Mrs. B. invested $30,000; part at 5%, and part at 8%.

Question

Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate?

A. x = $1,000, y = $29,000.
B. x = $5,000, y = $25,000.
C. x = $10,000, y = $20,000.
D. x = $15,000, y = $15,000.
E. x = $20,000, y = $10,000.

Solution

1) Total investment: x+y=30,000
2) Total interest .05x+.08y=2,100
Solving simultaneously: 
The solutions are: x = $10,000, y = $20,000.

This is an easy algebra problem and I was trying to solve it using the concept of weighted averages. But somehow, my calculations don't add up:

1. The interest of $2,100 is 7% of the total principal.
2. This means that given the rates say, x=5% and y=8% are in the ratio of x:y::1:2
3. Following logically, the ratios compared to the total interest of $2,100 give us a 700x multiplier.
4. So the interest breakup for the two rates is $700 and $1400
5. Similarly, the principal breakup is in the ratio 1:2 with a 10,000x multiplier and we get $10,000 and $20,000

The problem is that the interest earned of $700 and $1400 don't match the respective principles at their respective rates.

My question is, what am I doing wrong/mistakenly assuming? I feel that it is something glaringly obvious that will lead to a 'doh' moment later....

mandyrhtdm · Answer

1. The interest of $2,100 is 7% of the total principal.
2. This means that given the rates say, x=5% and y=8% are in the ratio of x:y::1:2
3. Following logically, the ratios compared to the total interest of $2,100 give us a 700x multiplier.
4. So the interest breakup for the two rates is $700 and $1400
5. Similarly, the principal breakup is in the ratio 1:2 with a 10,000x multiplier and we get $10,000 and $20,000

The problem is that the interest earned of $700 and $1400 don't match the respective principles at their respective rates.

My question is, what am I doing wrong/mistakenly assuming? I feel that it is something glaringly obvious that will lead to a 'doh' moment later....

What's the logic here ???
3. Following logically, the ratios compared to the total interest of $2,100 give us a 700x multiplier.
4. So the interest breakup for the two rates is $700 and $1400

Cause that logic whatever it is is flawed. Interest doesn't have to be in the ratio of the money invested
Interest will be in ratio of the product of money invested and the rate of interest
So here the interest will be in the ration .. 1*5 : 2*8 Clearly gives you 500 & 1600 as interest.

macjas · Answer

hey mandyrhtdm, can you elaborate on this? I'm trying to understand the relationship between the interest percentage ratio and the actual interest amount?

mandyrhtdm · Answer

Case one Interest rate are different and amount are same
Lets say you invest two sums of Rs 100 & Rs 100 on interest rations of 5 and 10
The ration of interest will be the same as the ratio of Interest rate : that is 5 : 10

No suppose you invest two sums of Rs 50 & Rs 100 on interest ratios of 10 & 5 
The ration of interest rate = 10: 5, ratio of interest payout = 5 : 5 . . . . . ??

How come ?? Think about it!
Interest = Rate x Principal how can it be simply dependent on the rate of interest !!!!

macjas · Answer

Thanks for the reply, but what you're saying is obvious and I see that, I just needed to put it into the context of weighted mean. Anyways I figured it out; my problem was that I was switching the weights with the data points. The interest rates are the weights and the principal amounts are the data points. You cannot extrapolate the ratios of the interest amounts from the weights, only the ratio of the data points (principal amounts).

mandyrhtdm · Answer

You will always need both, the weights and the data points to find the ratio of the terms they represent.

metallicafan · Answer

+1 C

The best way to solve this problem is evaluating each choice. The algebra takes more time in this case.

PareshGmat · Answer

Answer = C

2100 = (30000-x) 5/100 + 8x/100

Solving, we get x = 20000

So, answer is 10000 & 20000

Pushpinder · Answer

You can use the concept of allegation on it. (Let me show you how amazing it is)
Overall Interest rate earned: 2100/30000 = 0.07 or 7 %

5% 8%
 7%
1 : 2

Money was invested in 1 : 2 which means it was divided as 10000 and 20000.

If you want to learn allegation, you can watch my video on allegation from this link:

https://www.youtube.com/watch?v=7P4UrnDnrZQ
 
I hope it helps!!!

mau5 · Answer

Assuming each part was invested at the same rate say 8%, then the total investment comes at 2400. That is 300 more than what is given. Thus, the part where she invested at 5%, 3 % of that amount should give 300. That is only in option C

C.

C.

Mrosiji · Answer

Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate?

A. x = $1,000, y = $29,000.
B. x = $5,000, y = $25,000.
C. x = $10,000, y = $20,000.
D. x = $15,000, y = $15,000.
E. x = $20,000, y = $10,000.

The total interest on investment: $2,100 represents 7% of the total investment ($30,000).
[2,100/30,000 = 7/100 or 7%]

This means that part of the investment at 5% plus part at 8% add up to 7% average.

Since 7% is closer to 8% than 5%, this means that the 5% part weighs less than the 8% part.

Looking at the answer choices, I can eliminate D and E. I can also elmininate A (it seems too extreme).

A. x = $1,000, y = $29,000.
B. x = $5,000, y = $25,000.
C. x = $10,000, y = $20,000.
D. x = $15,000, y = $15,000.
E. x = $20,000, y = $10,000.

The next step is to figure out the exact weight of the two investments.

Putting it together; 5% part x+ 8% part y = 7% (part X+ Part Y)

I need to find the ratio. I can make more calculations by looking at the difference between the two parts and the percentage of the final investment.

5% is 2% lower than the final investment, and 8% is 1% higher than the final investment.
We can say that Part x has a -2 differential (5-7=-2) and Part y has a +1 (8-7=1) differential.

We need to make these differentials cancel out therefore we should multiply both differntials by the different numbers so that the positive will cancel out the negative.

X(-2) + Y(+1) = 0

if X = 1 and Y = 2, then 1(-2) + 2 (1) = 0

Therefore the ratio is 1:2
This means there is ONE part at 5% for TWO parts at 8% to produce an average of 7%.

The only answer that matches this ratio is C

C. x = $10,000, y = $20,000

I hope this helps.

For more information: look for lessons on Data points and weighted averages.
I'd also recommend MGMAT's Word problem book p55 to p58. Took me a litte while to get my head around it.
But once you do, you gain a couple of seconds. :-D

holidayhero · Answer

Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate?

A. x = $1,000, y = $29,000.
B. x = $5,000, y = $25,000.
C. x = $10,000, y = $20,000.
D. x = $15,000, y = $15,000.
E. x = $20,000, y = $10,000.

I approached this question like a mixture problem. 
I don't like the wording "part" at 5% and "part" at 8% because we have variables x and y in the answer choices and I have no idea which one is x or y.

Anyway here's how I did it.

0.05(x) + 0.08(y) = 0.07(x+y) 
0.01(y) = 0.02(x)
 y = 2x

Note: 0.07(x+y) because $2,100 is 7% of 30,000

In which answer choice is y = 2(x)? 
Answer C

bumpbot · Answer

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Mrs. B. invested $30,000; part at 5%, and part at 8%.

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Mrs. B. invested $30,000; part at 5%, and part at 8%.

Prep Toolkit

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