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Sagot :
To solve this problem, we need to determine the number of 100, 50, and 20 denomination notes that Surajit has, given the total amount, the total number of notes, and the ratio between the number of 100 and 50 notes. Let's break it down step-by-step.
### Step 1: Define Variables
- Let [tex]\( x \)[/tex] be a common multiplier for the ratio of 100 and 50 denomination notes.
- Since the ratio of 100 to 50 denomination notes is 5:2, we have:
- Number of 100 denomination notes = [tex]\( 5x \)[/tex]
- Number of 50 denomination notes = [tex]\( 2x \)[/tex]
- Let [tex]\( y \)[/tex] be the number of 20 denomination notes.
### Step 2: Use Given Conditions to Form Equations
1. Total number of notes:
[tex]\[ 5x + 2x + y = 8 \][/tex]
Simplifying, we get:
[tex]\[ 7x + y = 8 \quad \text{(Equation 1)} \][/tex]
2. Total amount of money:
[tex]\[ 100(5x) + 50(2x) + 20y = 620 \][/tex]
Simplifying, we get:
[tex]\[ 500x + 100x + 20y = 620 \][/tex]
[tex]\[ 600x + 20y = 620 \quad \text{(Equation 2)} \][/tex]
### Step 3: Solve the System of Equations
From Equation 1:
[tex]\[ y = 8 - 7x \quad \text{(Substitute y into Equation 2)} \][/tex]
Substitute [tex]\( y \)[/tex] in Equation 2:
[tex]\[ 600x + 20(8 - 7x) = 620 \][/tex]
[tex]\[ 600x + 160 - 140x = 620 \][/tex]
[tex]\[ 460x + 160 = 620 \][/tex]
Subtract 160 from both sides:
[tex]\[ 460x = 460 \][/tex]
Divide by 460:
[tex]\[ x = 1 \][/tex]
### Step 4: Find [tex]\( y \)[/tex]
Substitute [tex]\( x = 1 \)[/tex] back into Equation 1:
[tex]\[ 7(1) + y = 8 \][/tex]
[tex]\[ 7 + y = 8 \][/tex]
[tex]\[ y = 1 \][/tex]
### Step 5: Calculate the Number of Each Denomination
- Number of 100 denomination notes = [tex]\( 5x = 5(1) = 5 \)[/tex]
- Number of 50 denomination notes = [tex]\( 2x = 2(1) = 2 \)[/tex]
- Number of 20 denomination notes = [tex]\( y = 1 \)[/tex]
### Step 6: Verify the Results
1. Total number of notes:
[tex]\[ 5 + 2 + 1 = 8 \quad \text{(correct)} \][/tex]
2. Total amount:
[tex]\[ 5(100) + 2(50) + 1(20) = 500 + 100 + 20 = 620 \quad \text{(correct)} \][/tex]
### Final Answer
Surajit has:
- 5 notes of 100 denomination
- 2 notes of 50 denomination
- 1 note of 20 denomination
### Step 1: Define Variables
- Let [tex]\( x \)[/tex] be a common multiplier for the ratio of 100 and 50 denomination notes.
- Since the ratio of 100 to 50 denomination notes is 5:2, we have:
- Number of 100 denomination notes = [tex]\( 5x \)[/tex]
- Number of 50 denomination notes = [tex]\( 2x \)[/tex]
- Let [tex]\( y \)[/tex] be the number of 20 denomination notes.
### Step 2: Use Given Conditions to Form Equations
1. Total number of notes:
[tex]\[ 5x + 2x + y = 8 \][/tex]
Simplifying, we get:
[tex]\[ 7x + y = 8 \quad \text{(Equation 1)} \][/tex]
2. Total amount of money:
[tex]\[ 100(5x) + 50(2x) + 20y = 620 \][/tex]
Simplifying, we get:
[tex]\[ 500x + 100x + 20y = 620 \][/tex]
[tex]\[ 600x + 20y = 620 \quad \text{(Equation 2)} \][/tex]
### Step 3: Solve the System of Equations
From Equation 1:
[tex]\[ y = 8 - 7x \quad \text{(Substitute y into Equation 2)} \][/tex]
Substitute [tex]\( y \)[/tex] in Equation 2:
[tex]\[ 600x + 20(8 - 7x) = 620 \][/tex]
[tex]\[ 600x + 160 - 140x = 620 \][/tex]
[tex]\[ 460x + 160 = 620 \][/tex]
Subtract 160 from both sides:
[tex]\[ 460x = 460 \][/tex]
Divide by 460:
[tex]\[ x = 1 \][/tex]
### Step 4: Find [tex]\( y \)[/tex]
Substitute [tex]\( x = 1 \)[/tex] back into Equation 1:
[tex]\[ 7(1) + y = 8 \][/tex]
[tex]\[ 7 + y = 8 \][/tex]
[tex]\[ y = 1 \][/tex]
### Step 5: Calculate the Number of Each Denomination
- Number of 100 denomination notes = [tex]\( 5x = 5(1) = 5 \)[/tex]
- Number of 50 denomination notes = [tex]\( 2x = 2(1) = 2 \)[/tex]
- Number of 20 denomination notes = [tex]\( y = 1 \)[/tex]
### Step 6: Verify the Results
1. Total number of notes:
[tex]\[ 5 + 2 + 1 = 8 \quad \text{(correct)} \][/tex]
2. Total amount:
[tex]\[ 5(100) + 2(50) + 1(20) = 500 + 100 + 20 = 620 \quad \text{(correct)} \][/tex]
### Final Answer
Surajit has:
- 5 notes of 100 denomination
- 2 notes of 50 denomination
- 1 note of 20 denomination
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