Problem Solving

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!

Thanks all...really helpful!

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.

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

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.