1. A three-digit integer is to be randomly created using each of the digits 2, 3 and 6 once. What is the probability that the number created is even? Express your answer as a common fraction.
2. How many multiples of 8 are between 100 and 175?
3. In the following arithmetic sequence, what is the value of m?
-2, 4, m, 16, . . .
4. A particular fraction is equivalent to 2/3. The sum of its numerator and denominator is 105. What is the numerator of the fraction?
5. What is the least natural number that has four distinct prime factors?
6. The perimeter of a particular rectangle is 24 inches. If its length and width are positive integers, how many distinct areas could the rectangle have?
It would be beneficial for you if you know your prime numbers well. First, list the prime numbers from 1 to 100. (Check your answers with the MATHCOUNTS Toolbox that I have provided.) Then, list the prime numbers from 100 to 200.
Nidhi, the number does not necessarily have to be 4 digits long. The problem is asking for the smallest number that has 4 different prime factors. Keep working on it.
1. There are 3! = 6 ways to arrange the digits 2, 3, and 6. To make the 3-digit integer even, we have 2 choices for the units digit (6 or 2), 2 choices for the hundreds digit, and 1 choice for the tens digit. Therefore, our successes are:
2 × 1 × 2 = 4
Hence, the probability is 4/6 = 2/3
2. We determine the boundaries.
8 × 13 = 104 8 × 21 = 168
168 – 104 = 64 64/8 = 8 8 + 1 = 9
3. We can easily see that 6 is added each time. Hence, we have 4 + 6 = 10
4. We call the fraction 2x/3x, where the numerator is 2x and the denominator is 3x.
2x + 3x = 105 x = 21 2x = 42
5. The least natural number that has 4 distinct, or different, prime factors is the product of the first 4 primes:
Problems
ReplyDelete1. A three-digit integer is to be randomly created using each of the digits 2, 3 and 6 once. What is the probability that the number created is even? Express your answer as a common fraction.
2. How many multiples of 8 are between 100 and 175?
3. In the following arithmetic sequence, what is the value of m?
-2, 4, m, 16, . . .
4. A particular fraction is equivalent to 2/3. The sum of its numerator and denominator is 105. What is the numerator of the fraction?
5. What is the least natural number that has four distinct prime factors?
6. The perimeter of a particular rectangle is 24 inches. If its length and width are positive integers, how many distinct areas could the rectangle have?
1) 4/6 of a chance
ReplyDelete2) 9 multiples
3) m=10
4) I think it is 63
5) I think it is 30
6) 6 distinct areas
Here are my answers
-Vishal
Well done, Vishal!
ReplyDeleteMake sure you reduce common fractoins to their lowest terms. Keep trying problems 4 and 5.
does distinct mean different?
ReplyDeleteYes, distinct means different.
ReplyDeleteThis comment has been removed by the author.
ReplyDelete5)210
ReplyDeleteI am working on 4.
This comment has been removed by the author.
ReplyDelete4)43
ReplyDeletei get what i did wrong
Vishal, I can see that you're trying hard. Good job!
ReplyDelete*Check Problem 4. You might have made a careless error.
Everyone,
ReplyDeleteIt would be beneficial for you if you know your prime numbers well. First, list the prime numbers from 1 to 100. (Check your answers with the MATHCOUNTS Toolbox that I have provided.) Then, list the prime numbers from 100 to 200.
See how many you can find!
For #5, does the number have to be 4 digits long?
ReplyDelete1) 2/3
ReplyDelete2) 9 multiples
3) m=10
4) The numerator is 42.
5) 1235(?)
6) 6 distinct areas
1)2/3
ReplyDelete2)9 multiples
3)m=10
4)numerator: 70
5)210
6)6 distinct areas
Agni,
ReplyDeleteWhen will you post the solutions?
:)
Nidhi, the number does not necessarily have to be 4 digits long. The problem is asking for the smallest number that has 4 different prime factors. Keep working on it.
ReplyDelete--------------------------------------------------
Maya, well done! Check your work for problem 4. I will be posting the solutions by Sunday evening.
Sorry, #4 should be 42. I must have typed it wrong. Did I get #5 right the 2nd time?
ReplyDeleteI tried it again and i got 210 for #5
ReplyDeleteVishal and Nidhi, you are correct. Start working on your prime numbers.
ReplyDelete1. 2/3
ReplyDelete2. 9 multiples
3. m = 10
4. 42
5. 210
6. 6 distinct areas.
Wait
ReplyDeleteSolutions
ReplyDelete1. There are 3! = 6 ways to arrange the digits 2, 3, and 6. To make the 3-digit integer even, we have 2 choices for the units digit (6 or 2), 2 choices for the hundreds digit, and 1 choice for the tens digit. Therefore, our successes are:
2 × 1 × 2 = 4
Hence, the probability is 4/6 = 2/3
2. We determine the boundaries.
8 × 13 = 104
8 × 21 = 168
168 – 104 = 64
64/8 = 8
8 + 1 = 9
3. We can easily see that 6 is added each time. Hence, we have 4 + 6 = 10
4. We call the fraction 2x/3x, where the numerator is 2x and the denominator is 3x.
2x + 3x = 105
x = 21
2x = 42
5. The least natural number that has 4 distinct, or different, prime factors is the product of the first 4 primes:
2 × 3 × 5 × 7 = 210
6. Length + Width = 24/2 = 12
There are 6 pairs:
1 → 11
2 → 10
3 → 9
4 → 8
5 → 7
6 → 6
Ooops, Avanthika! I didn't notice your comment. Just try the problems or review the solutions.
ReplyDeleteOh. I was on the November tab. No wonder I couldn't find the questions.
ReplyDeleteGrace, try these problems. Then move on to the problems on the post above (titled Problems of the Week - 12/13).
ReplyDelete1. 2/3
ReplyDelete2. 9
3. 10
4. 42
5. 16
6. 6
Good work, Grace! Check problem 5 with the solutions provided above.
ReplyDeleteI just read solutions. Did not know distinct meant they all had to be different. I did 2^4.
ReplyDelete1. 2/3 or 4/6
ReplyDelete2. 9 multiples
3. m = 10
4. 42
5. 210
6. 6 distinct areas
Its fine Agni, and my answers are all correct.
ReplyDeleteI am not sure how to do number six
ReplyDeleteThere are six areas on a rectangle, but does that mean it has six distinct areas?
ReplyDelete