Every 8th person received a free dessert and every 6th person received a free salad. Out of 100 people, how many received both?

Question

math for 6th grade

Answer 1

The common multiples of 8 and 6 (less than 100) are: 48 and 96. So four people got free salad and dessert.

Answer 2

There are several ways to do this. I'll show the long way then the short way. (I think the short way is easier to do, but the long way may be easier to understand) You can make a list of all the multiples of 6 up to 100 and all the multiples of 8 up to 100 and find which ones match:

 

Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96 (the next is 102, which is bigger than what we want) These are the people who get salad.

 

Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 (the next is 104, but that's too big) These are the people who get dessert.

 

Now you compare the lists and see which ones got both!

 

The short way is to know that you are looking for how many numbers (smaller than 100) have the same factors as 6 and 8. The smallest factors of 6 are 2, 3. The smallest factors of 8 are 2, 2 and 2. So any number that has at least three factors of 2 and one factor of 3 will have the same factors as both 6 and 8.

 

The first common multiple of 6 and 8 is then 2x2x2x3x(1)=24. The second is going to be 2x2x2x3x(2)=48 and the third will be 2x2x2x3x(3)=72 and the fourth will be 2x2x2x3x(4)=96. These are the same people that got both salad and dessert in the two lists above. The 24th, 48th, 72nd and 96th people got both. That means 4 people got both.