Ravi had a total of 80 pieces of $10 notes and $50 notes. He used 1/2 of his $10 notes and withdrew another 8 pieces of $50 from the bank. After which, the number of $50 notes he had became 2/5 of the number of $10 notes. Find the total value of the 80 pieces of notes that he had at first
Let's start by finding the number of $10 notes that Ravi had initially.
Let the number of $10 notes be x.
Since he used 1/2 of the $10 notes, he now has x - 1/2*x = 1/2*x notes remaining.
The total number of $10 notes remaining is 80 - 1/2*80 = 40.
He also withdrew 8 pieces of $50 notes from the bank, so the number of $50 notes he had now is 1/2*x + 8.
According to the problem, the number of $50 notes now is 2/5 of the number of $10 notes, so we can write the equation:
1/2*x + 8 = 2/5*x.
Multiplying both sides of the equation by 10 to get rid of the fractions, we have:
5*(1/2*x + 8) = 5*(2/5*x).
Simplifying the equation, we get:
5/2*x + 40 = 4/5*x.
Multiplying both sides of the equation by 10 to get rid of the fractions, we have:
25*x + 400 = 8*x.
Moving all the terms to one side of the equation, we have:
17*x = 400.
Dividing both sides of the equation by 17, we have:
x = 400/17 = 23.53 (rounded to two decimal places).
Since the number of notes must be a whole number, we round x to the nearest whole number, x = 24.
Therefore, Ravi initially had 24 $10 notes.
Since he had a total of 80 notes, the number of $50 notes he initially had is 80 - 24 = 56.
The total value of the 80 pieces of notes that Ravi had at first is 24*10 + 56*50 = $<<24*10+56*50=3320>>3320. Answer: \boxed{3320}.
Let's break down the problem step by step:
Step 1: Let's assume that Ravi initially had "x" $10 notes and "y" $50 notes.
Step 2: From the given information, we know that Ravi used 1/2 of his $10 notes, meaning he used (1/2)x $10 notes.
Step 3: After using the $10 notes, Ravi had (x - (1/2)x) = (1/2)x $10 notes left.
Step 4: Ravi also withdrew 8 pieces of $50 notes, so he now has (y + 8) $50 notes.
Step 5: According to the problem, the number of $50 notes he had after withdrawing became 2/5 of the number of $10 notes he had left. Based on this, we can write the equation: (1/2)x = (2/5)(1/2)x + 8
Step 6: Simplifying the equation: (1/2)x = (1/5)x + 8
Step 7: We can eliminate the fractions by multiplying the equation by 10 to get rid of the denominators: 10(1/2)x = 10(1/5)x + 10(8)
Step 8: Simplifying further, we get: 5x = 2x + 80
Step 9: Subtracting 2x from both sides of the equation, we have: 5x - 2x = 2x + 80 - 2x
Step 10: Simplifying the equation, we get: 3x = 80
Step 11: Dividing both sides of the equation by 3, we have: x = 80/3
Step 12: Since the number of notes cannot be a fraction, we round x up to the nearest whole number: x = 27
Step 13: Now that we know x = 27, we can substitute it into the equation to find y: 27 + y = 80
Step 14: Subtracting 27 from both sides, we get: y = 80 - 27
Step 15: Simplifying the equation, we have: y = 53
Step 16: Therefore, Ravi initially had 27 $10 notes and 53 $50 notes.
Step 17: The total value of the 80 pieces of notes that Ravi had at first can be calculated as: (27 x $10) + (53 x $50) = $270 + $2650 = $2920
Hence, the total value of the 80 pieces of notes that Ravi had at first is $2920.
To solve this problem and find the total value of the 80 pieces of notes that Ravi had at first, we need to set up equations and solve them step by step.
Let's assume that the initial number of $10 notes that Ravi had is x, and the initial number of $50 notes is y.
Based on the given information, we know that Ravi had a total of 80 pieces of notes, so we can write the equation:
x + y = 80 -- Equation 1
Next, we know that Ravi used 1/2 of his $10 notes, which means he had x/2 $10 notes left. Additionally, he withdrew another 8 pieces of $50 notes, so the new number of $50 notes became y + 8.
According to the problem, the number of $50 notes Ravi had became 2/5 of the number of $10 notes left. We can express this information as an equation:
y + 8 = (2/5)(x/2) = (1/5)x -- Equation 2
Now, we have a system of two equations (Equation 1 and Equation 2), and we can solve them to find the values of x and y.
We can start by isolating x in Equation 2:
y + 8 = (1/5)x
5(y + 8) = x
Substitute this value of x in Equation 1:
5(y + 8) + y = 80
6y + 40 = 80
6y = 40
y = 40/6
y = 6.67 (approximately)
Since we are dealing with a whole number of notes, we know that y must be a whole number, so the closest whole number to 6.67 is 7.
Now, we can substitute the value of y back into Equation 1 to find x:
x + 7 = 80
x = 80 - 7
x = 73
Therefore, the initial number of $10 notes (x) was 73, and the initial number of $50 notes (y) was 7.
Finally, to find the total value of the 80 pieces of notes Ravi had at first, we can calculate:
Total value = (Value of $10 notes)(Number of $10 notes) + (Value of $50 notes)(Number of $50 notes)
Total value = ($10)(73) + ($50)(7)
Total value = $730 + $350
Total value = $1080
Therefore, the total value of the 80 pieces of notes that Ravi had at first was $1080.