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The most interesting thing about some of the tough ACT Math word problems is that you can potentially solve them in more than one way! “Do the Math” refers to using traditional algebra to solve. “Picking Numbers” is an awesome strategy and can work on a wide range of questions: problems with unknown starting values, and problems with variables in the answer choices.  All you have to do is choose your own value, then solve! Let’s look at an example question and see how it could be solved two way!

Saul can paint a room in 6 hours, while Carrie can paint a room in 8 hours. They decide to paint three rooms of equal size together, but after five hours, Carrie gets tired and leaves. Approximately how many hours will it take Saul to finish painting the three rooms by himself?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

## The “Picking Numbers” Method

This is a hard word problem asking about rates. To make it easier, let’s choose a number to represent the amount of square footage needed to be painted in each room. And let’s choose a number that both 6 and 8 will go into evenly so that we don’t have to deal with decimals. Let’s say each room had 120 square feet to paint. Then Saul can paint at a rate of 120/6 = 20 feet per hour, and Carrie can paint at a rate of 120/8 = 15 feet per hour.

We know they are painting three rooms, so that will be 3 x 120 = 360 square feet that needs to be painted. They work for five hours together before Carrie stops, so in those five hours together they are painting 20 + 15 = 35 square feet per hour.

35 square feet x 5 hours = 175, so they have painted 175 square feet out of the 360 total when Carrie leaves. The question asks how long it will take Saul to finish. She has 360 – 175 = 185 square feet left, and we know she paints at a rate of 20 feet per hour, so it will take her 185/20 = 9.25 hours to finish. The closest “approximate” answer is (D).

## The “Do the Math” Method

Saul can do 1/6th of a room in 1 hour, while Carrie can do 1/8th of a room. Together, they can paint 1/6+ 1/8 = 4/24 + 3/24 = 7/24th of a room in 1 hour. In five hours, they paint 35/24th of a room, or 1 complete room and 11/24th of the second room. That leaves Saul to paint 13/24th of the second room, and the entire third room (24/24) by herself, or 13 + 24 = 37/24th. Since her hourly work-rate is 4/24, it will take her 37/4 = 9.25 hours to complete the job.

If you’re a strong ACT Math test-taker, look for opportunities in your practice to solve questions in more than one way! Sometimes it’s fun to try a strategy like picking numbers, and then also attempt to the math. No one way is “better”!