Problem 22 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A purse contains 4 copper coins and 3 silver coins, the second purse contains 6 copper coins and 2 silver coins. If a coin is drawn out of any purse, then the probability that it is a copper coin is (a) \(4 / 7\) (b) \(3 / 4\) (c) \(37 / 56\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (c) \( \frac{37}{56} \).

Step 1: Calculate Total Coins in Each Purse

In the first purse, there are 4 copper coins and 3 silver coins, totaling 7 coins. In the second purse, there are 6 copper coins and 2 silver coins, totaling 8 coins.

Step 2: Calculate Probability from Each Purse

The probability of drawing a copper coin from the first purse is \( \frac{4}{7} \) because there are 4 copper coins out of 7 total coins. Similarly, for the second purse, the probability is \( \frac{6}{8} = \frac{3}{4} \) because there are 6 copper coins out of 8 total coins.

Step 3: Calculate Total Probability for Either Purse

Assuming an equal chance of picking a coin from any of the two purses, assign each purse a probability of \( \frac{1}{2} \). The total probability of drawing a copper coin from either purse is calculated by the formula: \( \left( \frac{1}{2} \times \frac{4}{7} \right) + \left( \frac{1}{2} \times \frac{3}{4} \right) \).

Step 4: Simplify the Total Probability

First, calculate each part: \( \frac{1}{2} \times \frac{4}{7} = \frac{2}{7} \) and \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Now add these probabilities: \( \frac{2}{7} + \frac{3}{8} \).

Step 5: Find a Common Denominator and Add

The common denominator for 7 and 8 is 56. Convert both fractions: \( \frac{2}{7} = \frac{16}{56} \) and \( \frac{3}{8} = \frac{21}{56} \). The sum is \( \frac{16}{56} + \frac{21}{56} = \frac{37}{56} \).

Step 6: Validate the Probability

The probability \( \frac{37}{56} \) is one of the options (c), confirming this is the correct answer.

Key Concepts

Copper Coins

When thinking about copper coins, imagine shiny, reddish coins sitting inside a purse.
The first purse holds 4 such copper coins.
Copper coins are often used as a symbol in probability problems because of their different appearance from coins made of other metals.
The goal here is to calculate the likelihood of drawing one of these copper coins from the purse.

• First purse: 4 copper coins
• Second purse: 6 copper coins

When calculating probability, these are counted separately and then together to find the overall probability for either purse.

Silver Coins

Silver coins add a twist to the probability challenge.
Both purses in the exercise contain not just copper coins, but silver ones too.
The difference in number and type influences the probability outcome.

• First purse: 3 silver coins
• Second purse: 2 silver coins

The presence of silver coins affects the denominator in our probability equation, representing all possible outcomes when picking a coin.
This interplay between copper and silver coins is crucial in understanding how mixed items alter probabilities.

Probability Calculation

Calculating probability helps us understand chance.
In the discussed exercise, there are steps to determine how likely it is to draw a copper coin from the two purses.
Here's a simplified view:

• 1st purse probability: \[ rac{4}{7} \] - since there are 4 copper coins out of a total 7 coins
• 2nd purse probability: \[ rac{3}{4} \] - calculated from 6 copper coins in 8 total

The task is to find the total probability, assuming we have equal chances of selecting from either purse.
We introduced a factor of \( rac{1}{2} \) for each to account for this, leading to a combined chance of gaining a copper coin by adding those partial probabilities.

Common Denominator

Understanding how to find a common denominator is vital in probability problems dealing with fractions.
When combining probabilities from different sources, like purses filled with coins, identifying a common denominator allows us to add fractional probabilities together easily.
First, convert \( rac{2}{7} \) and \( rac{3}{8} \) into a compatible form by finding the least common multiple of 7 and 8, which is 56:

• \( rac{2}{7} = rac{16}{56} \)
• \( rac{3}{8} = rac{21}{56} \)

Adding them gives us the total probability of selecting a copper coin from either purse: \( rac{37}{56} \).
The correct calculation lets us confirm the answer as option (c).