# AMC8 Test 1

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#### How many subgroups of two elements may be eliminated from the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 so that the remaining numbers have a mean (average) of 6?

Correct! Wrong!

1st option

We know their sum is 54 because there would be 9 items after elimination and their mean is 6. We also know that the pre-removal total of the collection is 66. As a result, the sum of the two eliminated items is 66 – 54 = 12. There are only five subsets of two elements that add up to 12: {1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}.

2nd Option

We can easily remove five subsets of two numbers, leaving only six. The answer is 5, because the average of this one-number collection is still 6.

#### How many square yards of carpet are needed to cover a 12 foot long by 9 foot wide rectangular floor? (A yard is three feet long.)

Correct! Wrong!

Solution

We begin by multiplying 12 by 9. To do so, you'll need to cover 108 square feet of carpet. Because a square yard is 9 square feet, you divide 108 by 9 to get 12 square yards, which is our answer.

2nd Option

Because a yard has three feet, we divide 9 by 3 to get 3, and 12 by 3 to get 4. We multiply these two figures together to get 12 to find the carpet's area.

#### It's a two-digit number called N. The remainder after dividing N by 9 is one. The remainder after dividing N by ten is three. When you divide N by 11, what is the result?

Correct! Wrong!

We know the second digit to be 3 because of the second bullet point. Because dividing by 9 leaves a leftover of 1, the multiple of 9 must finish in a 2. This is the one we're looking for now:

9(1) = 9

9(2) = 18

9(3) = 27

9(4) = 36

9(5) = 45

9(6) = 54

9(7) = 63

9(8) = 72

Both conditions are satisfied by the number 72 + 1 = 73. To calculate the remainder, we subtract the largest multiple of 11 less than 73. As a result, 73 + 11(6) = 73 + 66 = 7.

#### Curt's father takes him to school in 20 minutes during rush hour traffic. There is no traffic one day, so his father can drive him to school in 12 minutes, 18 miles per hour faster. How far is it to school in miles?

Correct! Wrong!

Solution 1

We get d/v = 1/3 and d/v + 18 = 1/5

This equals d = 1/5v + 3.6 = 1/3v, which equals v = 27, which equals d = 9.

Solution 2

d = rt, d = 1/3 x r = 1/5 x (r + 18)

r/3 = r/5 + 18/5

2r/15 = 18/5

10r = 270 so r = 27, plug into the first one and it’s 9 miles to school

Solution 3

We devised a formula involving d, the distance, and x, the speed in miles per hour. We get d = (1/3)(x) = (1/5)(x + 18) as a result.

(5)(x) = (3)(x + 18)

5x = 3x + 54

2x = 54

x = 27

Hence, d = 27/3 = 9.

Solution 4

Because it takes him 3/5 of the time to get to school in no traffic, the speed must be 5/3 of the speed in traffic, or 2/3 more. Using x as the distance he can drive in an hour with traffic, we get 2x/3 = 18 miles per hour. When we solve for x, we get 27 miles per hour. The distance would be 9 miles because 20 minutes is a third of an hour.

#### What is the total of 2016's distinct prime integer divisors?

Correct! Wrong!

Solution 1

2016 = 25 x 32 x 7 is the prime factorization. We have 2, 3, and 7 because the problem simply asks for different prime factors. As a result, their intended total is 12.

2nd Option

Because 2 + 0 + 1 + 6 = 9, and 9|9, we observe that 9|2016. We can reach 224 by multiplying 2016 by 9. As 4|24, this is divisible by four. We get 56 when we divide 224 by four. This is plainly divisible by seven, resulting in the number eight. We have 2016 = 9 4 7 8 as our number. 4 and 8 are both multiples of 2, 9 is 32, and 7 is prime, as we know. This suggests that the prime factors 2, 3, and 7 are distinct. Their total is twelve.

#### Ethan and Anthony are going swimming at a one-mile pool near their home. They both leave at the same time. Anthony travels to the pool on her bicycle at a consistent speed of ten miles per hour. Ethan walks at a consistent speed of 4 miles per hour to the pool. Anthony will come in how many minutes before Ethan?

Correct! Wrong!

We can create an equation for when Anthony gets at the pool using d = rt:

10t = 1

When we solve for t, we find that Anthony arrives in the pool in 1/10 of an hour, or 6 minutes. Using the same method for Ethan, we discover that he arrives in the pool in 1/4 of an hour, or 15 minutes. As a result, Anthony will have to wait for Ethan to arrive at the pool for 15 – 6 = 9 minutes.

#### Determine how many two-digit numbers satisfy the following property: the sum of the number plus the number produced by reversing the digits is 132.

Correct! Wrong!

The two-digit number can be written as 10a + b; the inverse of 10a + b is 10b + a. The total of those figures is:
(10a + b) + (10b + a) = 132

11a + 11b = 132

a + b = 12

We can obtain order pairings (a,b) such that a + b = 12 using brute force. Because a and b are both digits, they must both be integers smaller than ten. As a result, our ordered pairings are (3,9); (4,8); (5,7); (6,6); (7,5); (8,4); (9,3) or 7 ordered pairs.

#### Carlo's car consumes a gallon of gas every 35 miles and has a gas tank capacity of 14 gallons when full. Karl started with a full tank of gas, traveled 350 miles, stopped to buy 8 gallons of gas, and then drove the rest of the way to his destination. His gas tank was half full when he arrived. Karl drove how many miles that day?

Correct! Wrong!

He had consumed 350/35 = 10 gallons of gas after 350 miles since he uses a gallon of gas every 35 miles. As a result, he had 14 – 10 = 4 gallons of gas left after the first leg of his journey. Then he went out and bought another 8 gallons of gas, bringing his total gas tank capacity to 12 gallons. He had 1/2 14 = 7 gallons of gas when he arrived. As a result, on the second part of his journey, he used 5 gallons of gas. As a result, the second leg of his journey covered 5 35 = 175 miles. When we add this to the 350 miles, we get 350 + 175 = 525 miles.

Correct! Wrong!