OG 12th Problem Solving 81-100

Question 81 – “Data for a certain…”
OG Explanation is fine.

Question 82 – “If n is an integer greater than 6….”
OG Explanation is okay. Alternate Approach (More advanced): Every third number is divisible by 3. If we have three consecutive integers, then the product must be divisible by 3. We can express 3 consecutive integers as:
(n-1)(n)(n+1).
A) Now, (n-4) has the same remainder as (n-1) when divided by 3. CORRECT
B) (n+2) does not have the same remainder as (n+1) when divided by 3. INCORRECT
C) n, (n+3) have the same remainder when divided by 3, so if they have a remainder of 1, then n-5 has a remainder of 2. INCORRECT
D) n(n+4)(n-2) is equivalent to n(n+1)(n-2) in terms of remainders and (n-2) does not have the same remainder as (n-1). INCORRECT
E) n(n+5)(n-6), n and (n-6) have the same remainder. INCORRECT

If this is not clear:
(n-1)(n)(n+1) must be divisible
Every whole number when divided by 3 has a remainder of 0, 1, or 2. As long as we have one-term with a remainder of 0, the product of that term with any other whole numbers must be divisible by 3.

If (n-1) has a remainder of 2, then n has a remainder of 0, and (n+1) has a remainder of 1
If (n-1) has a remainder of 1, then n has a remainder of 2, and (n+1) has a remainder of 0
If (n-1) has a remainder of 0, then the product is divisible by 3.

Question 83 – “The total cost for company X….”
OG Explanation is fine.

Question 84 – “A dealer originally bought 100 identical batteries….”
OG Explanation is fine. Alternatively, you could pick numbers. If q=$200, each battery was purchased at $2. If it was sold for 50% more, it was sold for $3.
Answer A: (3)(200)/200 = $3 CORRECT
Answer B: (3)(200)/2 = $300 INCORRECT
Answer C: 150(200) = way too big INCORRECT
Answer D: (200)/100 + 50 = 52 INCORRECT
Answer E: 150/200 = 0.75 INCORRECT
The correct answer is A.
Khan Academy Link OG 11th #85, 6:40

Question 85 – “In an increasing sequence of 10 consecutive integers the”
OG Explanation is fine. The OG Explanation solves for the first integer and manually calculates. Alternatively, you could think of it like this:
If x is the first number of the sequence, then the first 5 numbers are:
x
x+1
x+2
x+3
x+4

The next five numbers are each five more than the corresponding term.
x+5 = x + 5
x+6 = x+1 + 5
x+7 = x+2 + 5
x+8 = x+3 + 5
x+9 = x +4 + 5
From the problem, we know the first five numbers: x, x+1, x+2, x+3, x+4 sum to 560. The next 5 numbers based on the columns is 560 + 5(5) = 585.

Khan Academy Link OG 11th #86, 9:05

Question 86 – “Machine A produces 100 parts twice as fast….”
OG Explanation is fine.
Khan Academy Link #87 – 0:00

Question 87 – “A necklace is made by stringing N individual beads together in the repeating pattern…”
OG Explanation is mediocre. I think it is easier to recognize the pattern.

Because we have a repeating pattern with an interval of 5, it becomes clear that you can only have a white bead when the units digits is 3 or 8. Only answer choice E ends in a 3 or 8.

Khan Academy Link OG 11th #88 1:05

Question 88 – “In the xy-coordinate system, if(a,b) and (a+3, b+k) are two points on the line defined….”
OG Explanation is okay. Alternatively, you can pick numbers but you must state with x and y as points a and b are defined by the equation x = 3y-7 . If y=3, x=2. Let (a,b) = (2,3). To find, (a+3, b+k), we need to find out what the y coordinate would be when x = a+3 or 5. When x = 5:
5 = 3y – 7
12 = 3y
y = 4
going back
b + k = 4
b was originally 3
3 + k = 4
k = 1

The correct answer is D.
Khan Academy Link OG 11th #89 3:30

Question 89 – “Is s is the product of the integers…”
OG Explanation is okay.

Question 90 – “If Jakes loses 8 pounds, he will weigh twice as much as his sister…”
OG Explanation is fine. This problem is similar to the age problems that are like “in 4 years, John will be twice the age….”

Question 91 – “A certain store sells all maps….”
OG Explanation is fine.

Question 92 – “A store reported total sales of $385….”
OG Explanation is fine.

Question 93 – “If the median of the numbers in list I”
OG Explanation is fine. Remember that when there are an odd number of terms in a list of numbers, the median is the middle number.
Khan Academy Link OG 11th #93 –  Begins at 1:40

Question 94 – “In a certain city, 60 percent of the registered…”
OG Explanation is fine. Alternatively, you could pick numbers. Since this is a percent problem, 100 is a good choice. There are 100 people in the city, 60% or 60 are Democrats and the rest, 40% or 40, are Republicans. 75% of the Democrats (i.e. 75% of 60) or 45 Democrats will vote for Candidate A. 20% of the Republicans (i.e. 20% of 40) or 8 Republican will vote for Candidate A. There will 45+8 or 53 people voting for Candidate A out of 100. Therefore, 53% are expected to vote for A.
Khan Academy Link OG 11th #94 – 3:15

Question 95 –“1/2 + ….”
OG Explanation is fine.
Khan Academy Link OG 11th #96 Begins at 0:00

Question #96
OG Explanation is fine. If the ratio by mass is 2 to 16, then for every 18 grams of water we have 16 grams of oxygen. 144/18 = 8, So we have 8 sets of 18 grams of water. Each set has 16 grams of oxygen; 8 sets multiplied by 16 grams of oxygen per set = 128 grams of oxygen.
Khan Academy Link OG 11th #97 – 1:20

Question 97 – “If x(2x+1) = 0…..”
OG Explanation is fine.
Khan Academy Link OG 11th #98 – 4:55

Question 98 – “On a scale that measures…”
OG Explanation is too short. If we start with a reading of 3, a reading of 4 is 10 times more intense,. A reading of 5 is 10 or 101 times more intense that a reading of 4 and a 100 or 102 times more intense than a reading of 3. This pattern continues until a reading of 8 is 105 more intense than a reading of 3. (Trivia: for science people out there, this is how a pH scale works).
Khan Academy Link OG 11th #99 – 7:40

Question 99 – “For the positive numbers,”
OG Explanation is okay. I think it is easier to plug in numbers. Let n=1,
The sequence is 1, 2, 3, 5, 9. The median is 3. To calculate the mean:

The mean is 4 so the mean is 1 greater than the median.

Question 100 – “T = (5/9)(K-32)…”
OG Explanation is fine.
Khan Academy Link OG 11th #102 – 4:15


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