Math Problems G12-1

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Problem of the Week Problem E and Solution The Quest For Ann Sirs? Problem Ann Sirs has already written three mathematics tests, worth 100 marks each. The average of her marks on her rst two tests is calculated and then averaged with her third test mark giving an average of 86%. The average of her marks on her second and third tests is calculated and then averaged with her rst test mark giving an average of 79.5%. The average of her marks on her rst and third tests is calculated and then averaged with her second test mark giving an average of 81.5%. Her fourth mathematics test is tomorrow and it is also out of 100. What mark does Ann need on her test to achieve an overall average of 85% in her mathematics course? Solution Let  a represent Ann’s rst mark,  b  represent Ann’s second mark,  c  represent Ann’s third mark and  d represent Ann’s fourth mark. When the average of her rst and second marks is averaged with her third test mark, the new average is 86 so a+b 2  +c 2  = 86. Multiplying b y 2 gi ve s  a+b 2  + c  = 172. Multiplying by 2 again yields  a + b  + 2c = 344. (1) When the average of her second and third marks is averaged with her rst test mark, the new average is 79.5 so b+c 2  +a 2  = 79.5. Multiplying by 2 gives  b+c 2  + a  = 159. Multiplying by 2 again yields  b + c + 2a = 318. (2) When the average of her rst and third marks is averaged with her second test mark, the new average is 81.5 so a+c 2  +b 2  = 81.5. Multiplying by 2 gives  a+c 2  + b  = 163. Multiplying by 2 again yields  a + c + 2b = 326. (3) By adding equations (1), (2) and (3) we obtain 4 a + 4b + 4c = 988 which simplies to a + b  + c  = 247 after dividing by 4. This means that the sum of her rst three marks is 247. To obtain an average of 85% over four tests, Ann needs a total of 85 × 4 = 340 marks . In othe r words,  a + b + c + d = 340. We know that  a + b + c  = 247 so 247 + d  = 340 and d = 93 follows.  to obtain an average of 85%, Ann needs 93 on tomorrow’s test. It would be a straight forward process to determine Ann’s rst three test marks but our quest did not ask us to do this. However, for the curious, on her rst test Ann got 71, on her second test Ann got 79, and on her third test Ann got 97.

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Problem of the Week

Problem E and Solution

The Quest For Ann Sirs?

Problem

Ann Sirs has already written three mathematics tests, worth 100 marks each. The average of her marks on her rst two tests is calculated and then averaged with her third test mark givingan average of 86%. The average of her marks on her second and third tests is calculated andthen averaged with her rst test mark giving an average of 79.5%. The average of her marks onher rst and third tests is calculated and then averaged with her second test mark giving anaverage of 81.5%. Her fourth mathematics test is tomorrow and it is also out of 100. Whatmark does Ann need on her test to achieve an overall average of 85% in her mathematicscourse?

Solution

Let a represent Ann’s rst mark, b represent Ann’s second mark, c represent Ann’s third markand d represent Ann’s fourth mark.

When the average of her rst and second marks is averaged with her third test mark, the newaverage is 86 so

a + b

2 + c

2 = 86. Multiplying by 2 gives a + b

2 + c = 172. Multiplying by 2 againyields a + b + 2 c = 344. (1)

When the average of her second and third marks is averaged with her rst test mark, the new

average is 79.5 so

b + c

2 + a

2 = 79.5. Multiplying by 2 gives

b+ c

2 + a

= 159. Multiplying by 2 againyields b + c + 2 a = 318. (2)

When the average of her rst and third marks is averaged with her second test mark, the newaverage is 81.5 so

a + c

2 + b

2 = 81 .5. Multiplying by 2 gives a + c

2 + b = 163. Multiplying by 2 againyields a + c + 2 b = 326. (3)

By adding equations (1), (2) and (3) we obtain 4 a + 4 b + 4 c = 988 which simplies toa + b + c = 247 after dividing by 4. This means that the sum of her rst three marks is 247.

To obtain an average of 85% over four tests, Ann needs a total of 85 × 4 = 340 marks. In otherwords, a + b + c + d = 340. We know that a + b + c = 247 so 247 + d = 340 and d = 93 follows.

to obtain an average of 85%, Ann needs 93 on tomorrow’s test.

It would be a straight forward process to determine Ann’s rst three test marks but our questdid not ask us to do this. However, for the curious, on her rst test Ann got 71, on her secondtest Ann got 79, and on her third test Ann got 97.