Word problems are often solved by translating textual information into an arithmetic or algebraic representation. For example, an “odd integer” can be represented by the expression where n is an integer; and the statement “the cost of a taxi trip is $3.00, plus $1.25 for each mile” can be represented by the expression . More generally, translation occurs when you understand a word problem in mathematical terms in order to model the problem mathematically.
• This strategy is used in the following two sample questions.
This is a Multiple-Choice – Select One Answer Choice Question.
A car got 33 miles per gallon using gasoline that cost $2.95 per gallon. Approximately what was the cost, in dollars, of the gasoline used in driving the car 350 miles?
Scanning the answer choices indicates that you can do at least some estimation and still answer confidently. The car used gallons of gasoline, so the cost was dollars. You can estimate the product by estimating a little low, 10, and estimating 2.95 a little high, 3, to get approximately dollars. You can also use the calculator to compute a more exact answer and then round the answer to the nearest 10 dollars, as suggested by the answer choices. The calculator yields the decimal which rounds to 30 dollars. Thus, the correct answer is Choice C, $30.
This is a Numeric Entry Question.
Working alone at its constant rate, machine A produces k liters of a chemical in 10 minutes. Working alone at its constant rate, machine B produces k liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce k liters of the chemical?
Machine A produces liters per minute, and machine B produces liters per minute. So when the machines work simultaneously, the rate at which the chemical is produced is the sum of these two rates, which is liters per minute. To compute the time required to produce k liters at this rate, divide the amount k by the rate to get Therefore, the correct answer is 6 minutes (or equivalent).
One way to check that the answer of 6 minutes is reasonable is to observe that if the slower rate of machine B were the same as machine A's faster rate of k liters in 10 minutes, then the two machines, working simultaneously, would take half the time, or 5 minutes, to produce the k liters. So the answer has to be greater than 5 minutes. Similarly, if the faster rate of machine A were the same as machine B's slower rate of k liters in 15 minutes, then the two machines, would take half the time, or 7.5 minutes, to produce the k liters. So the answer has to be less than 7.5 minutes. Thus, the answer of 6 minutes is reasonable compared to the lower estimate of 5 minutes and the upper estimate of 7.5 minutes.
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