Chad buys peanuts in 2 pound bags. He repackages them into bags that hold 5/6 pound of peanuts. How many 2 pound bags of peanuts should Chad buy so that he can fill the 5/6 pound bags without having any peanuts left over?
let the number of bags needed be n
so 2n ÷ (5/6) must be a whole number
= 2n(6/5)
= 12n/5
The smallest value of n is 5
So he should buy 5 of the 2 lbs bags
Check:
if n=5 , he will have 10 lbs.
number of smaller bags he can fill
= 10 ÷ (5/6)
= 10(6/5)
= 12 ---> exact number of bags, no peanuts left over.
3 bags
in 3 bags he will have 6 pounds. If for each bag he needs 5/6 of a pound this means he will have 1/6 x 6 leftover, which is exactly another bag.
To find out how many 2-pound bags of peanuts Chad should buy, we need to determine how many 2-pound bags are needed to fill a 5/6 pound bag without any leftovers.
First, we need to find a common denominator for 2 and 5/6. The least common multiple of 2 and 6 is 6.
Now, let's convert the 2-pound bags into a fraction with a denominator of 6:
2 pounds = (2 * 6) / 6 = 12/6 pounds
So, a 2-pound bag is equivalent to 12/6 pounds.
Next, let's divide 5/6 pound by 12/6 pound:
(5/6) / (12/6) = (5/6) * (6/12) = 30/72 = 5/12
This means that one bag with a capacity of 5/6 pound can be filled with 5/12 of a 2-pound bag.
Since Chad wants to avoid leftovers, he needs to buy enough 2-pound bags to fill each 5/6-pound bag without any remainders. Thus, Chad should buy the reciprocal of 5/12, which is 12/5.
Therefore, Chad should buy 12/5 or 2.4 (rounded up to 3) 2-pound bags of peanuts to fill the 5/6-pound bags without having any peanuts left over.
To solve this problem, we need to find the least common multiple (LCM) of the given weights - 2 pounds and 5/6 pounds. The LCM will give us the smallest weight that can evenly divide both weights.
First, let's convert 5/6 pounds to a common denominator of 6:
5/6 pounds = (5/6) * 6/6 = 30/36 pounds.
Now, let's find the LCM of 2 and 30/36.
Prime factorize 2: 2 = 2^1.
Prime factorize 30/36: 30/36 = (2 * 3 * 5) / (2^2 * 3^2) = 5/4.
Now, write down the prime factors and their highest powers:
2^1, 3^0, 5^1.
To find the LCM, we take the highest power of each prime factor:
2^1, 3^0, 5^1.
Multiply these factors together:
2^1 * 3^0 * 5^1 = 2 * 5 = 10.
Therefore, the LCM of 2 pounds and 5/6 pounds is 10 pounds.
Now, we need to figure out how many 2 pound bags of peanuts will give us a total weight of 10 pounds.
10 pounds / 2 pounds per bag = 5 bags.
So, Chad should buy 5 bags of peanuts so that he can fill the 5/6 pound bags without having any peanuts left over.