ID: 70feb725
(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)
Comment: Some form of a system of equations is fitting to model this word problem, but some quick arithmetic is just as effective.
Method 1: Since the question asks for a single unknown—how many miles Morgan biked—and the correct answer must be one of the four numbers listed, it might be fastest to simply work backwards to corroborate known information. The process takes little more than mental math and good organization. Because 100 would split the total number of miles down the middle, it might be a good place to start.
100 miles at 10 mph = 10 hours (biking)
100 miles at 5 mph = 20 hours (running)
This might look like "b: 100 @10 = 10; r: 100 @5 = 20," something codified that would make sense to you, but not to the person sitting next to you. (You need not write everything out.) In any case, since the passage tells us that Morgan biked for twice as many hours as she ran, this answer choice cannot be correct. Importantly, though, we can rule out (A): if she had biked for 80 miles, then she would have had to run the remaining 120 miles, and we know, once again, that she spent more time biking. One of the remaining answers must be correct, and it does not matter which one we choose: either it will confirm what the passage tells us, or we will know that the other answer choice must be correct. How about answer choice (C)?
120 miles at 10 mph = 12 hours (biking)
80 miles at 5 mph = 16 hours (running)
Clearly, the balance is still tilted in the wrong direction, and (D) must be the correct answer. You could quickly check if you wanted, given that the process above may have taken less than a minute up to this point.
160 miles at 10 mph =16 hours (biking)
40 miles at 5 mph = 8 hours (running)
Of course, this is the only option that tells us the same thing we know already, that Morgan spent twice as much time biking as she did running.
Method 2: It is easy enough to model the equation x + y = 200, and it might help you see what would have to be true, as opposed to working out all the mental math above.
To get that table, click on the + symbol in the top-left corner, above the equation, and type in the answer choices for x. To make those points visible, hover over each one in the graph and click to make it stick.
Method 3: As always, there is nothing wrong with an algebraic solution, even if it might not be the fastest way in this case. The first sentence provides information on rates and distance, leaving us with an equation that would yield an answer in time.
Equation 1: r miles at 5 mph + b miles at 10 mph = total time
(r mi)(1 h/5 mi) = r/5 hours and (b mi)(1 h/10 mi) = b/10 hours
The second sentence starts by giving us the total number of running and biking miles, 200. Thus,
Equation 2: run (r) + bike (b) = 200
After the comma, we learn that Morgan biked for twice as many hours as she ran. Keep this simple. In numerical terms, if she ran 1 hour, she biked 2; if she ran 3 hours, she biked 6. Algebraically, then, using the figures from above,
hours biked = 2 * hours run
(b/10) = 2(r/5)
b/10 = 2r/5
b = 4r
Substituting into our second equation above,
r + (4r) = 200
5r = 200
r = 40
If Morgan ran for 40 miles, then she must have biked the remaining 160 miles, and the correct answer is (D).
Note that these are not the only ways to solve this problem, but any method above should suffice to help you solve the question comfortably in about a minute and a half or less.
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