Let's analyze the problem step-by-step.
Problem Restatement
- Rani distributes chocolates to her friends on her birthday.
- If she gives chocolates to each friend, one friend will get only 3 chocolates.
- Also, if she gives 3 chocolates to each friend , she will be left with 22 chocolates.
- We need to find how many friends she has.
Understanding the Problem
Let's denote:
- nnn = number of friends
- TTT = total number of chocolates Rani has
From the problem:
- When she distributes chocolates, if she gives chocolates equally to each friend, one friend will get only 3 chocolates. This implies that the chocolates are not enough to give more than 3 to each friend, or the distribution is uneven, and the smallest share is 3 chocolates.
- If she gives exactly 3 chocolates to each friend , she will be left with 22 chocolates.
Step 1: Express the chocolates given when 3 chocolates are given to each
friend
If she gives 3 chocolates to each friend, the total chocolates given are:
3×n3\times n3×n
She is left with 22 chocolates, so:
T=3n+22T=3n+22T=3n+22
Step 2: Analyze the first condition
The first condition says:
- If she gives chocolates to each friend (probably equally), one friend will get only 3 chocolates.
This can be interpreted as: when distributing the chocolates equally among friends, the minimum number of chocolates any friend gets is 3. Since when she gives 3 chocolates each, she has leftover chocolates (22), it means the total chocolates are not a multiple of nnn. If she distributes all chocolates equally (without leftover), the number of chocolates per friend would be:
chocolates per friend=Tn\text{chocolates per friend}=\frac{T}{n}chocolates per friend=nT
Since one friend gets only 3 chocolates, the minimum share is 3, so:
Tn=3(or very close to 3)\frac{T}{n}=3\quad \text{(or very close to 3)}nT=3(or very close to 3)
But since leftover chocolates exist when giving 3 chocolates per friend, the actual average per friend is slightly more than 3.
Step 3: Use the leftover chocolates to find nnn
From Step 1:
T=3n+22T=3n+22T=3n+22
If she tries to give chocolates equally to nnn friends, the number of chocolates per friend is:
Tn=3n+22n=3+22n\frac{T}{n}=\frac{3n+22}{n}=3+\frac{22}{n}nT=n3n+22=3+n22
Since one friend gets only 3 chocolates, the leftover chocolates (22) must be less than nnn times the difference between the average and 3. For one friend to get only 3 chocolates, the leftover chocolates must be less than nnn, so:
22<n22<n22<n
Step 4: Find nnn such that 22<n22<n22<n and T=3n+22T=3n+22T=3n+22
Since nnn must be an integer greater than 22, let's check values starting from 23 upwards.
- For n=23n=23n=23:
T=3×23+22=69+22=91T=3\times 23+22=69+22=91T=3×23+22=69+22=91
Check if TTT divided by nnn gives a minimum of 3 chocolates per friend:
9123≈3.956\frac{91}{23}\approx 3.9562391≈3.956
Since the average is about 3.956, one friend could get only 3 chocolates if the distribution is uneven.
- For n=24n=24n=24:
T=3×24+22=72+22=94T=3\times 24+22=72+22=94T=3×24+22=72+22=94
9424≈3.916\frac{94}{24}\approx 3.9162494≈3.916
- For n=25n=25n=25:
T=3×25+22=75+22=97T=3\times 25+22=75+22=97T=3×25+22=75+22=97
9725=3.88\frac{97}{25}=3.882597=3.88
Since the problem states "if she gives chocolates to each friend, one friend will get only 3 chocolates," the smallest integer nnn greater than 22 that fits the leftover condition is 23.
Final Answer:
Rani has 23 friends.
Summary:
- Total chocolates T=3n+22T=3n+22T=3n+22
- Number of friends n>22n>22n>22
- The smallest such nnn is 23
Therefore, Rani has 23 friends.
