Finley Fogg is a student at Drudgery High. After every test he figures his cumulative average, which he always rounds to the nearest whole percent. (So 85.49 would round down to 85, but 85.50 would round up to 86.)
Today he had two tests. First he got 75 in French, which dropped his average by 1 point. Then he got 83 in History, which lowered his average another 2 points.
What is his average now?
Note: The individual test scores correspond to integer values.
x=original average. Any number from 0 to (presumably) 100
(This is an unstated assumption I'll make for this solution)
n=original number of tests. Any whole number.
also xn=the sum of the test scores must also be a whole number.
Since we don't know x or n. Make a slider for n and see what happens.
f(x)=round(x). The original average reported to the nearest whole percent.
g(x)=round((xn+75)/(n+1))+1. Adding a 75. The +1 makes them equal again.
h(x)=round((xn+158)/(n+2))+3. Adding another 83. Since the average dropped 2 more the +3 will make them equal again.
To see where the step functions overlap, make each graph a different thickness and contrasting color.
The smallest n where they overlap is n=11 from x=98.4545 to 98.4999.
This works since 1082.9995<xn<1083.4989 means xn=1083
***
Analysis:
xn=1083. x=98.45. Round to 98.
xn+75=1158. 1158/12=96.5 which rounds to 97. (Down by 1)
xn+158=1241. 1241/13=95.461 which rounds to 95. (Down by 2)
A SOLUTION.
Answer to the question: his new average is 95.
Are there more solutions?
The next smallest is n=12 from x=99.396 to 99.499
This works 1192.752<xn<1193.988 means xn=1193
***
Analysis:
12 test total 1193. Average=1193/12=99.417 rounded to 99.
13 test total 1268. Average=1268/13=97.538 rounded to 98.
14 test total 1351. Average=1351/14=96.5 rounded to 97.
A near miss!
When n=12 there is another overlap but it's above 100 so it would require test results above 100.
When n>12 all of the overlaps are above 100.
Here's a portion of the graph of the solution when n=11.
https://www.desmos.com/calculator/mdacdiwyun
I'm in awe of whomever was able to fine tune this problem to have exactly one solution. Finding 75 and 83 must have took some hard thinking.
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Posted by Jer
on 2023-12-30 17:54:18 |
Solution by graphing
