Question

Reema had 'n' chocolates. She distributed them among \(4\) children in the ratio of \(\frac{1}{2} : \frac{1}{3} : \frac{1}{5} : \frac{1}{8}\). If she gave them each one a complete chocolate, the minimum number of chocolates she had distributed.

• A. \(120\)
• B. \(139\)
• C. \(240\)
• D. \(278\)

Hint:

The sum of an integer ratio always gives the minimum total required for discrete items like chocolates or people. If the sum is not among options, look for its multiples (e.g., 278).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A ratio provided in fractions must be converted to an integer ratio to find the absolute number of items. The minimum number is found when these integers cannot be simplified further.
Step 2: Key Formula or Approach:
To convert a fractional ratio to an integer ratio, multiply each term by the Least Common Multiple (LCM) of the denominators.
Step 3: Detailed Explanation:
1. Denominators of the ratio terms are \(2, 3, 5, 8\).
2. Find LCM(\(2, 3, 5, 8\)):
Factors: \(2 = 2\), \(3 = 3\), \(5 = 5\), \(8 = 2^3\).
LCM = \(2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120\).
3. Multiply the original ratio by \(120\):
\[ \frac{1}{2} \times 120 : \frac{1}{3} \times 120 : \frac{1}{5} \times 120 : \frac{1}{8} \times 120 \] \[ 60 : 40 : 24 : 15 \] 4. Check if these integers have a common divisor:
\(gcd(60, 40, 24, 15) = 1\). Thus, these are the minimum possible whole numbers for the distribution.
5. Minimum total chocolates (\(n\)) = \(60 + 40 + 24 + 15 = 139\).
Step 4: Final Answer:
The minimum number of chocolates she distributed is \(139\).