sum of ₹ 8400 is made up of 50,20,10 and 5 rupees notes. the number of 10 rupee notes is five times the number of 5 rupee notes, four times the number of 20 rupee notes and ten times the number of 50 rupee notes. what is the number of notes in each denomination ?

Ans

Let's assume:

The number of 50 rupee notes = x
The number of 20 rupee notes = y
The number of 10 rupee notes = z
The number of 5 rupee notes = w

We are given:
x + y + z + w = 8400 ---(1)
z = 10 * x ---(2)
z = 4 * y ---(3)
z = 5 * w ---(4)

We can solve this system of equations to find the values of x, y, z, and w.

From equation (2), we can substitute the value of z in equation (3) and (4):

10 * x = 4 * y ---(5)
10 * x = 5 * w ---(6)

Divide equation (5) by 2:
5 * x = 2 * y

Now, we can substitute the value of y in terms of x in equation (3):
4 * (5 * x / 2) = z
20 * x = z ---(7)

Substitute the value of z in terms of x in equation (1):
x + y + (20 * x) + (5 * w) = 8400
x + (5 * x) + (20 * x) + (5 * w) = 8400
26 * x + 5 * w = 8400 ---(8)

Now, we have two equations:
20 * x = z ---(7)
26 * x + 5 * w = 8400 ---(8)

To simplify, let's calculate both equations in terms of x:

Multiply equation (7) by 5:
100 * x = 5 * z ---(9)

Multiply equation (8) by 20:
520 * x + 100 * w = 168000 ---(10)

Now substitute the value of z from equation (7) in equation (9):
100 * x = 5 * (20 * x)
100 * x = 100 * x

Now, substitute the value of z from equation (7) and w from equation (10) in equation (8):
520 * x + 100 * w = 168000
520 * x + 100 * (100 * x / 520) = 168000
520 * x + 10000 * x / 520 = 168000
(520^2 + 10000) * x / 520 = 168000
275600 * x / 520 = 168000
275600 * x = 520 * 168000
275600 * x = 87360000
x = 87360000 / 275600
x = 316

Now substitute the value of x in equation (7) to find z:
20 * x = z
20 * 316 = z
z = 6320

Now substitute the value of z in equation (4) to find w:
z = 5 * w
6320 = 5 * w
w = 6320 / 5
w = 1264

Now substitute the value of x, y, z, and w in equation (1) to find the total number of notes:
x + y + z + w = 8400
316 + y + 6320 + 1264 = 8400
y = 8400 - 316 - 6320 - 1264
y = 1500

Therefore, the number of notes in each denomination is:
Number of 50 rupee notes = x = 316
Number of 20 rupee notes = y = 1500
Number of 10 rupee notes = z = 6320
Number of 5 rupee notes = w = 1264

To solve this problem, we need to set up equations based on the given information and solve for the unknowns. Let's denote the number of 50 rupee notes as 'x', the number of 20 rupee notes as 'y', the number of 10 rupee notes as 'z', and the number of 5 rupee notes as 'w'.

1. The sum of the denominations: The total sum of money is given as ₹8400. We can write this as an equation:
50x + 20y + 10z + 5w = 8400

2. The relation between the number of notes: We are given that the number of 10 rupee notes is five times the number of 5 rupee notes, four times the number of 20 rupee notes, and ten times the number of 50 rupee notes. We can write these relationships as equations:
z = 5w (equation 1)
z = 4y (equation 2)
z = 10x (equation 3)

Now, we have a system of equations. We can solve the system to find the values of 'x', 'y', 'z', and 'w' by substitution.

From equation 2, we can substitute 'z' in equation 1 and equation 3:
5w = 4y (from equation 2)
10x = 4y (from equation 3)

Now, we have a relation between 5w and 10x:
5w = 2(10x)
w = 2x (equation 4)

Substituting equation 4 into equation 1:
z = 5w
z = 5(2x)
z = 10x (equation 5)

Equations 3 and 5 give us the same relation for 'z'. Therefore, we can eliminate 'z' from the equations and focus on finding 'x', 'y', and 'w'.

To apply substitution, we can substitute the value of 'z' from equation 5 into equation 3:
10x = 4y (from equation 3)
10x = 4y (from equation 5)

Now, we have a relation between 10x and 4y:
10x = 4y

Simplifying this equation:
5x = 2y (dividing both sides by 2)

Now, we have a relation between 'x' and 'y'.

From equation 4, we know that w = 2x. And from the given sum of denominations, we need to find their quantities in terms of the above variables.

Thus, the quantities of notes in each denomination are:
The number of 50 rupee notes (x)
The number of 20 rupee notes (y)
The number of 10 rupee notes (z)
The number of 5 rupee notes (w = 2x)

just write down the facts. If there are a,b,c,d of 5,10,20,50 notes, then

b=5a
b=4c
b=10d
5a+10b+20c+50d = 8400

Now just plug and chug