Difference between revisions of "1999 AHSME Problems/Problem 14"
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==Solution== | ==Solution== | ||
| − | {{ | + | Alina and Tina must sing more than <math>4</math>, but less than <math>7</math>, songs. Therefore, Alina sang <math>5</math> or <math>6</math> songs, and Tina sang <math>5</math> or <math>6</math> songs, with <math>4</math> possible combinations. |
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| + | However, since every song is a trio, if you add up all the numbers of songs a person sang for all four singers, it must be divisible by <math>3</math>. Thus, <math>7 + 4 + a + t</math> must be divisible by <math>3</math>, and <math>a+t - 1</math> must be divisible by <math>3</math>. | ||
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| + | Since <math>10 \le a+t \le 12</math> by the bounds of <math>5 \le a, t \le 6</math>, we have <math>9 \le a + t - 1 \le 11</math>. Because the middle number must be a multiple of <math>3</math>, we set <math>a + t - 1 = 9</math>, leading to <math>a + t = 10</math> and <math>a = t = 5</math>. | ||
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| + | So, the four girls sange <math>4, 5, 5</math> and <math>7</math> songs in trios. That means there were <math>\frac{4 + 5 + 5 + 7}{3} = 7</math> songs sung, and the answer is <math>\boxed{A}</math>. | ||
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==See Also== | ==See Also== | ||
{{AHSME box|year=1999|num-b=13|num-a=15}} | {{AHSME box|year=1999|num-b=13|num-a=15}} | ||
Revision as of 11:02, 8 August 2011
Problem
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 
songs, which was more than any other girl, and Mary sang 
songs, which was fewer than any other girl. How many songs did these trios sing?
Solution
Alina and Tina must sing more than , but less than , songs. Therefore, Alina sang 
or 
songs, and Tina sang or songs, with possible combinations.
However, since every song is a trio, if you add up all the numbers of songs a person sang for all four singers, it must be divisible by 
. Thus, 
must be divisible by , and 
must be divisible by .
Since 
by the bounds of 
, we have 
. Because the middle number must be a multiple of , we set 
, leading to 
and 
.
So, the four girls sange 
and songs in trios. That means there were 
songs sung, and the answer is 
.
See Also
| 1999 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
