Solution for the cost of pencils, Problem for maximum value
Thanks everyone for trying out the problem and posting in the comments. Even if you didn't get the problem right, it's great that you tried and I encourage you to do the other problems! If you keep doing them you'll eventually get a lot better with these types of problems. :)
A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the
pencils was $17.71. What was the cost of a pencil in cents?
This problem may seem hard when you first look at it because you're hardly given any information. All you basically know is
cost of pencil in cents > # of pencils > 1
# of students who bought the pencil > 15
(because it stated a majority of the 30 students)
Also, you can figure out that the products of all the unknown variables will equal 1771
(# of pencils)*(cost in cents)*(# of students) = 1771
All you can do now is start trying to find the prime factors of 1771. If you know your divisibility rules for numbers 1-10 you'll realize pretty quickly that 7 is the lowest prime factor of 1771.
Now we have to find the factors of 253. Once again, we can use the divisibility rules and go through the numbers rather quickly. You find out that 11 is the next prime factor of 253.
Now, we know (7)(11)(23)=1771 From our inequality equations that we made earlier we can see that 23 is the number of students in the class, the cost of pencils in cents is 11, and the number of pencils each student bought is 7 pencils.
Therefore, each pencils costs 11 cents which makes it the answer.
If you didn't know the divisibility rules then the only way you could do it is just divide each number. It would only just take a little bit longer to do the problem. Here are the divisibility rules for numbers 1-10
1: Any number (15)
2: If the number is even (64)
3: Add up the digits in the number and if the sum is divisible by 3 (3453)
4: If the last 2 digits is divisible by 4 (236)
5: If the last digit is 0 or 5 (35)
6: If the number is divisible by 2 and 3 (1542)
7: Multiply the digits from the right hand side with numbers 1, 3, 2, 6, 4, 5, and add the products. If the sum is divisible by 7 then the number is divisible by 7 (2016)
8: If the last 3 digits is divisible by 8 (5248)
9: Add up the digits in the number and if the sum is divisible by 9 (855)
10: If the number ends in 0
Hopefully, this can help you do simple division problems much faster and may help you on tests in school or other things.
Here's the next problem!
Suppose that |x + y| + |x - y| = 2. What is the maximum possible value of x^2 - 6x + y^2?
Good luck! and remember, calculators are not allowed.