Solving Math Problems: How Many Chocolates Will Mr. Smith Have Left Over?
Have you ever faced a math problem that seemed complex at first glance but can be solved with a few simple steps? Let's dive into the story of Mr. Smith and solve the math problem he encounters when purchasing and grouping chocolates.
Understanding the Problem
Mr. Smith buys chocolates in boxes of 48 and then groups them into sets of 11. If he buys 93 boxes, how many chocolates will he have left over?
Step-by-Step Solution
To solve this problem, we need to calculate the total number of chocolates Mr. Smith has, then determine how many full sets of 11 he can make and how many will be left over. Here's a detailed breakdown:
Calculate the Total Number of Chocolates
The first step is to find out the total number of chocolates Mr. Smith has:
93 boxes each with 48 chocolates 4464 chocolates
Determine How Many Full Sets of 11 Can Be Formed
Next, we need to find out how many full sets of 11 chocolates Mr. Smith can form. We can do this by dividing the total number of chocolates by 11 and finding the remainder:
4464 / 11 405 remainder 9
This tells us that Mr. Smith can form 405 full sets of 11 chocolates. The remainder, 9, represents the number of chocolates that will be left over.
Alternative Method
Another way to find the remainder is to use modular arithmetic. We can use the properties of numbers and modular operations to simplify the calculation:
93 boxes 11 * 8 5 48 chocolates 11 * 4 4Therefore, the product of the number of boxes and chocolates can be written as:
93 * 48 ≡ 5 * 4 ≡ 20 ≡ 9 (mod 11)
This shows that when the total number of chocolates is divided by 11, the remainder is 9. So, 9 chocolates will be left over.
Conclusion
To find out how many chocolates Mr. Smith will have left over, we can conclude that he will have 9 chocolates left over after grouping them into sets of 11.
Additional Practice
Let's practice with another similar problem. Suppose Mr. Gardose buys chocolates in boxes of 48 and groups them into sets of 11. If he buys 405 boxes, how many chocolates will he have left over?
Using the step-by-step method:
Calculate the total number of chocolates:405 * 48 19440 chocolatesDetermine how many full sets of 11 can be formed:
19440 / 11 1767 remainder 3The remainder, 3, is the number of chocolates left over.
Thus, Mr. Gardose will have 3 chocolates left over after grouping them into sets of 11.
Summary
Through the steps outlined above, we can solve math problems involving remainders using basic arithmetic and modular operations. This method can be applied to various similar problems to find the number of items left over after grouping them into sets.