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As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first six bicycles he sells, $12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true?
I. y > 2x
II. y > x
III. y > 3
I. y > 2x
II. y > x
III. y > 3
- a)I only
- b)II only
- c)I and II
- d)II and III
- e)I, II, and III
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
As a bicycle salesperson, Norman earns a fixed salary of $20 per week ...
. UNCERTAIN: It depends on how many bicycles Norman sold.
For example, if x = 4, then Norman earned $44 [= $20 + (4 × $6)] last week. In order to double his earnings, he would have to sell a minimum of 9 bicycles this week (y = 9), making $92 [= $20 + (6 × $6) + (3 × $12)]. In that case, y > 2x.
However, if x = 6 and y = 11, then Norman would have earned $56 [= $20 + (6 × $6)] last week and $116 [= $20 + (6 × $6) + (5 × $12)] this week. In that case, $116 > 2 × $56, yet y < 2x.
So, it is possible for Norman to more than double his earnings without selling twice as many bicycles.
II. TRUE: In order to earn more money this week, Norman must sell more bicycles.
III. TRUE: If Norman did not sell any bicycles at all last week ( x = 0), then he would have earned the minimum fixed salary of $20. So he must have earned at least $40 this week. If y = 3, then Norman earned $38 [= $20 + (3 × $6)] this week. If y = 4, then Norman earned $44 [= $20 + (4 × $6)] this week. Therefore, Norman must have sold at least 4 bicycles this week, which can be expressed y > 3.
For example, if x = 4, then Norman earned $44 [= $20 + (4 × $6)] last week. In order to double his earnings, he would have to sell a minimum of 9 bicycles this week (y = 9), making $92 [= $20 + (6 × $6) + (3 × $12)]. In that case, y > 2x.
However, if x = 6 and y = 11, then Norman would have earned $56 [= $20 + (6 × $6)] last week and $116 [= $20 + (6 × $6) + (5 × $12)] this week. In that case, $116 > 2 × $56, yet y < 2x.
So, it is possible for Norman to more than double his earnings without selling twice as many bicycles.
II. TRUE: In order to earn more money this week, Norman must sell more bicycles.
III. TRUE: If Norman did not sell any bicycles at all last week ( x = 0), then he would have earned the minimum fixed salary of $20. So he must have earned at least $40 this week. If y = 3, then Norman earned $38 [= $20 + (3 × $6)] this week. If y = 4, then Norman earned $44 [= $20 + (4 × $6)] this week. Therefore, Norman must have sold at least 4 bicycles this week, which can be expressed y > 3.
Most Upvoted Answer
As a bicycle salesperson, Norman earns a fixed salary of $20 per week ...
Y > 2x
II. y > 12
III. y > x
To determine which of these statements must be true, we can start by setting up the equation for Norman's earnings last week and this week.
Last week's earnings: $20 + $6x
This week's earnings: $20 + $6(6) + $12(6) + $18(y-12)
We know that this week's earnings are more than twice last week's earnings, so we can set up the inequality:
$20 + $6(6) + $12(6) + $18(y-12) > 2($20 + $6x)
Simplifying this inequality, we get:
$20 + $36 + $72 + $18y - $216 > $40 + $12x
Combine like terms:
$92 + $18y - $216 > $40 + $12x
Simplify further:
$18y - $124 > $12x
Now let's analyze each statement:
I. y > 2x
If y > 2x, then we can substitute 2x for y in the inequality above:
$18(2x) - $124 > $12x
$36x - $124 > $12x
$24x > $124
x > $5.17
Since x represents the number of bicycles sold last week, it cannot be a fractional value. Therefore, statement I cannot be true.
II. y > 12
This statement does not provide any information about Norman's earnings, so it cannot be determined from the given information. Therefore, statement II cannot be true.
III. y > x
If y > x, then we can substitute x for y in the inequality above:
$18(x) - $124 > $12x
$18x - $124 > $12x
$6x > $124
x > $20.67
Since x represents the number of bicycles sold last week, it cannot be a fractional value. Therefore, statement III cannot be true.
In conclusion, none of the statements must be true based on the given information.
II. y > 12
III. y > x
To determine which of these statements must be true, we can start by setting up the equation for Norman's earnings last week and this week.
Last week's earnings: $20 + $6x
This week's earnings: $20 + $6(6) + $12(6) + $18(y-12)
We know that this week's earnings are more than twice last week's earnings, so we can set up the inequality:
$20 + $6(6) + $12(6) + $18(y-12) > 2($20 + $6x)
Simplifying this inequality, we get:
$20 + $36 + $72 + $18y - $216 > $40 + $12x
Combine like terms:
$92 + $18y - $216 > $40 + $12x
Simplify further:
$18y - $124 > $12x
Now let's analyze each statement:
I. y > 2x
If y > 2x, then we can substitute 2x for y in the inequality above:
$18(2x) - $124 > $12x
$36x - $124 > $12x
$24x > $124
x > $5.17
Since x represents the number of bicycles sold last week, it cannot be a fractional value. Therefore, statement I cannot be true.
II. y > 12
This statement does not provide any information about Norman's earnings, so it cannot be determined from the given information. Therefore, statement II cannot be true.
III. y > x
If y > x, then we can substitute x for y in the inequality above:
$18(x) - $124 > $12x
$18x - $124 > $12x
$6x > $124
x > $20.67
Since x represents the number of bicycles sold last week, it cannot be a fractional value. Therefore, statement III cannot be true.
In conclusion, none of the statements must be true based on the given information.
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