Number theory problems and solutions

Revised SAT Math Problems with Solutions

By Dr. Steve Warner Last week I supplied you with two math problems for the revised SAT. To see that post click the following link: SAT Math Problems for the Revised SAT Today I will solve the first of those two problems, and I will solve the second one next week.

Before I give the solution, I would just like to remind you that if you buy any of my SAT prep books for the current SAT, you will be given access to download the new problems and solutions I am creating for the revised SAT. See last week’s post for details. And if you have any questions about this, simplysend me an email: steve@SATPrepGet800.com You can find all of my books here: http://satprepget800.com/products/ As promised here is a solution to the first question I gave you last week.

Level 2 Number Theory 1.A chemist has a supply of 5.2 liter bottles of a certain solvent that must be shipped to a central warehouse. The warehouse can accept the solvent at the rate of 3 hectoliters per minute for a maximum of 8 hours per day. If 1 hectoliter equals 100 liters, what is the maximum number of bottles that the warehouse could receive from the chemist each day.

(A) 461 (B) 462 (C) 27,692 (D) 83,200 The warehouse can receive 3 ∙60 ∙8 hectoliters of the solvent per day, or equivalently, 1440 ∙100 = 144,000 liters of the solvent per day. Therefore the number of bottles that can be accepted each day is 144,000/5.2 which is approximately 27,692.30769. The maximum number of bottles that the warehouse can accept in one day is therefore 27,692, choice (C).

Notes: (1) Since there are 60 minutes in an hour, “3 hectoliters per minute” is the same as 3 ∙60 = 180 hectoliters per hour.

Similarly, since the warehouse can accept the solvent for a maximum of 8 hours per day, “180 hectoliters per hour” is equivalent to a maximum of 180∙ 8 = 1440 hectoliters per day.

(2) In the above solution we combined the two conversions given in note 1 into a single conversion: “3 hectoliters per minute” is equivalent to a maximum of 3 ∙60 ∙8 = 1440 hectoliters per day.

(3) Since 1 hectoliter equals 100 liters, we can convert hectoliters to liters by multiplying by 100. So 1440 hectoliters is equal to 1440 ∙100 = 144,000 liters.

(4) We can convert between hectoliters and liters more formally by setting up a ratio.

The two things being compared are “liters” and “hectoliters.” liters hectoliters

100 1

x 1440

Now draw in the division symbols and equal sign, cross multiply and divide the corresponding ratio to find the unknown quantity x.

100/1 =x/1440 1x = 100 ∙1440 x = 144,000 (5) Instead of converting 1440 hectoliters to 144,000 liters, and then dividing by 5.2, we can instead convert 5.2 liters to hectoliters, and then divide 1440/.052, giving the same answer of 27,692, choice (C).

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