975 + 325 = 1300$

Altogether, he invested $1300.

Answer 2

Answer:

Step-by-step explanation:

Let x represent the amount invested in the account paying 14% interest.

Let y represent the amount invested in the account paying 5% interest.

He invests three times as much in an account paying 14% as he does in an account paying 5%. This means that

x = 3y

The formula for simple interest is expressed as

I = (PRT)/100

Considering the account earning 14% interest,

I = (x × 14 × 1)/100 = 0.14x

Considering the account earning 5% interest,

I = (y × 5 × 1)/100 = 0.05y

If he earns $152.75 in interest in one year from both accounts combined, it means that

0.14x + 0.05y = 152.75 - - - - - - - - - -1

Substituting x = 3y into equation 1, it becomes

0.14(3y) + 0.05y = 152.75

0.42y + 0.05y = 152.75

0.47y = 152.75

y = 152.75!0.47

y = 325

x = 3y = 3 × 325

x = $975

Total amount of money invested is

975 + 325 = $1300