There are 90 coins, comprising 5 and 10 paisa coins. The value of all the coins is Rs. 7. How many 5 paisa coins are there?
A. 50
B. 45
C. 40
D. 35

Question

There are 90 coins, comprising 5 and 10 paisa coins. The value of all the coins is Rs. 7. How many 5 paisa coins are there?A. 50B. 45C. 40 D. 35

Prepp Team · Accepted Answer

Solving the Coin Problem: Finding the Number of 5 Paisa Coins

This problem involves a collection of coins with different denominations and a total value. We need to find out exactly how many coins of a specific denomination are present. We can solve this by setting up a system of linear equations.

Understanding the Coin Problem

We are given:

Total number of coins: 90
Denominations of coins: 5 paisa and 10 paisa
Total value of all coins: Rs. 7

The goal is to find the number of 5 paisa coins.

Setting Up Equations for the Coin Problem

Let's use variables to represent the unknown quantities:

Let \(x\) be the number of 5 paisa coins.
Let \(y\) be the number of 10 paisa coins.

From the given information, we can form two equations:

The total number of coins is 90.
This gives us the equation:
\[ x + y = 90 \quad \text{(Equation 1)} \]
The total value of the coins is Rs. 7.
First, convert the total value to paisa: Rs. 7 = \(7 \times 100\) paisa = 700 paisa.

The value from \(x\) five paisa coins is \(5x\) paisa.

The value from \(y\) ten paisa coins is \(10y\) paisa.

The total value in paisa is the sum of these values:
\[ 5x + 10y = 700 \quad \text{(Equation 2)} \]

Now we have a system of two linear equations with two variables \(x\) and \(y\).

\[ x + y = 90 \] \[ 5x + 10y = 700 \]

Solving the System of Equations to Find 5 Paisa Coins

We can solve this system using the substitution method. From Equation 1, we can express \(y\) in terms of \(x\):

\[ y = 90 - x \]

Now, substitute this expression for \(y\) into Equation 2:

\[ 5x + 10(90 - x) = 700 \]

Distribute the 10 on the left side:

\[ 5x + 900 - 10x = 700 \]

Combine the \(x\) terms:

\[ 900 - 5x = 700 \]

Subtract 900 from both sides of the equation:

\[ -5x = 700 - 900 \] \[ -5x = -200 \]

Divide both sides by -5 to solve for \(x\):

\[ x = \frac{-200}{-5} \] \[ x = 40 \]

So, the number of 5 paisa coins is 40.

We can also find the number of 10 paisa coins (\(y\)) using Equation 1:

\[ y = 90 - x = 90 - 40 = 50 \]

There are 40 five paisa coins and 50 ten paisa coins. Let's check the total value:

\((40 \text{ coins} \times 5 \text{ paisa/coin}) + (50 \text{ coins} \times 10 \text{ paisa/coin}) = 200 \text{ paisa} + 500 \text{ paisa} = 700 \text{ paisa}\).

Since 700 paisa is equal to Rs. 7, our solution is correct.

The number of 5 paisa coins is 40.

Revision Table: Key Concepts for Coin Problems

Concept	Description	Relevance to Coin Problems
System of Linear Equations	A set of two or more linear equations involving the same variables.	Coin problems often require two equations (one for quantity, one for value) to solve for two unknowns (number of each coin type).
Variables	Symbols (like \(x\) and \(y\)) used to represent unknown quantities.	Used to represent the number of each type of coin.
Setting up Equations	Translating words from the problem into mathematical expressions and equations.	Crucial step to model the relationships between coin quantities and values.
Solving Methods	Techniques like substitution or elimination used to find the values of variables in a system of equations.	Necessary to find the actual number of each coin type.

Additional Information on Solving Word Problems

Coin problems are a common type of word problem that can be solved using linear equations. Here are some related concepts:

Elimination Method: Another way to solve a system of equations is by eliminating one variable. This involves multiplying one or both equations by a constant so that the coefficients of one variable are opposites, and then adding the equations together.
Unit Conversion: Always ensure all values are in the same units (e.g., all in paisa or all in rupees) before setting up the value equation.
Checking the Solution: After finding the values of the variables, always plug them back into the original word problem statements (total coins and total value) to verify the solution is correct.