Pamela and Joe invested some money at annual simple interest

I was also confused at the beginning about whether the interest paid would be like 2 x (0,06) and 2 x (0,07) or
(1.06)^2 + (1,07)^2. But here is an explaination what to do when you encounter this type of questions.

Simple Interest vs. Compound Interest:
Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. The interest, typically expressed as a percentage, can be either simple or compounded. Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period. Simple interest is calculated only on the principal amount of a loan or deposit, so it is easier to determine than compound interest.


The question clearly states simple annual interest, thus we go for 2 x (0,06)=0,12 and 2 x (0,07)=0,14, and with X be the initial amount of pamela and Y be the initial amount of joe, it would look like this: 0,12X + 0,14Y= $354

Given: Pamela and Joe invested some money at annual simple interest rates of 6% and 7% respectively. At the end of 2 years they found that together they received a sum of $354 as interest.
Let P = the amount of money Pamela invested
Let J = the amount of money Joe invested

When it comes to simple interest, we have: Interest earned = (annual interest rate)(number of years)(principle)
So, Pamela's interest = (6%)(2)(P) = (0.06)(2)(P) = 0.12P
And Joe's interest = (7%)(2)(J) = (0.07)(2)(J) = 0.14J

We can now write: 0.12P + 0.14J = 354

Target question: What is the value of P + J

Statement 1: One fourth of Pamela’s initial investment is equal to one-fifth of the money invested by Joe.
We can write: (1/4)P = (1/5)J, which is the same as 0.25P = 0.2J, which means we now have the following system of equations:
0.12P + 0.14J = 354
0.25P = 0.2J
Since we have a system of two different linear equations with two variables, we COULD solve the system for P and J, which means we COULD determine the value of P + J - although we would never waste valuable time on test day actually solving the system
Since we could answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: Joe invested $300 more than the sum invested by Pamela.
We can write: J = P + 300, which means we now have the following system of equations:
0.12P + 0.14J = 354
J = P + 300
Once again, we could solve this system for P and J, which means we COULD determine the value of P + J
Since we could answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

avigutman

Could a reasoning-based approach work for such a question? If so, how would you go about it?

Took me over 4 mins to go through the math, only to get it wrong

Certainly, no need for pen and paper here achloes.

The free info describes a weighted average scenario: the $354 must lie somewhere in between 12% and 14% of the total amount invested (consider the extremes - if Pamela invested nothing then $354 is 14% of the total investment, and if Joe invested nothing then $354 is 12% of the total investment).

We are asked whether its possible to find the total amount invested by the two people, and we can already say that it's somewhere in between those two extremes ($354/14% and $354/12%).

Statement (1) gives us the ratio of the two investments (1/4 of P = 1/5 of J, so P:J is 4:5). Would different ratios imply different total amounts of investment? Of course. Any change you make to the ratio in Pamela's favour, for example, would require a greater overall investment to keep the $354 in place, and vice versa. So, if we know what the ratio is, that can only correspond with one particular total amount invested. Sufficient.

Statement (2) gives us the difference between the two investments. Given this $300 difference, any change in overall investment would change the $354 (an upward change would push the interest up and vice versa). There can be only one overall investment that leads to exactly $354, given that the difference is set at $300.

If this isn't intuitive for you, think of it this way: Joe and Pamela invest the exact same amount, Joe at 7% annual and Pamela at 6% annual, for a total of 13% annual. Then, Joe decides to add ANOTHER $300 at 7% annual. If we just look at that extra $300 separately, and deduct the interest (14% of $300) from $354, the remainder ($312) must be 13% of (the total amount invested - $300) so of course we can find what that total amount invested is.

The weighted average perspective is brilliant and really all I needed - thanks so much, Avi!

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