'Average' problem
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 neelgandham
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Sam earned a $2,000 commission on a big sale, raising his average commission by $100. If Sam's new average commission is $900, how many sales has he made?
Let total number of sales prior to the big sale be S and
the average commission prior to the big sale be A
Total commission earned prior to the big sale = A*S
Total commission earned after the big sale = A*S + 2000
Total number of sales including the big sale = S+1
Question can be rephrased to What is the value of S+1 ?
900 = Average commission prior to the big sale + 100 = A+100
Implies A = 800
Average commission after the big sale be A = Total commission earned after the big sale/Total number of sales including the big sale
=> (A*S + 2000) / (S+1) = 900
=> (800S + 2000)/(S+1) = 900
=> 800S + 2000 = 900S + 900
=> 100S = 1100
=> S = 11
S+1 = 12 = Total number of sales he made !
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Let total number of sales prior to the big sale be S and
the average commission prior to the big sale be A
Total commission earned prior to the big sale = A*S
Total commission earned after the big sale = A*S + 2000
Total number of sales including the big sale = S+1
Question can be rephrased to What is the value of S+1 ?
900 = Average commission prior to the big sale + 100 = A+100
Implies A = 800
Average commission after the big sale be A = Total commission earned after the big sale/Total number of sales including the big sale
=> (A*S + 2000) / (S+1) = 900
=> (800S + 2000)/(S+1) = 900
=> 800S + 2000 = 900S + 900
=> 100S = 1100
=> S = 11
S+1 = 12 = Total number of sales he made !
p.s: For quick responses,
Please post PS questions here: https://www.beatthegmat.com/problemsolvingf6.html
Please post DS questions here: https://www.beatthegmat.com/datasufficiencyf7.html
Anil Gandham
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You could also do this using allegations.
New commission = Old + 100 = 900.
Old = 800. Resultant = 900 and Addition is 2000. ( 8 and 20 mixed to give 9)
So, (209) / (98) = 11/1. ( 11 of 8 and 1 of 20)
So, 11 + 1= 12 commissions.
New commission = Old + 100 = 900.
Old = 800. Resultant = 900 and Addition is 2000. ( 8 and 20 mixed to give 9)
So, (209) / (98) = 11/1. ( 11 of 8 and 1 of 20)
So, 11 + 1= 12 commissions.
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Solution:
Since his new average commission is $900, his old average commission must be $800. We can create the equation where n is the number of sales Sam has made before the big sale.
(800n + 2000) / (n + 1) = 900
800n + 2000 = 900n + 900
1100 = 100n
11 = n
Including the big sale, Sam has made 12 sales.
Answer: 12
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Note: If Sam's average commission increased by $100 to $900, then his FORMER average commission was $800 and his NEW average commission is $900
When it comes to averages, we know that average value = (sum of n values)/n
We can rewrite this into a useful formula: sum of n values = (average value)(n)
Let n = the number of sales Sam made to calculate his FORMER average commission.
When we apply the above formula, we get: sum of FORMER commissions = 800n
Once Sam collects his $2,000 commission, the NEW sum of commissions = 800n + 2000
At this point, n + 1 = the total number of commissions (since we just added the $2000 commission)
Since Sam's NEW average commission is $900, we can write: 800n + 2000/(n + 1) = $900
Multiply both sides of the equation by (n + 1) to get: 800n + 2000 = 900(n + 1)
Expand the right side: 800n + 2000 = 900n + 900
Subtract 800n from both sides: 2000 = 100n + 900
Subtract 900 from both sides: 1100 = 100n
Solve: n = 11
The question asks us to determine how many sales Sam HAS made, which means we must include the latest sale.
Since n represents the number of sales Sam made to calculate his FORMER average commission, we must add the latest sale (the one that landed Sam a $2,000 commission).
So the total number of sales made = 11 + 1 = 12
Answer: 12
New average = $ 900
Since average is increased by $100,
Old average = $ 800
If the total sales were x(before the big sale),
Then, Total commission = $ 800x
By adding the new commission of $ 2000,
Total comission(after the big sale) = $ 800x + 2000
Also, total sales will be x+1.
Thus, $ 2000 + 800x = $ 900(x+1)
100x = 1100
x = 11.
Total sales = x + 1 = 12
Since average is increased by $100,
Old average = $ 800
If the total sales were x(before the big sale),
Then, Total commission = $ 800x
By adding the new commission of $ 2000,
Total comission(after the big sale) = $ 800x + 2000
Also, total sales will be x+1.
Thus, $ 2000 + 800x = $ 900(x+1)
100x = 1100
x = 11.
Total sales = x + 1 = 12