Alfredo invested a total of $33,000 in 2 accounts, A and B, with

is there a logical way of doing this?

Sort of.

Since the interest earned for A is double that of B, but the interest rate for A (5%) is a little less than double that of B (3%), this means A needs a slightly bigger piece of the $33,000 in order for its interest earned to be double.

From here, you can take a number that is slightly more than half of $33,000 to test. $18,000 for A and $15,000 are clean numbers and good places to start. From here, you can quickly see that $18,000*0.05 would yield $900 and $15,000*0.03 would yield $450—perfectly double.

$900 + $450 = $1,350

Since the interest earned by Account A was twice the interest earned by Account B and their interest rates were 5% and 3% respectively, it is obvious that Alfred had to invest more than 50% in Account A, otherwise Account A would earn 5%/3% = 1.66 or 66% more than Account B.
Now we could approximate how many more dollars Alfredo was supposed to deposit to Account A:
2 (times more earned)/1.66 (difference in interest rates)=1.2 or 20% more money went to Account A.
Then you could start playing with numbers: more than 50% of total 33k (let's assume 18k) went to Account A, then 15k went to Account B.
18k more than 15k by exactly 20% (3k/15k).
All is left is just to calculate interest earned by each account:
18k*0.05=900 and 15k*0.03=450 that gives us $1350 in total.

Let, a & b the principal of the respective of accounts,

Given ,
5a= 2*3b
Then principal ratio, a : b = 6 :5
By weighted average concept, effective % = 6*5+5*3/ 6+5 = 45/11 %
Total interest= 45/11% of 33000= 1350 (Ans)

 
­Some of these other solutions involve guessing the numbers 18,000 and 15,000 for A and B, which is an interesting method, but I think a lot of people will have trouble getting to these numbers quickly.

In this case, although it's longer than most questions, I think the algebra is your best bet, and it does benefit you to get very quick with your translations and algebra.

However, for many other mixture questions, I do encourage the Teeter Totter method. (It doesn't work well on this one, because we don't have the weights for the Principals; rather, we have a ratio of the amount of interest.)

Here's a playlist with 4 basic examples going over the process for this Teeter Totter method: https://www.youtube.com/playlist?list=PL2exXfCUscn8Hvafet5-IPH1eNNLSjQBP
 ­

We can directly use Simple interest formula and skip some steps to arrive at below ratio:
Pa/Pb= 2*3/5= 6/5

Therefore, Pa= (6/6+5)* 33k= 6*3k= 18k
Similarly, Pb= 5/11*33k= 15k

SIa= 18k*5*1/100
SIb= 15k*3*1/100

Combined SI= (90k+45k)/100= 1350.

There is, actually.

You can start by seeing that if equal amounts of money were invested in A and B, then the cumulative interest rate would be the average of 5% and 3% = 4%.
Since in reality a little more money is invested in A than in B, this means that the average will be skewed toward 5%.
In other words, the average interest rate will be slightly larger than 4%, but less than 5%.

Next, calculate what 4% of $33,000 is.

10% of 33,000 = 3,300

Therefore 5% of 33,000 = 3,300/2 = 1,650

1% of 33,000 = 330

So 4% of 33,000 = 5% of 33,000 – 1% of 33,000 = 1,650 – 330 = 1,320

You can immediately discard some answer choices based on this information.

For instance, the last answer (2,475) is larger than 1,650. This would be achieved only if the average interest rate were higher than 5%, which is is impossible in our case.

The first answer choice (1,290) is less than 1,320. This corresponds to an average interest rate less than 4%. Our analysis above indicated that the interest rate in greater than 4%, so we can exclude this answer choice as well.

The second answer choice (1,320) corresponds exactly to the interest rate of 4%. But since we are looking for an interest rate slightly greater than 4%, this answer choice is also incorrect.

A quick look at the fourth answer choice (1,430) reveals that it is exactly 110 more than 1,320, which is 4% of 33,000. Note that 110 = 1/3 of 330 and 330 is 1% of 33,000. So 1,430 is 1/3 of a percent (i.e. 0.33%) greater than 4%. In other words, 1,430 corresponds to 4.33%.
Although this is greater than 4% (as desired), it is still unconvincing.
The correct answer should be very very close to 4%, since the amount by which the average is tilted isn't that significant.
The question states that the interest earned on Account A should be 2x the interest earned on Account B. But 5% is already almost double 3%. So the difference in the money invested should be really slight, which in turn makes the difference in the average interest rate almost negligible.

The fourth answer choice (1,430) deviates too much, so the only answer choice left is the third (1,350), which is way closer to (but still greater than) our target of 4%.

Another way to solve this without solving for B is going from what you need to solve back-track to what you are given:

- You need to solve this (5A+ 3B) / 100. You can rewrite this into: (3A + 2A + 3B) / 100 = 3(A+B)/100 + 2A/100. You know you are given A + B = 33000, then you substitute it into the equation: 3*33000/100 + A/50 = 990 + A/50

- Next, you are also given that (I am just going to copy what others provided) 5A - 6B = 0 and again, A + B = 33000. Stop here and think: Can you solve for A without having to solve for B? Yes, you can. Multiply the second equation by 6, then add equation 1 and 2 together, you will get: 11A = 6*33000 or A =18000

- A/ 50 is now equal 360

- What do you need to solve now? 990 + 360 = $1,350

Harsha

Alfredo invested a total of $33,000 in 2 accounts, A and B, with annual interest rates of 5% and 3%, respectively. For the first year, the interest earned by Account A was twice the interest earned by Account B. What was the total interest earned by the 2 accounts for the first year?

We can keep this simple. In fact, with the right approach, we can keep the numbers in this question simple enough to do the whole thing in our head.

To do so, we can first determine the ratio of A to B in the following way, which ignores the decimals in the percentages.

\(5A = 2(3B)\)

\(5A = 6B\)

\(A = \frac{6}{5}B\)

Then, to find A and B, we can do the following, which ignores the thousands in 33,000.

\(\frac{6}{5}B + B = 33\)

\(\frac{11}{5}B = 33\)

\(\frac{1}{5}B = 3\)

So, \(B = 15,000\), and \(A = 18,000\).

Then, ignoring the decimals and the thousands, we can do the following.

For A: \(5 × 18 = 90\)

For B: \(3 × 15 = 45\)

\(90 + 45 = 135\)

A. $1,290
B. $1,320
C. $1,350
D. $1,430
E. $2,475

Scanning the choices, we see that (C) matches.

Correct answer: C

KarishmaB
hmmm....I'm not following what's wrong with the approach of the weighted average:

(3-x)/(5-x) = 2/1 -> x = 13/3

13/3*33 = 1430

When averaging interest rates, we cannot use interest earned as the weights. The amount invested is the weight, but we do not have that.

Check: https://anaprep.com/arithmetic-weights- ... d-average/

Since 5% rate gives twice the interest compared with 3%, it means a little more than half of the amount will be invested at 5%. An obvious split to try will be 18k and 15k which works. If that doesn't come to mind, use algebra. Ignore '000s

Amounts invested are x and 33 - x

5% of x = 2 * 3% of (33-x)

5x = 198 - 6x

x = 18

Interest earned at 5% = 5/100 * 18,000 = 900
Interest earned at 3% will be half = 450

Total interest earned = 1350

Answer (C)

5% of A is equal to twice 3% of B.
Thus:
\(5A = 2(3B)\\
\frac{A}{B} = \frac{6}{5}\)

Since A and B sum to 33,000, we get:
\(\frac{A}{B} = \frac{6}{5} = \frac{6000}{5000} = \frac{18000}{15000}\)

Since A earns 5% interest, while B earns 3% interest, we get:
total interest = \((\frac{5}{100} * 18000) + (\frac{3}{100} * 15000) = 900 + 450 = 1350\)

Here, we can create two expressions:

1) A + B = 33000

2) (5/100)A = 2(3/100)B

The second expression basically becomes:

5A = 6B

We can then "add" different forms of the original two expressions (e.g. by multiplying the first expression by 6)

5 A - 6B = 0
+ (6A + 6B) = 6(33000)


11A = 6(33000)
A = 6(3000)
A = 18000

Since A + B = 33000, then B would be 15000

The total interest would therefore be:

5% of 18000 + 3% of 15000

1350

(C) is your answer