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16 Oct 2019, 00:14
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
72% (02:14) correct
28%
(02:56)
wrong
based on 87
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In a certain supermarket, a triangular display of cans is arranged in 8 rows, numbered 1 through 8 from top to bottom. Each successively numbered row contains 3 more cans than the row immediately above it. If there are fewer than 95 cans in the entire display, how many cans are in the fifth row?
A. 10
B. 11
C. 13
D. 14
E. 16
A. 10
B. 11
C. 13
D. 14
E. 16
16 Oct 2019, 00:22
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Bunuel
Let the number of cans in the first row = x
Each row below has 3 more than the row above it
--> Total number of cans in 8 rows = x + (x+3) + (x+6) + . . . . . . (x+21) = 8x + 84
Total is less than 95
--> 8x + 84 < 95
--> 8x < 11
--> Only possible value of x = 1
--> Number of cans in 5th row = x + 12 = 1 + 12 = 13
IMO Option C
20 Oct 2019, 12:36
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Solution
Given
- • In a certain supermarket, a triangular display of cans is arranged.
• The display has 8 rows, numbered 1 through 8 from top to bottom.
• Each successively numbered row contains 3 more cans than the row immediately above it.
To find
- • The number of cans in the fifth row if there are fewer than 95 cans in the entire display.
Approach and Working out
Let row 1 has ‘a’ cans.
- • Hence, row 2 has ‘a + 3’ cans.
• Row 3 = ‘a + 6’ cans
• Similarly, each successive row has 3 more cans.
- o Hence, cans are in an arithmetic progression with:
o a = being the number of cans in the first row and,
o 3= being the common difference,
o 8 = number of rows
• Therefore, total cans =\(\frac{8}{2}\)(2a + (8-1)3) = 4(2a + 21)
- o 4(2a + 21) < 95
o 2a + 21 < \(\frac{95}{4}\)
o 2a < \(\frac{95}{4}\) – 21
o 2a < \(\frac{11}{4}\)
o a < \(\frac{11}{8}\)
o a < 1 + \(\frac{3}{8}\)
• Since the number of balls has to be a positive integer, a must be 1.
• Thus, number of balls in the 5th row = 1 + 4*3 = 13
Thus, option C is the correct answer.
Correct Answer: Option C
