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OG 2016 PS #9: Dividing by Decimals

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by southsideslim03 » Sat Feb 06, 2016 5:35 pm
In the problem, we are to determine the value of an item when tax is paid at 7% on anything over $1,000. The total tax that was paid is $87.50. The solution says this:

7% of value in excess of $1000 = 87.50
0.07(x-1000) = 87.50
x-1000 = 1,250
x=2,250

My question is, are there any shortcuts or quick tips that could be used to solve the section of the solution in which you divide 87.50 by .07? I know the trick on dividing by 5 from Magoosh is multiply N by 2 and then divide by 10, but that gets me 1750 which is no where close to 1250.

Any help is greatly appreciated.

Thanks!
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by [email protected] » Sat Feb 06, 2016 6:03 pm
Hi southsideslim03,

You can approach this question by TESTing THE ANSWERS. Instead of setting this up algebraically, you can use the 'spread' of the answer choices to your advantage.

Here, if the total value of the item were $2,000, then the tax would be $1,000 x .07 = $70. Since the tax is supposed to be $87.50, we know that the answer has to be GREATER than $2,000.

Try TESTing ANSWER D: $2,400. How much tax would that generate? If it's too high or too low, what answer would have to be correct?

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by Brent@GMATPrepNow » Sat Feb 06, 2016 6:18 pm
When Leo imported a certain item, he paid a 7% import tax on the portion of the total value of the item in excess of $1000. If the amount of the import tax that Leo paid was $87.50 what was the total value of the item?
A $1600
B $1850
C $2250
D $2400
E $2750
Here's the original question and a step-by-step algebraic solution.

Let T = the TOTAL value of the item.

Leo paid a 7% import tax on the portion of the total value of the item in EXCESS of $1000
So, Leo pays tax on the amount that's GREATER then $1000
So, Leo pays 7% tax on (T - 1000)
We can write: import tax = 7% of (T - 1000)

The amount of the import tax that Leo paid was $87.50
So, we write: $87.50 = 7% of (T - 1000)
Or: $87.50 = 0.07(T - 1000)
Expand to get: 87.50 = 0.07T - 70
Add 70 to both sides to get: 157.5 = 0.07T

NOTE: At this point, you might just plug in the answer choices to see which one makes the above equation true.
Or....
Divide both sides by 0.07 to get: 157.5/0.07 = T
Solve: 2250 = T

Answer: C

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by southsideslim03 » Sat Feb 06, 2016 7:19 pm
Thanks all...really helpful!
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by MartyMurray » Sat Feb 06, 2016 9:43 pm
When Leo imported a certain item, he paid a 7% import tax on the portion of the total value of the item in excess of $1000. If the amount of the import tax that Leo paid was $87.50 what was the total value of the item?
A $1600
B $1850
C $2250
D $2400
E $2750
In working with percentages, converting them into 1% and 10% increments can be useful. For instance you can start off calculating 10% of any number by dividing by 10, and then use 10% increments to make your work easier. For instance 10% of 1260 is 126. Depending on the situation, having that 126 to work with can be pretty handy.

So here's a shortcut for this question, one that you could use to do all the work in your head if you felt like it.

We are looking for a total quantity, 1000 + x, such that 87.5 is 7% of x.

Divide 87.5 by 7 to get 1% of x.

87.5/7 = 70/7 + 17.5/7 = 10 + 2.5 = 12.5

So 12.5 = 1% of x, and x = 1250

So 1000 + x = 2250.

The correct answer is C.
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by regor60 » Tue Feb 09, 2016 10:52 am
It can be useful to be able to recognize what multiples of certain percentages equal 100.

For example, 14 x .07 is roughly 100. 7 x 14%, likewise. 7 x 15%, only 5% in excess of 100.

In the problem, you can see that a $2000 price would generate $70 bill based on the 7% given. Since that's less than the $87.50, it establishes a minimum.

Similarly, you can hypothesize a tax bill of $100, which by remembering the 14 times multiple above, would suggest a $2400 price as a maximum.

So your looking for an answer between the two, and there is only one choice
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by Matt@VeritasPrep » Thu Feb 11, 2016 5:23 pm
You could also just say that

7% of something = 87.5

.07 * something = 87.5

7 * something = 8750

something = 1250

Since we only pay tax above $1000, we must be $1250 ABOVE $1000, or $1000 + $1250, or $2250.
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