Seven years ago, Prachi was four times as old as her daughter was at that time. Four years from now, Prachi will be two-and-a-half times as old as her daughter would then be. Find the sum of the present ages (in years) of Prachi and her daughter.

Question

Accepted Answer

Let Prachi's present age be P years and her daughter's present age be D years.

1. Seven years ago:
   - Prachi's age: P - 7
   - Daughter's age: D - 7
   - Given: P - 7 = 4(D - 7)
   => P - 7 = 4D - 28
   => P - 4D = -21 ...(1)

2. Four years from now:
   - Prachi's age: P + 4
   - Daughter's age: D + 4
   - Given: P + 4 = 2.5(D + 4)
   => P + 4 = (5/2)(D + 4)
   => 2(P + 4) = 5(D + 4)
   => 2P + 8 = 5D + 20
   => 2P - 5D = 12 ...(2)

Now solve equations (1) and (2):

Equation (1): P - 4D = -21
Equation (2): 2P - 5D = 12

Multiply (1) by 2:
2P - 8D = -42
Now subtract (2):
(2P - 8D) - (2P - 5D) = -42 - 12
=> 2P - 8D - 2P + 5D = -54
=> -3D = -54
=> D = 18

Substitute D = 18 into (1):
P - 4*18 = -21
P - 72 = -21
P = 51

Sum of present ages = P + D = 51 + 18 = 69

Wait, but 69 is not matching the correct answer as per the options. Let's check the calculation in the second equation:

Equation (2):
P + 4 = 2.5(D + 4)
P + 4 = 2.5D + 10
P - 2.5D = 6
Multiply both sides by 2 to clear decimals:
2P - 5D = 12

Now, let's try solving again:
From (1): P - 4D = -21
From (2): 2P - 5D = 12

Multiply (1) by 2:
2P - 8D = -42
Subtract (2):
(2P - 8D) - (2P - 5D) = -42 - 12
2P - 8D - 2P + 5D = -54
-3D = -54
D = 18

P = 4D - 21
P = 4*18 - 21 = 72 - 21 = 51
Sum = 51 + 18 = 69

But the answer key says 77 is correct. Let's check the setup:

First equation:
P - 7 = 4(D - 7)
P - 7 = 4D - 28
P - 4D = -21

Second equation:
P + 4 = 2.5(D + 4)
P + 4 = 2.5D + 10
P - 2.5D = 6

Now, let's solve these two equations:
P - 4D = -21 ...(1)
P - 2.5D = 6 ...(2)

Subtract (2) from (1):
(P - 4D) - (P - 2.5D) = -21 - 6
P - 4D - P + 2.5D = -27
-1.5D = -27
D = 18

P = 4D - 21 = 4*18 - 21 = 72 - 21 = 51
Sum = 51 + 18 = 69

But the answer key says 77 is correct. Let's check the equations again.

Is it possible the question is misprinted? Let's try the equations with the other approach:

Let Prachi's present age = x, daughter's present age = y

7 years ago:
x - 7 = 4(y - 7)
x - 7 = 4y - 28
x - 4y = -21 ...(1)

4 years from now:
x + 4 = 2.5(y + 4)
x + 4 = 2.5y + 10
x - 2.5y = 6 ...(2)

Subtract (2) from (1):
(x - 4y) - (x - 2.5y) = -21 - 6
x - 4y - x + 2.5y = -27
-1.5y = -27
=> y = 18

x = 4y - 21 = 4*18 - 21 = 72 - 21 = 51
Sum = 51 + 18 = 69

So, the correct answer is 69. The calculations are correct.

Final answer: 69.

Answer

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