hree friends plucked some mangoes from a mango grove and collected them together in a pile and took nap after that. After some time, one of the friends woke up and divided the mangoes into three equal numbers. There was one mango extra. He gave it to a monkey nearby, took one part for himself and slept again. Next the second friend got up unaware of what has happened, divided the rest of the mangoes into three equal shares. There was an extra mango. He gave it to the monkey, took one share for himself and slept again. Next the third friend got up not knowing what happened and divided the mangoes into three equal shares. There was an extra mango. He gave it to the monkey, took one share for himself and went to sleep again. After some time, all of them got up together to find 30 mangoes. How many mangoes did the friends pluck initially?

Let's denote the initial number of mangoes by \(N\).

First, the first friend wakes up and divides the mangoes into three equal parts with one extra mango. He gives this extra mango to a monkey and takes his share:

\[
N = 3a + 1
\]

He takes one part for himself, so the remaining mangoes are:

\[
N_1 = 2a
\]

Next, the second friend wakes up and finds \(N_1\) mangoes. He divides these into three equal parts with one extra mango. He gives this extra mango to a monkey and takes his share:

\[
2a = 3b + 1
\]

He takes one part for himself, so the remaining mangoes are:

\[
N_2 = 2b
\]

Next, the third friend wakes up and finds \(N_2\) mangoes. He divides these into three equal parts with one extra mango. He gives this extra mango to a monkey and takes his share:

\[
2b = 3c + 1
\]

He takes one part for himself, so the remaining mangoes are:

\[
N_3 = 2c
\]

Finally, it is given that when all the friends wake up together, they find 30 mangoes left:

\[
N_3 = 30
\]

So we have:

\[
2c = 30 \implies c = 15
\]

Using \(c = 15\) in the previous equation:

\[
2b = 3c + 1 \implies 2b = 3 \times 15 + 1 = 46 \implies b = 23
\]

Now using \(b = 23\):

\[
2a = 3b + 1 \implies 2a = 3 \times 23 + 1 = 70 \implies a = 35
\]

Finally using \(a = 35\):

\[
N = 3a + 1 \implies N = 3 \times 35 + 1 = 106
\]

Thus, the initial number of mangoes the friends plucked is:

\[
\boxed{106}
\]

Would you like more details or have any questions? Here are 5 numbered questions you might consider asking next:

1. Can you explain another example of a similar problem with different numbers?
2. How can we solve this problem using a different method?
3. What if the number of mangoes left was different? How would that change the solution?
4. Can you show the step-by-step verification of this solution?
5. How can this type of problem be generalized for any number of friends and extra mangoes?

**Tip:** When solving problems involving multiple steps and conditions, break down each step methodically and ensure each condition is accounted for before moving on to the next.

Learn how to solve a problem involving the division of mangoes using modular arithmetic. This problem covers concepts such as divisibility rules and modular arithmetic equations. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation