greenwich wrote:Bob visited a book fair and bought a certain no. of books. he estimated the total cost on these books by rounding the cost on each of these books to the nearest dollars. was the actual cost minus the estimated cost less than 15 dollars?
1) Bob bought a total of 25 books.
2) the estimated price of 80% of the books is greater than the actual price
If the actual cost is GREATER than the estimated cost, then Bob UNDERESTIMATED the actual cost.
Question stem, rephrased:
Did Bob underestimate the actual cost by less than $15?
Any price LESS THAN HALFWAY between consecutive dollar amounts is rounded DOWN.
Any price EXACTLY HALFWAY OR MORE THAN HALFWAY between consecutive dollar amounts is rounded UP.
Case 1: actual book cost = $1.49
Since the cost is less than halfway between $1 and $2, the estimated cost is yielded by rounding down to $1.
Since the actual cost is $1.49 and the estimated cost is $1, Bob underestimates the actual cost by 49 cents.
Case 2: actual book cost = $7.50
Since the actual cost is exactly halfway between $7 and $8, the estimated cost is yielded by rounding up to $8.
Since the actual cost is $7.50 and the estimated cost is $8, Bob overestimates the actual cost by 50 cents.
Case 3: actual book cost = $25.99
Since the actual cost is more than halfway between $25 and $26, the estimated cost is yielded by rounding up to $26.
Since the actual cost is $25.99 and the estimated cost is $26, Bob overestimates the actual cost by 1 cent.
As Case 1 illustrates, Bob can underestimate the actual cost of each book by AT MOST 49 CENTS.
Statement 1: Bob bought a total of 25 books
Since there are 25 books, and Bob can underestimate the cost of each book by at most 49 cents, he can underestimate the total cost by at most the following value:
(25)(0.49) = less than 12.5.
Thus, Bob underestimated the total cost by less than $15.
SUFFICIENT.
Statement 2: The estimated price of 80% of the books is greater than the actual price
In other words, Bob overestimates the cost of 80% of the books, implying that he could underestimate the cost of 20% of the books.
Case 1: Total number of books = 10.
In this case, the cost of 2 of the books could be underestimated, implying that the total amount of underestimation could not exceed the following value:
(2)(0.49) = 0.98.
Case 2: Total number of books = 1000.
In this case, the cost of 200 books could be underestimated, while the cost of 800 books is overestimated.
Bob could underestimate the cost of 200 books by at most the following value:
(200)(0.49) = 98.
If Bob overestimates the cost of the other 800 books by only 1 cent each, then he overestimates the total cost of 800 books by the following value:
(800)(.01) = 8.
Net underestimation for the total cost of the 1000 books = 98-8 = 90.
Since Bob could underestimate the actual cost by less than $15 or by more than $15, INSUFFICIENT.
The correct answer is
A.
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