John and Ingrid pay 30% and 40% tax annually, respectively.

Question

John and Ingrid pay 30% and 40% tax annually, respectively. If John makes $56000 and Ingrid makes $72000, what is their combined tax rate?

A. 32%
B. 34.4%
C. 35%
D. 35.6%
E. 36.4%

It is one of my favorite problems from the Veritas Arithmetic book.. I like to use Weighted Averages here...

KarishmaB · Accepted Answer

John and Ingrid pay 30% and 40% tax annually, respectively. If John makes $56000 and Ingrid makes $72000, what is their combined tax rate?

a. 32%
b. 34.4%
c. 35%
d. 35.6%
e. 36.4%

Great solution everyone! Yes, the answer is (D) 35.6.

Now let me explain why I like this question. The big numbers - 56000 and 72000 have limited utility. I could have as well given 86380 and 111060 as their salaries; I would have still obtained the same answer!
We need to find the weighted average of their tax rate. The important thing is their individual tax rates and the ratio of their salaries 7:9 (56000:72000), not the actual salaries

Attachment:

Ques1.jpg [ 3.44 KiB | Viewed 25957 times ]

The length between 30% and 40% i.e. 10% is split in the ratio 9:7.
So the average will be at (9/16)*10 + 30 = 35.6%.

So you don't have to do any long calculations. If there is a doubt in this method, check out this link:
https://gmatclub.com/forum/tough-ds-105651.html#p828579

When we want to find the average of lets say age of 2 groups - one group has average age of 30 yrs and there are 10 people in it and the other group has average age of 40 yrs and there are 40 people in it, all we need is the relative weights (relative number of people) in the two groups i.e. 1:4. It doesn't matter whether there are 10 and 40 people or 100 and 400 people... It's exactly the same concept here.

Let me know if there is a doubt in this concept.

and when we say, "A makes $100,000," we mean gross by default. Taxes are a part of your salary just like your other expenses.

diebeatsthegmat · Answer

(0.3*56000+0.4*72000)/(72000+56000)=0.35625
D
thank you

craky · Answer

It would be good to specify whether they the salary they make is already taxed or not.

But anyway, is there any way to solve

(0.3*56000+0.4*72000)/(72000+56000)=0.35625

in hand?

santosh18 · Answer

Income TAX
J - 56,000 16,800
I - 72,000 28,800
-----------------------
T- 128,000 45,600

Part X
----- = ------
Whole 100

45,600 X
-------- = -----
128,000 100

45,600 * 100
X = --------------
 128,000

4560
X = ------ = 35.6
 128

short cut

4560 4560 456 
X = ------ =(approx) ------- = ---- = 35.07 
 128 130 13

As denominator has been increased answer choice should be greater than 35.07

So the Ans is 35.6%

mehtakaustubh · Answer

i go with 
D
(0.3*56000+0.4*72000)/(72000+56000)=0.35625

dkj1984 · Answer

Hi Karishma,

I really love the weighted averages shortcut method you use every time,but I get confused in using it.

In the problem above,please advise,why you have used 9:7 instead of 7:9?

Thanks!

regards,

KarishmaB · Answer

When we use the formula of weighted averages, we say that one quantity is A1 and its weight is w1. Another is A2 and its weight is w2. We have to find the average Aavg. 
The formula is w1/w2 = (A2 - Aavg)/(Aavg - A1)

Notice that you have w1 in numerator on LHS and A2 in numerator on RHS.

dcastan2 · Answer

I am lost at (9/16)*10 + 30 = 35.6%. where YOU add 30. I want to learn how to use this method completely so I can have it in my back pocket ready to use. I feel that we already found the ratio and then flipped it so it would be the distance and could put it in the chart. We divided 9/16 to find the middle number that was missing and multiplied by 10 because that was the weight? Is the only reason because 10+30 = the original 40?

KarishmaB · Answer

The distance between 30 and 40 i.e. 10 has to be split in the ratio 9:7. The average is 9 (of these 16 units) to the right of 30. So you find 9/16th of 10 which is 90/16 and add it to 30.

Before you work on this question, check out the links mentioned in the post above. I have discussed simpler numbers in them. They will give you a feel for the method. Thereafter, you can solve any question using this method.

mbaiseasy · Answer

Using weighted averages...

(1) When 30 and 40 has equal weight or weight = 1/2, the answer would be 35.
(2) When 40 has larger weight than 30, the answer would be in between 35 and 40. Unfortunately, we have 2 answer choices D and E that fit that condition so we need to narrow down our range.
(3) Get 72000/128000 = 72/128 = 9/16. 9/16 is a little above 8/16 = 1/2. Thus, our answer is just a little above 35.

Answer: D

For details on Weighted Averages:  Weighted Averages

Narenn · Answer

Here is another approach

Asifpirlo · Answer

(0.3 * 56000 + 0.4 * 72000) / (72000 + 56000) = 0.356 = 35.6% = D (Answer)

jlgdr · Answer

Ratio is 56/72 = 8/9 Weighted average is therefore ((30)(8) + (40)(9))/17 = 600/17=35.6% Hence answer is D Hope it helps Cheers! J

Raihanuddin · Answer

ratio of their salaries 7:9 (56000:72000)
30 40
 X
40-x x-30
(40-x):(x-30) = 7: 9
X= 35.6%

mvictor · Answer

I always get messed up when trying to use this method...ended up the direct one...
56,000 = 100%
5,600=10%
11,200 = 20%
28,000 = 50%
16800 = 30%

72,000 = 100%
36,000 = 50%
7,200 = 10%
28,800 = 40%

now..
28,800+16,800=45600
total amount: 128,000

45,600/128,000 = 456/1280 = 228/640 = 114/320 = 57/160. clearly it is more than 33% since 60/180 is 1/3 or 33%. A is out right away...
now...long division....
57/160 = 0.35625

so 35.625%

closest one: D

Kritisood · Answer

Diagrammatic approach

Basshead · Answer

56,000 : 72,000
7: 9

72,000 is weighted more heavily so we can eliminate A, B, and C right off the bat.

9/16 * 10 = 5.625

30 + 5.625 = 35.625

Answer is D.

rajatjoshi · Answer

A better approach could be to first take the ratio of salaries which is 7:9 then find the weighted average of tax rate that is  /(7 + 9) = 35.6%.

Hope this helps

BKD5050 · Answer

Hi Karishma,

When to use alligation method and when to use scaling method or is it that both works on all such q types? Also how to quickly sense that we can use scaling method in the given question?

John and Ingrid pay 30% and 40% tax annually, respectively.

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