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Sagot :
Sure, let's solve the problem step-by-step.
### Part i: Finding Number of Rs 2 Coins in Mani's Purse
1. Understanding the given conditions:
- Mani has 36 coins in her purse.
- The coins are either Rs 2 coins or Rs 5 coins.
2. Setting up equations:
- Let [tex]\( x \)[/tex] be the number of Rs 2 coins.
- Let [tex]\( y \)[/tex] be the number of Rs 5 coins.
We have two conditions:
- The total number of coins is 36: [tex]\( x + y = 36 \)[/tex]
- The total amount of money is Rs 84: [tex]\( 2x + 5y = 84 \)[/tex]
3. Solving the system of equations:
[tex]\[ \begin{cases} x + y = 36 \quad \text{(Equation 1)}\\ 2x + 5y = 84 \quad \text{(Equation 2)} \end{cases} \][/tex]
First, solve Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 36 - x \][/tex]
Next, substitute [tex]\( y \)[/tex] in Equation 2:
[tex]\[ 2x + 5(36 - x) = 84 \][/tex]
Simplify this equation:
[tex]\[ 2x + 180 - 5x = 84 \][/tex]
Combine like terms:
[tex]\[ -3x + 180 = 84 \\ -3x = 84 - 180 \\ -3x = -96 \\ x = 32 \][/tex]
So, the number of Rs 2 coins ([tex]\( x \)[/tex]) is 32.
To find [tex]\( y \)[/tex]:
[tex]\[ y = 36 - 32 \\ y = 4 \][/tex]
Verification:
[tex]\[ 2(32) + 5(4) = 64 + 20 = 84 \][/tex]
The calculations are correct.
- Number of Rs 2 coins in Mani's purse: 32
### Part ii: Finding the Number of Coins of Rs 2 in Moni's Purse
1. Understanding the given conditions:
- Moni has Rs 85 with coins of Rs 1 and Rs 2.
- The ratio of Rs 1 coins to Rs 2 coins is 5:6.
2. Setting up the ratio:
- Let the number of Rs 1 coins be [tex]\( 5k \)[/tex].
- Let the number of Rs 2 coins be [tex]\( 6k \)[/tex].
3. Forming the equation using the total amount:
[tex]\[ 5k + 12k = 85 \][/tex]
Simplify this:
[tex]\[ 17k = 85 \\ k = 5 \][/tex]
So, the number of Rs 1 coins:
[tex]\[ 5k = 5 \times 5 = 25 \][/tex]
And the number of Rs 2 coins:
[tex]\[ 6k = 6 \times 5 = 30 \][/tex]
Verification:
[tex]\[ 1(25) + 2(30) = 25 + 60 = 85 \][/tex]
The calculations are correct.
- Number of Rs 2 coins in Moni's purse: 30
### Part iii: Writing the Equation for Finding the Number of Coins in Mahi's Purse
Given that Mahi has a situation similar to Mani’s in having Rs 2 and Rs 5 coins, we can model the number of coins in Mahi's purse as we did for Mani.
If [tex]\( x \)[/tex] is the number of Rs 2 coins and [tex]\( y \)[/tex] is the number of Rs 5 coins in Mahi's purse, and she also has a total of 36 coins worth a specific total value (unknown here), the core equation to find the number of coins of each type is:
1. Total coins equation:
[tex]\[ x + y = 36 \][/tex]
2. Value equation:
[tex]\[ 2x + 5y = \text{[Total Amount in Rupees]} \][/tex]
These are the fundamental equations for solving the problem related to Mahi's purse assuming the format remains the same as Mani's.
### Part i: Finding Number of Rs 2 Coins in Mani's Purse
1. Understanding the given conditions:
- Mani has 36 coins in her purse.
- The coins are either Rs 2 coins or Rs 5 coins.
2. Setting up equations:
- Let [tex]\( x \)[/tex] be the number of Rs 2 coins.
- Let [tex]\( y \)[/tex] be the number of Rs 5 coins.
We have two conditions:
- The total number of coins is 36: [tex]\( x + y = 36 \)[/tex]
- The total amount of money is Rs 84: [tex]\( 2x + 5y = 84 \)[/tex]
3. Solving the system of equations:
[tex]\[ \begin{cases} x + y = 36 \quad \text{(Equation 1)}\\ 2x + 5y = 84 \quad \text{(Equation 2)} \end{cases} \][/tex]
First, solve Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 36 - x \][/tex]
Next, substitute [tex]\( y \)[/tex] in Equation 2:
[tex]\[ 2x + 5(36 - x) = 84 \][/tex]
Simplify this equation:
[tex]\[ 2x + 180 - 5x = 84 \][/tex]
Combine like terms:
[tex]\[ -3x + 180 = 84 \\ -3x = 84 - 180 \\ -3x = -96 \\ x = 32 \][/tex]
So, the number of Rs 2 coins ([tex]\( x \)[/tex]) is 32.
To find [tex]\( y \)[/tex]:
[tex]\[ y = 36 - 32 \\ y = 4 \][/tex]
Verification:
[tex]\[ 2(32) + 5(4) = 64 + 20 = 84 \][/tex]
The calculations are correct.
- Number of Rs 2 coins in Mani's purse: 32
### Part ii: Finding the Number of Coins of Rs 2 in Moni's Purse
1. Understanding the given conditions:
- Moni has Rs 85 with coins of Rs 1 and Rs 2.
- The ratio of Rs 1 coins to Rs 2 coins is 5:6.
2. Setting up the ratio:
- Let the number of Rs 1 coins be [tex]\( 5k \)[/tex].
- Let the number of Rs 2 coins be [tex]\( 6k \)[/tex].
3. Forming the equation using the total amount:
[tex]\[ 5k + 12k = 85 \][/tex]
Simplify this:
[tex]\[ 17k = 85 \\ k = 5 \][/tex]
So, the number of Rs 1 coins:
[tex]\[ 5k = 5 \times 5 = 25 \][/tex]
And the number of Rs 2 coins:
[tex]\[ 6k = 6 \times 5 = 30 \][/tex]
Verification:
[tex]\[ 1(25) + 2(30) = 25 + 60 = 85 \][/tex]
The calculations are correct.
- Number of Rs 2 coins in Moni's purse: 30
### Part iii: Writing the Equation for Finding the Number of Coins in Mahi's Purse
Given that Mahi has a situation similar to Mani’s in having Rs 2 and Rs 5 coins, we can model the number of coins in Mahi's purse as we did for Mani.
If [tex]\( x \)[/tex] is the number of Rs 2 coins and [tex]\( y \)[/tex] is the number of Rs 5 coins in Mahi's purse, and she also has a total of 36 coins worth a specific total value (unknown here), the core equation to find the number of coins of each type is:
1. Total coins equation:
[tex]\[ x + y = 36 \][/tex]
2. Value equation:
[tex]\[ 2x + 5y = \text{[Total Amount in Rupees]} \][/tex]
These are the fundamental equations for solving the problem related to Mahi's purse assuming the format remains the same as Mani's.
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